def test_6():
test = [
[(S(1) + a*x)**(S(3)/S(2))/sqrt(S(1) - a*x), x, S(4), S(3)/S(2)*arcsin(a*x)/a - S(1)/S(2)*(S(1) + a*x)**(S(3)/S(2))*sqrt(S(1) - a*x)/a - S(3)/S(2)*sqrt(S(1) - a*x)*sqrt(S(1) + a*x)/a],
[(S(1) - x)**(S(1)/S(2))/(S(1) + x)**(S(1)/S(2)), x, S(3), arcsin(x) + sqrt(S(1) - x)*sqrt(S(1) + x)],
[S(1)/((S(1) - x)**(S(1)/S(2))*(S(1) + x)**(S(3)/S(2))), x, S(1), - sqrt(S(1) - x)/sqrt(S(1) + x)],
[(a + a*x)**(S(5)/S(2))*(c - c*x)**(S(5)/S(2)), x, S(5), S(5)/S(24)*a*c*x*(a + a*x)**(S(3)/S(2))*(c - c*x)**(S(3)/S(2)) + S(1)/S(6)*x*(a + a*x)**(S(5)/S(2))*(c - c*x)**(S(5)/S(2)) + S(5)/S(8)*a**(S(5)/S(2))*c**(S(5)/S(2))*arctan(sqrt(c)*sqrt(a + a*x)/(sqrt(a)*sqrt(c - c*x))) + S(5)/S(16)*a**S(2)*c**S(2)*x*sqrt(a + a*x)*sqrt(c - c*x)],
[S(1)/((a + a*x)**(S(5)/S(2))*(c - c*x)**(S(5)/S(2))), x, S(2), S(1)/S(3)*x/(a*c*(a + a*x)**(S(3)/S(2))*(c - c*x)**(S(3)/S(2))) + S(2)/S(3)*x/(a**S(2)*c**S(2)*sqrt(a + a*x)*sqrt(c - c*x))],
[(S(3) - x)**(S(1)/S(2))*( - S(2) + x)**(S(1)/S(2)), x, S(5), - S(1)/S(8)*arcsin(S(5) - S(2)*x) - S(1)/S(2)*(S(3) - x)**(S(3)/S(2))*sqrt( - S(2) + x) + S(1)/S(4)*sqrt(S(3) - x)*sqrt( - S(2) + x)],
[S(1)/(sqrt(a + b*x)*sqrt( - a*d + b*d*x)), x, S(2), S(2)*arctanh(sqrt(d)*sqrt(a + b*x)/sqrt( - a*d + b*d*x))/(b*sqrt(d))],
[S(1)/((a - I*a*x)**(S(7)/S(4))*(a + I*a*x)**(S(1)/S(4))), x, S(1), - S(2)/S(3)*I*(a + I*a*x)**(S(3)/S(4))/(a**S(2)*(a - I*a*x)**(S(3)/S(4)))],
[(a + b*x)**S(2)*(a*c - b*c*x)**n, x, S(2), - S(4)*a**S(2)*(a*c - b*c*x)**(S(1) + n)/(b*c*(S(1) + n)) + S(4)*a*(a*c - b*c*x)**(S(2) + n)/(b*c**S(2)*(S(2) + n)) - (a*c - b*c*x)**(S(3) + n)/(b*c**S(3)*(S(3) + n))],
[(a + b*x)**S(4)*(c + d*x), x, S(2), S(1)/S(5)*(b*c - a*d)*(a + b*x)**S(5)/b**S(2) + S(1)/S(6)*d*(a + b*x)**S(6)/b**S(2)],
[(a + b*x)*(c + d*x), x, S(2), a*c*x + S(1)/S(2)*(b*c + a*d)*x**S(2) + S(1)/S(3)*b*d*x**S(3)],
[(a + b*x)**S(5)/(c + d*x), x, S(2), b*(b*c - a*d)**S(4)*x/d**S(5) - S(1)/S(2)*(b*c - a*d)**S(3)*(a + b*x)**S(2)/d**S(4) + S(1)/S(3)*(b*c - a*d)**S(2)*(a + b*x)**S(3)/d**S(3) - S(1)/S(4)*(b*c - a*d)*(a + b*x)**S(4)/d**S(2) + S(1)/S(5)*(a + b*x)**S(5)/d - (b*c - a*d)**S(5)*log(c + d*x)/d**S(6)],
[(a + b*x)/(c + d*x)**S(3), x, S(1), S(1)/S(2)*(a + b*x)**S(2)/((b*c - a*d)*(c + d*x)**S(2))],
[(a + b*x)**S(5)*(c + d*x)**(S(1)/S(2)), x, S(2), - S(2)/S(3)*(b*c - a*d)**S(5)*(c + d*x)**(S(3)/S(2))/d**S(6) + S(2)*b*(b*c - a*d)**S(4)*(c + d*x)**(S(5)/S(2))/d**S(6) - S(20)/S(7)*b**S(2)*(b*c - a*d)**S(3)*(c + d*x)**(S(7)/S(2))/d**S(6) + S(20)/S(9)*b**S(3)*(b*c - a*d)**S(2)*(c + d*x)**(S(9)/S(2))/d**S(6) - S(10)/S(11)*b**S(4)*(b*c - a*d)*(c + d*x)**(S(11)/S(2))/d**S(6) + S(2)/S(13)*b**S(5)*(c + d*x)**(S(13)/S(2))/d**S(6)],
[(c + d*x)**(S(1)/S(2))/(a + b*x)**S(2), x, S(3), - d*arctanh(sqrt(b)*sqrt(c + d*x)/sqrt(b*c - a*d))/(b**(S(3)/S(2))*sqrt(b*c - a*d)) - sqrt(c + d*x)/(b*(a + b*x))],
[(a + b*x)**S(4)/(c + d*x)**(S(1)/S(2)), x, S(2), - S(8)/S(3)*b*(b*c - a*d)**S(3)*(c + d*x)**(S(3)/S(2))/d**S(5) + S(12)/S(5)*b**S(2)*(b*c - a*d)**S(2)*(c + d*x)**(S(5)/S(2))/d**S(5) - S(8)/S(7)*b**S(3)*(b*c - a*d)*(c + d*x)**(S(7)/S(2))/d**S(5) + S(2)/S(9)*b**S(4)*(c + d*x)**(S(9)/S(2))/d**S(5) + S(2)*(b*c - a*d)**S(4)*sqrt(c + d*x)/d**S(5)],
[(a + b*x)**S(2)/(c + d*x)**(S(1)/S(2)), x, S(2), - S(4)/S(3)*b*(b*c - a*d)*(c + d*x)**(S(3)/S(2))/d**S(3) + S(2)/S(5)*b**S(2)*(c + d*x)**(S(5)/S(2))/d**S(3) + S(2)*(b*c - a*d)**S(2)*sqrt(c + d*x)/d**S(3)],
[(S(1) - x)**(S(1)/S(3))/(S(1) + x), x, S(5), S(3)*(S(1) - x)**(S(1)/S(3)) + S(3)*log(S(2)**(S(1)/S(3)) - (S(1) - x)**(S(1)/S(3)))/S(2)**(S(2)/S(3)) - log(S(1) + x)/S(2)**(S(2)/S(3)) - S(2)**(S(1)/S(3))*arctan((S(1) + S(2)**(S(2)/S(3))*(S(1) - x)**(S(1)/S(3)))/sqrt(S(3)))*sqrt(S(3))],
[(c + d*x)**(S(1)/S(2))/(a + b*x)**(S(1)/S(2)), x, S(3), (b*c - a*d)*arctanh(sqrt(d)*sqrt(a + b*x)/(sqrt(b)*sqrt(c + d*x)))/(b**(S(3)/S(2))*sqrt(d)) + sqrt(a + b*x)*sqrt(c + d*x)/b],
[(a + b*x)**(S(1)/S(2))*(c + d*x)**(S(3)/S(2)), x, S(5), S(1)/S(3)*(a + b*x)**(S(3)/S(2))*(c + d*x)**(S(3)/S(2))/b - S(1)/S(8)*(b*c - a*d)**S(3)*arctanh(sqrt(d)*sqrt(a + b*x)/(sqrt(b)*sqrt(c + d*x)))/(b**(S(5)/S(2))*d**(S(3)/S(2))) + S(1)/S(4)*(b*c - a*d)*(a + b*x)**(S(3)/S(2))*sqrt(c + d*x)/b**S(2) + S(1)/S(8)*(b*c - a*d)**S(2)*sqrt(a + b*x)*sqrt(c + d*x)/(b**S(2)*d)],
[(a + b*x)**(S(1)/S(2))/(c + d*x)**(S(1)/S(2)), x, S(3), - (b*c - a*d)*arctanh(sqrt(d)*sqrt(a + b*x)/(sqrt(b)*sqrt(c + d*x)))/(d**(S(3)/S(2))*sqrt(b)) + sqrt(a + b*x)*sqrt(c + d*x)/d],
[S(1)/((a + b*x)**(S(1)/S(2))*(c + d*x)**(S(5)/S(2))), x, S(2), S(2)/S(3)*sqrt(a + b*x)/((b*c - a*d)*(c + d*x)**(S(3)/S(2))) + S(4)/S(3)*b*sqrt(a + b*x)/((b*c - a*d)**S(2)*sqrt(c + d*x))],
[(a + b*x)**m*(c + d*x)**(S(1) + S(2)*n - S(2)*(S(1) + n)), x, S(2), (a + b*x)**(S(1) + m)*hypergeom([S(1), S(1) + m], [S(2) + m], - d*(a + b*x)/(b*c - a*d))/((b*c - a*d)*(S(1) + m))],
[a + b*x + c*x**S(2) + d*x**S(3), x, S(1), a*x + S(1)/S(2)*b*x**S(2) + S(1)/S(3)*c*x**S(3) + S(1)/S(4)*d*x**S(4)],
[a + d/x**S(3) + c/x**S(2) + b/x, x, S(1), - S(1)/S(2)*d/x**S(2) - c/x + a*x + b*log(x)],
]
for i in test:
r = rubi_integrate(i[0], i[1])
if len(i) == 5:
assert rubi_test(r, i[1], i[3], expand=True, _diff=True) or rubi_test(r, i[1], i[4], expand=True, _diff=True)
else:
assert rubi_test(r, i[1], i[3], expand=True, _diff=True)
def test_3():
test = [
[x**m*(a + b*x), x, S(2), a*x**(S(1) + m)/(S(1) + m) + b*x**(S(2) + m)/(S(2) + m)],
[x**(S(5)/S(2))*(a + b*x), x, S(2), S(2)/S(7)*a*x**(S(7)/S(2)) + S(2)/S(9)*b*x**(S(9)/S(2))],
[x**(S(5)/S(2))/(a + b*x), x, S(5), - S(2)/S(3)*a*x**(S(3)/S(2))/b**S(2) + S(2)/S(5)*x**(S(5)/S(2))/b - S(2)*a**(S(5)/S(2))*arctan(sqrt(b)*sqrt(x)/sqrt(a))/b**(S(7)/S(2)) + S(2)*a**S(2)*sqrt(x)/b**S(3)],
[x**(S(3)/S(2))/(a + b*x), x, S(4), S(2)/S(3)*x**(S(3)/S(2))/b + S(2)*a**(S(3)/S(2))*arctan(sqrt(b)*sqrt(x)/sqrt(a))/b**(S(5)/S(2)) - S(2)*a*sqrt(x)/b**S(2)],
[x**(S(5)/S(2))/( - a + b*x), x, S(5), S(2)/S(3)*a*x**(S(3)/S(2))/b**S(2) + S(2)/S(5)*x**(S(5)/S(2))/b - S(2)*a**(S(5)/S(2))*arctanh(sqrt(b)*sqrt(x)/sqrt(a))/b**(S(7)/S(2)) + S(2)*a**S(2)*sqrt(x)/b**S(3)],
[x**(S(5)/S(2))*sqrt(a + b*x), x, S(6), - S(5)/S(64)*a**S(4)*arctanh(sqrt(b)*sqrt(x)/sqrt(a + b*x))/b**(S(7)/S(2)) - S(5)/S(96)*a**S(2)*x**(S(3)/S(2))*sqrt(a + b*x)/b**S(2) + S(1)/S(24)*a*x**(S(5)/S(2))*sqrt(a + b*x)/b + S(1)/S(4)*x**(S(7)/S(2))*sqrt(a + b*x) + S(5)/S(64)*a**S(3)*sqrt(x)*sqrt(a + b*x)/b**S(3)],
[x**(S(3)/S(2))*sqrt(a + b*x), x, S(5), S(1)/S(8)*a**S(3)*arctanh(sqrt(b)*sqrt(x)/sqrt(a + b*x))/b**(S(5)/S(2)) + S(1)/S(12)*a*x**(S(3)/S(2))*sqrt(a + b*x)/b + S(1)/S(3)*x**(S(5)/S(2))*sqrt(a + b*x) - S(1)/S(8)*a**S(2)*sqrt(x)*sqrt(a + b*x)/b**S(2)],
[x**(S(5)/S(2))/sqrt(a + b*x), x, S(5), - S(5)/S(8)*a**S(3)*arctanh(sqrt(b)*sqrt(x)/sqrt(a + b*x))/b**(S(7)/S(2)) - S(5)/S(12)*a*x**(S(3)/S(2))*sqrt(a + b*x)/b**S(2) + S(1)/S(3)*x**(S(5)/S(2))*sqrt(a + b*x)/b + S(5)/S(8)*a**S(2)*sqrt(x)*sqrt(a + b*x)/b**S(3)],
[sqrt(x)/sqrt(a + b*x), x, S(3), - a*arctanh(sqrt(b)*sqrt(x)/sqrt(a + b*x))/b**(S(3)/S(2)) + sqrt(x)*sqrt(a + b*x)/b],
[x**(S(2)/S(3))*(a + b*x), x, S(2), S(3)/S(5)*a*x**(S(5)/S(3)) + S(3)/S(8)*b*x**(S(8)/S(3))],
[x**(S(1)/S(3))*(a + b*x), x, S(2), S(3)/S(4)*a*x**(S(4)/S(3)) + S(3)/S(7)*b*x**(S(7)/S(3))],
[x**(S(5)/S(3))/(a + b*x), x, S(6), - S(3)/S(2)*a*x**(S(2)/S(3))/b**S(2) + S(3)/S(5)*x**(S(5)/S(3))/b - S(3)/S(2)*a**(S(5)/S(3))*log(a**(S(1)/S(3)) + b**(S(1)/S(3))*x**(S(1)/S(3)))/b**(S(8)/S(3)) + S(1)/S(2)*a**(S(5)/S(3))*log(a + b*x)/b**(S(8)/S(3)) - a**(S(5)/S(3))*arctan((a**(S(1)/S(3)) - S(2)*b**(S(1)/S(3))*x**(S(1)/S(3)))/(a**(S(1)/S(3))*sqrt(S(3))))*sqrt(S(3))/b**(S(8)/S(3))],
[x**(S(4)/S(3))/(a + b*x), x, S(6), - S(3)*a*x**(S(1)/S(3))/b**S(2) + S(3)/S(4)*x**(S(4)/S(3))/b + S(3)/S(2)*a**(S(4)/S(3))*log(a**(S(1)/S(3)) + b**(S(1)/S(3))*x**(S(1)/S(3)))/b**(S(7)/S(3)) - S(1)/S(2)*a**(S(4)/S(3))*log(a + b*x)/b**(S(7)/S(3)) - a**(S(4)/S(3))*arctan((a**(S(1)/S(3)) - S(2)*b**(S(1)/S(3))*x**(S(1)/S(3)))/(a**(S(1)/S(3))*sqrt(S(3))))*sqrt(S(3))/b**(S(7)/S(3))],
[(S(1) - x)**(S(1)/S(4))/(S(1) + x), x, S(5), S(4)*(S(1) - x)**(S(1)/S(4)) - S(2)*S(2)**(S(1)/S(4))*arctan((S(1) - x)**(S(1)/S(4))/S(2)**(S(1)/S(4))) - S(2)*S(2)**(S(1)/S(4))*arctanh((S(1) - x)**(S(1)/S(4))/S(2)**(S(1)/S(4)))],
[x**m*(a + b*x)**S(2), x, S(2), a**S(2)*x**(S(1) + m)/(S(1) + m) + S(2)*a*b*x**(S(2) + m)/(S(2) + m) + b**S(2)*x**(S(3) + m)/(S(3) + m)],
]
for i in test:
r = rubi_integrate(i[0], i[1])
if len(i) == 5:
assert rubi_test(r, i[1], i[3], expand=True, _diff=True) or rubi_test(r, i[1], i[4], expand=True, _diff=True)
else:
assert rubi_test(r, i[1], i[3], expand=True, _diff=True)
def test_3():
test = [
[x**m*(a + b*x), x, S(2), a*x**(S(1) + m)/(S(1) + m) + b*x**(S(2) + m)/(S(2) + m)],
[x**(S(5)/S(2))*(a + b*x), x, S(2), S(2)/S(7)*a*x**(S(7)/S(2)) + S(2)/S(9)*b*x**(S(9)/S(2))],
[x**(S(5)/S(2))/(a + b*x), x, S(5), - S(2)/S(3)*a*x**(S(3)/S(2))/b**S(2) + S(2)/S(5)*x**(S(5)/S(2))/b - S(2)*a**(S(5)/S(2))*arctan(sqrt(b)*sqrt(x)/sqrt(a))/b**(S(7)/S(2)) + S(2)*a**S(2)*sqrt(x)/b**S(3)],
[x**(S(3)/S(2))/(a + b*x), x, S(4), S(2)/S(3)*x**(S(3)/S(2))/b + S(2)*a**(S(3)/S(2))*arctan(sqrt(b)*sqrt(x)/sqrt(a))/b**(S(5)/S(2)) - S(2)*a*sqrt(x)/b**S(2)],
[x**(S(5)/S(2))/( - a + b*x), x, S(5), S(2)/S(3)*a*x**(S(3)/S(2))/b**S(2) + S(2)/S(5)*x**(S(5)/S(2))/b - S(2)*a**(S(5)/S(2))*arctanh(sqrt(b)*sqrt(x)/sqrt(a))/b**(S(7)/S(2)) + S(2)*a**S(2)*sqrt(x)/b**S(3)],
[x**(S(5)/S(2))*sqrt(a + b*x), x, S(6), - S(5)/S(64)*a**S(4)*arctanh(sqrt(b)*sqrt(x)/sqrt(a + b*x))/b**(S(7)/S(2)) - S(5)/S(96)*a**S(2)*x**(S(3)/S(2))*sqrt(a + b*x)/b**S(2) + S(1)/S(24)*a*x**(S(5)/S(2))*sqrt(a + b*x)/b + S(1)/S(4)*x**(S(7)/S(2))*sqrt(a + b*x) + S(5)/S(64)*a**S(3)*sqrt(x)*sqrt(a + b*x)/b**S(3)],
[x**(S(3)/S(2))*sqrt(a + b*x), x, S(5), S(1)/S(8)*a**S(3)*arctanh(sqrt(b)*sqrt(x)/sqrt(a + b*x))/b**(S(5)/S(2)) + S(1)/S(12)*a*x**(S(3)/S(2))*sqrt(a + b*x)/b + S(1)/S(3)*x**(S(5)/S(2))*sqrt(a + b*x) - S(1)/S(8)*a**S(2)*sqrt(x)*sqrt(a + b*x)/b**S(2)],
[x**(S(5)/S(2))/sqrt(a + b*x), x, S(5), - S(5)/S(8)*a**S(3)*arctanh(sqrt(b)*sqrt(x)/sqrt(a + b*x))/b**(S(7)/S(2)) - S(5)/S(12)*a*x**(S(3)/S(2))*sqrt(a + b*x)/b**S(2) + S(1)/S(3)*x**(S(5)/S(2))*sqrt(a + b*x)/b + S(5)/S(8)*a**S(2)*sqrt(x)*sqrt(a + b*x)/b**S(3)],
[sqrt(x)/sqrt(a + b*x), x, S(3), - a*arctanh(sqrt(b)*sqrt(x)/sqrt(a + b*x))/b**(S(3)/S(2)) + sqrt(x)*sqrt(a + b*x)/b],
[x**(S(2)/S(3))*(a + b*x), x, S(2), S(3)/S(5)*a*x**(S(5)/S(3)) + S(3)/S(8)*b*x**(S(8)/S(3))],
[x**(S(1)/S(3))*(a + b*x), x, S(2), S(3)/S(4)*a*x**(S(4)/S(3)) + S(3)/S(7)*b*x**(S(7)/S(3))],
[x**(S(5)/S(3))/(a + b*x), x, S(6), - S(3)/S(2)*a*x**(S(2)/S(3))/b**S(2) + S(3)/S(5)*x**(S(5)/S(3))/b - S(3)/S(2)*a**(S(5)/S(3))*log(a**(S(1)/S(3)) + b**(S(1)/S(3))*x**(S(1)/S(3)))/b**(S(8)/S(3)) + S(1)/S(2)*a**(S(5)/S(3))*log(a + b*x)/b**(S(8)/S(3)) - a**(S(5)/S(3))*arctan((a**(S(1)/S(3)) - S(2)*b**(S(1)/S(3))*x**(S(1)/S(3)))/(a**(S(1)/S(3))*sqrt(S(3))))*sqrt(S(3))/b**(S(8)/S(3))],
[x**(S(4)/S(3))/(a + b*x), x, S(6), - S(3)*a*x**(S(1)/S(3))/b**S(2) + S(3)/S(4)*x**(S(4)/S(3))/b + S(3)/S(2)*a**(S(4)/S(3))*log(a**(S(1)/S(3)) + b**(S(1)/S(3))*x**(S(1)/S(3)))/b**(S(7)/S(3)) - S(1)/S(2)*a**(S(4)/S(3))*log(a + b*x)/b**(S(7)/S(3)) - a**(S(4)/S(3))*arctan((a**(S(1)/S(3)) - S(2)*b**(S(1)/S(3))*x**(S(1)/S(3)))/(a**(S(1)/S(3))*sqrt(S(3))))*sqrt(S(3))/b**(S(7)/S(3))],
[(S(1) - x)**(S(1)/S(4))/(S(1) + x), x, S(5), S(4)*(S(1) - x)**(S(1)/S(4)) - S(2)*S(2)**(S(1)/S(4))*arctan((S(1) - x)**(S(1)/S(4))/S(2)**(S(1)/S(4))) - S(2)*S(2)**(S(1)/S(4))*arctanh((S(1) - x)**(S(1)/S(4))/S(2)**(S(1)/S(4)))],
[x**m*(a + b*x)**S(2), x, S(2), a**S(2)*x**(S(1) + m)/(S(1) + m) + S(2)*a*b*x**(S(2) + m)/(S(2) + m) + b**S(2)*x**(S(3) + m)/(S(3) + m)],
]
for i in test:
r = rubi_integrate(i[0], i[1])
if len(i) == 5:
assert rubi_test(r, i[1], i[3], expand=True, _diff=True) or rubi_test(r, i[1], i[4], expand=True, _diff=True)
else:
assert rubi_test(r, i[1], i[3], expand=True, _diff=True)
def test_6():
test = [
[(S(1) + a*x)**(S(3)/S(2))/sqrt(S(1) - a*x), x, S(4), S(3)/S(2)*arcsin(a*x)/a - S(1)/S(2)*(S(1) + a*x)**(S(3)/S(2))*sqrt(S(1) - a*x)/a - S(3)/S(2)*sqrt(S(1) - a*x)*sqrt(S(1) + a*x)/a],
[(S(1) - x)**(S(1)/S(2))/(S(1) + x)**(S(1)/S(2)), x, S(3), arcsin(x) + sqrt(S(1) - x)*sqrt(S(1) + x)],
[S(1)/((S(1) - x)**(S(1)/S(2))*(S(1) + x)**(S(3)/S(2))), x, S(1), - sqrt(S(1) - x)/sqrt(S(1) + x)],
[(a + a*x)**(S(5)/S(2))*(c - c*x)**(S(5)/S(2)), x, S(5), S(5)/S(24)*a*c*x*(a + a*x)**(S(3)/S(2))*(c - c*x)**(S(3)/S(2)) + S(1)/S(6)*x*(a + a*x)**(S(5)/S(2))*(c - c*x)**(S(5)/S(2)) + S(5)/S(8)*a**(S(5)/S(2))*c**(S(5)/S(2))*arctan(sqrt(c)*sqrt(a + a*x)/(sqrt(a)*sqrt(c - c*x))) + S(5)/S(16)*a**S(2)*c**S(2)*x*sqrt(a + a*x)*sqrt(c - c*x)],
[S(1)/((a + a*x)**(S(5)/S(2))*(c - c*x)**(S(5)/S(2))), x, S(2), S(1)/S(3)*x/(a*c*(a + a*x)**(S(3)/S(2))*(c - c*x)**(S(3)/S(2))) + S(2)/S(3)*x/(a**S(2)*c**S(2)*sqrt(a + a*x)*sqrt(c - c*x))],
[(S(3) - x)**(S(1)/S(2))*( - S(2) + x)**(S(1)/S(2)), x, S(5), - S(1)/S(8)*arcsin(S(5) - S(2)*x) - S(1)/S(2)*(S(3) - x)**(S(3)/S(2))*sqrt( - S(2) + x) + S(1)/S(4)*sqrt(S(3) - x)*sqrt( - S(2) + x)],
[S(1)/(sqrt(a + b*x)*sqrt( - a*d + b*d*x)), x, S(2), S(2)*arctanh(sqrt(d)*sqrt(a + b*x)/sqrt( - a*d + b*d*x))/(b*sqrt(d))],
[S(1)/((a - I*a*x)**(S(7)/S(4))*(a + I*a*x)**(S(1)/S(4))), x, S(1), - S(2)/S(3)*I*(a + I*a*x)**(S(3)/S(4))/(a**S(2)*(a - I*a*x)**(S(3)/S(4)))],
[(a + b*x)**S(2)*(a*c - b*c*x)**n, x, S(2), - S(4)*a**S(2)*(a*c - b*c*x)**(S(1) + n)/(b*c*(S(1) + n)) + S(4)*a*(a*c - b*c*x)**(S(2) + n)/(b*c**S(2)*(S(2) + n)) - (a*c - b*c*x)**(S(3) + n)/(b*c**S(3)*(S(3) + n))],
[(a + b*x)**S(4)*(c + d*x), x, S(2), S(1)/S(5)*(b*c - a*d)*(a + b*x)**S(5)/b**S(2) + S(1)/S(6)*d*(a + b*x)**S(6)/b**S(2)],
[(a + b*x)*(c + d*x), x, S(2), a*c*x + S(1)/S(2)*(b*c + a*d)*x**S(2) + S(1)/S(3)*b*d*x**S(3)],
[(a + b*x)**S(5)/(c + d*x), x, S(2), b*(b*c - a*d)**S(4)*x/d**S(5) - S(1)/S(2)*(b*c - a*d)**S(3)*(a + b*x)**S(2)/d**S(4) + S(1)/S(3)*(b*c - a*d)**S(2)*(a + b*x)**S(3)/d**S(3) - S(1)/S(4)*(b*c - a*d)*(a + b*x)**S(4)/d**S(2) + S(1)/S(5)*(a + b*x)**S(5)/d - (b*c - a*d)**S(5)*log(c + d*x)/d**S(6)],
[(a + b*x)/(c + d*x)**S(3), x, S(1), S(1)/S(2)*(a + b*x)**S(2)/((b*c - a*d)*(c + d*x)**S(2))],
[(a + b*x)**S(5)*(c + d*x)**(S(1)/S(2)), x, S(2), - S(2)/S(3)*(b*c - a*d)**S(5)*(c + d*x)**(S(3)/S(2))/d**S(6) + S(2)*b*(b*c - a*d)**S(4)*(c + d*x)**(S(5)/S(2))/d**S(6) - S(20)/S(7)*b**S(2)*(b*c - a*d)**S(3)*(c + d*x)**(S(7)/S(2))/d**S(6) + S(20)/S(9)*b**S(3)*(b*c - a*d)**S(2)*(c + d*x)**(S(9)/S(2))/d**S(6) - S(10)/S(11)*b**S(4)*(b*c - a*d)*(c + d*x)**(S(11)/S(2))/d**S(6) + S(2)/S(13)*b**S(5)*(c + d*x)**(S(13)/S(2))/d**S(6)],
[(c + d*x)**(S(1)/S(2))/(a + b*x)**S(2), x, S(3), - d*arctanh(sqrt(b)*sqrt(c + d*x)/sqrt(b*c - a*d))/(b**(S(3)/S(2))*sqrt(b*c - a*d)) - sqrt(c + d*x)/(b*(a + b*x))],
[(a + b*x)**S(4)/(c + d*x)**(S(1)/S(2)), x, S(2), - S(8)/S(3)*b*(b*c - a*d)**S(3)*(c + d*x)**(S(3)/S(2))/d**S(5) + S(12)/S(5)*b**S(2)*(b*c - a*d)**S(2)*(c + d*x)**(S(5)/S(2))/d**S(5) - S(8)/S(7)*b**S(3)*(b*c - a*d)*(c + d*x)**(S(7)/S(2))/d**S(5) + S(2)/S(9)*b**S(4)*(c + d*x)**(S(9)/S(2))/d**S(5) + S(2)*(b*c - a*d)**S(4)*sqrt(c + d*x)/d**S(5)],
[(a + b*x)**S(2)/(c + d*x)**(S(1)/S(2)), x, S(2), - S(4)/S(3)*b*(b*c - a*d)*(c + d*x)**(S(3)/S(2))/d**S(3) + S(2)/S(5)*b**S(2)*(c + d*x)**(S(5)/S(2))/d**S(3) + S(2)*(b*c - a*d)**S(2)*sqrt(c + d*x)/d**S(3)],
[(S(1) - x)**(S(1)/S(3))/(S(1) + x), x, S(5), S(3)*(S(1) - x)**(S(1)/S(3)) + S(3)*log(S(2)**(S(1)/S(3)) - (S(1) - x)**(S(1)/S(3)))/S(2)**(S(2)/S(3)) - log(S(1) + x)/S(2)**(S(2)/S(3)) - S(2)**(S(1)/S(3))*arctan((S(1) + S(2)**(S(2)/S(3))*(S(1) - x)**(S(1)/S(3)))/sqrt(S(3)))*sqrt(S(3))],
[(c + d*x)**(S(1)/S(2))/(a + b*x)**(S(1)/S(2)), x, S(3), (b*c - a*d)*arctanh(sqrt(d)*sqrt(a + b*x)/(sqrt(b)*sqrt(c + d*x)))/(b**(S(3)/S(2))*sqrt(d)) + sqrt(a + b*x)*sqrt(c + d*x)/b],
[(a + b*x)**(S(1)/S(2))*(c + d*x)**(S(3)/S(2)), x, S(5), S(1)/S(3)*(a + b*x)**(S(3)/S(2))*(c + d*x)**(S(3)/S(2))/b - S(1)/S(8)*(b*c - a*d)**S(3)*arctanh(sqrt(d)*sqrt(a + b*x)/(sqrt(b)*sqrt(c + d*x)))/(b**(S(5)/S(2))*d**(S(3)/S(2))) + S(1)/S(4)*(b*c - a*d)*(a + b*x)**(S(3)/S(2))*sqrt(c + d*x)/b**S(2) + S(1)/S(8)*(b*c - a*d)**S(2)*sqrt(a + b*x)*sqrt(c + d*x)/(b**S(2)*d)],
[(a + b*x)**(S(1)/S(2))/(c + d*x)**(S(1)/S(2)), x, S(3), - (b*c - a*d)*arctanh(sqrt(d)*sqrt(a + b*x)/(sqrt(b)*sqrt(c + d*x)))/(d**(S(3)/S(2))*sqrt(b)) + sqrt(a + b*x)*sqrt(c + d*x)/d],
[S(1)/((a + b*x)**(S(1)/S(2))*(c + d*x)**(S(5)/S(2))), x, S(2), S(2)/S(3)*sqrt(a + b*x)/((b*c - a*d)*(c + d*x)**(S(3)/S(2))) + S(4)/S(3)*b*sqrt(a + b*x)/((b*c - a*d)**S(2)*sqrt(c + d*x))],
[(a + b*x)**m*(c + d*x)**(S(1) + S(2)*n - S(2)*(S(1) + n)), x, S(2), (a + b*x)**(S(1) + m)*hypergeom([S(1), S(1) + m], [S(2) + m], - d*(a + b*x)/(b*c - a*d))/((b*c - a*d)*(S(1) + m))],
[a + b*x + c*x**S(2) + d*x**S(3), x, S(1), a*x + S(1)/S(2)*b*x**S(2) + S(1)/S(3)*c*x**S(3) + S(1)/S(4)*d*x**S(4)],
[a + d/x**S(3) + c/x**S(2) + b/x, x, S(1), - S(1)/S(2)*d/x**S(2) - c/x + a*x + b*log(x)],
]
for i in test:
r = rubi_integrate(i[0], i[1])
if len(i) == 5:
assert rubi_test(r, i[1], i[3], expand=True, _diff=True) or rubi_test(r, i[1], i[4], expand=True, _diff=True)
else:
assert rubi_test(r, i[1], i[3], expand=True, _diff=True)
def test_2():
test = [
[(a + b*x)/x, x, S(2), b*x + a*log(x)],
[x**S(5)/(a + b*x), x, S(2), a**S(4)*x/b**S(5) - S(1)/S(2)*a**S(3)*x**S(2)/b**S(4) + S(1)/S(3)*a**S(2)*x**S(3)/b**S(3) - S(1)/S(4)*a*x**S(4)/b**S(2) + S(1)/S(5)*x**S(5)/b - a**S(5)*log(a + b*x)/b**S(6)],
[S(1)/(a + b*x)**S(2), x, S(1), ( - S(1))/(b*(a + b*x))],
[S(1)/(x*(a + b*x)**S(3)), x, S(2), S(1)/S(2)/(a*(a + b*x)**S(2)) + S(1)/(a**S(2)*(a + b*x)) + log(x)/a**S(3) - log(a + b*x)/a**S(3)],
[S(1)/(S(2) + S(2)*x), x, S(1), S(1)/S(2)*log(S(1) + x)],
[S(1)/(x*(S(1) + b*x)), x, S(3), log(x) - log(S(1) + b*x)],
[x**S(3)*sqrt(a + b*x), x, S(2), - S(2)/S(3)*a**S(3)*(a + b*x)**(S(3)/S(2))/b**S(4) + S(6)/S(5)*a**S(2)*(a + b*x)**(S(5)/S(2))/b**S(4) - S(6)/S(7)*a*(a + b*x)**(S(7)/S(2))/b**S(4) + S(2)/S(9)*(a + b*x)**(S(9)/S(2))/b**S(4)],
[(a + b*x)**(S(3)/S(2)), x, S(1), S(2)/S(5)*(a + b*x)**(S(5)/S(2))/b],
[x**S(4)/sqrt(a + b*x), x, S(2), - S(8)/S(3)*a**S(3)*(a + b*x)**(S(3)/S(2))/b**S(5) + S(12)/S(5)*a**S(2)*(a + b*x)**(S(5)/S(2))/b**S(5) - S(8)/S(7)*a*(a + b*x)**(S(7)/S(2))/b**S(5) + S(2)/S(9)*(a + b*x)**(S(9)/S(2))/b**S(5) + S(2)*a**S(4)*sqrt(a + b*x)/b**S(5)],
[S(1)/sqrt(a + b*x), x, S(1), S(2)*sqrt(a + b*x)/b],
[S(1)/(x*(a + b*x)**(S(3)/S(2))), x, S(3), - S(2)*arctanh(sqrt(a + b*x)/sqrt(a))/a**(S(3)/S(2)) + S(2)/(a*sqrt(a + b*x))],
[S(1)/(x**S(2)*( - a + b*x)**(S(3)/S(2))), x, S(4), - S(3)*b*arctan(sqrt( - a + b*x)/sqrt(a))/a**(S(5)/S(2)) + ( - S(2))/(a*x*sqrt( - a + b*x)) - S(3)*sqrt( - a + b*x)/(a**S(2)*x)],
[x**S(3)*(a + b*x)**(S(1)/S(3)), x, S(2), - S(3)/S(4)*a**S(3)*(a + b*x)**(S(4)/S(3))/b**S(4) + S(9)/S(7)*a**S(2)*(a + b*x)**(S(7)/S(3))/b**S(4) - S(9)/S(10)*a*(a + b*x)**(S(10)/S(3))/b**S(4) + S(3)/S(13)*(a + b*x)**(S(13)/S(3))/b**S(4)],
[x**S(2)*(a + b*x)**(S(2)/S(3)), x, S(2), S(3)/S(5)*a**S(2)*(a + b*x)**(S(5)/S(3))/b**S(3) - S(3)/S(4)*a*(a + b*x)**(S(8)/S(3))/b**S(3) + S(3)/S(11)*(a + b*x)**(S(11)/S(3))/b**S(3)],
[x**S(2)/(a + b*x)**(S(1)/S(3)), x, S(2), S(3)/S(2)*a**S(2)*(a + b*x)**(S(2)/S(3))/b**S(3) - S(6)/S(5)*a*(a + b*x)**(S(5)/S(3))/b**S(3) + S(3)/S(8)*(a + b*x)**(S(8)/S(3))/b**S(3)],
[x**S(3)/( - a + b*x)**(S(1)/S(3)), x, S(2), S(3)/S(2)*a**S(3)*( - a + b*x)**(S(2)/S(3))/b**S(4) + S(9)/S(5)*a**S(2)*( - a + b*x)**(S(5)/S(3))/b**S(4) + S(9)/S(8)*a*( - a + b*x)**(S(8)/S(3))/b**S(4) + S(3)/S(11)*( - a + b*x)**(S(11)/S(3))/b**S(4)],
]
for i in test:
r = rubi_integrate(i[0], i[1])
if len(i) == 5:
assert rubi_test(r, i[1], i[3], expand=True, _diff=True) or rubi_test(r, i[1], i[4], expand=True, _diff=True)
else:
assert rubi_test(r, i[1], i[3], expand=True, _diff=True)
def test_2():
test = [
[(a + b*x)/x, x, S(2), b*x + a*log(x)],
[x**S(5)/(a + b*x), x, S(2), a**S(4)*x/b**S(5) - S(1)/S(2)*a**S(3)*x**S(2)/b**S(4) + S(1)/S(3)*a**S(2)*x**S(3)/b**S(3) - S(1)/S(4)*a*x**S(4)/b**S(2) + S(1)/S(5)*x**S(5)/b - a**S(5)*log(a + b*x)/b**S(6)],
[S(1)/(a + b*x)**S(2), x, S(1), ( - S(1))/(b*(a + b*x))],
[S(1)/(x*(a + b*x)**S(3)), x, S(2), S(1)/S(2)/(a*(a + b*x)**S(2)) + S(1)/(a**S(2)*(a + b*x)) + log(x)/a**S(3) - log(a + b*x)/a**S(3)],
[S(1)/(S(2) + S(2)*x), x, S(1), S(1)/S(2)*log(S(1) + x)],
[S(1)/(x*(S(1) + b*x)), x, S(3), log(x) - log(S(1) + b*x)],
[x**S(3)*sqrt(a + b*x), x, S(2), - S(2)/S(3)*a**S(3)*(a + b*x)**(S(3)/S(2))/b**S(4) + S(6)/S(5)*a**S(2)*(a + b*x)**(S(5)/S(2))/b**S(4) - S(6)/S(7)*a*(a + b*x)**(S(7)/S(2))/b**S(4) + S(2)/S(9)*(a + b*x)**(S(9)/S(2))/b**S(4)],
[(a + b*x)**(S(3)/S(2)), x, S(1), S(2)/S(5)*(a + b*x)**(S(5)/S(2))/b],
[x**S(4)/sqrt(a + b*x), x, S(2), - S(8)/S(3)*a**S(3)*(a + b*x)**(S(3)/S(2))/b**S(5) + S(12)/S(5)*a**S(2)*(a + b*x)**(S(5)/S(2))/b**S(5) - S(8)/S(7)*a*(a + b*x)**(S(7)/S(2))/b**S(5) + S(2)/S(9)*(a + b*x)**(S(9)/S(2))/b**S(5) + S(2)*a**S(4)*sqrt(a + b*x)/b**S(5)],
[S(1)/sqrt(a + b*x), x, S(1), S(2)*sqrt(a + b*x)/b],
[S(1)/(x*(a + b*x)**(S(3)/S(2))), x, S(3), - S(2)*arctanh(sqrt(a + b*x)/sqrt(a))/a**(S(3)/S(2)) + S(2)/(a*sqrt(a + b*x))],
[S(1)/(x**S(2)*( - a + b*x)**(S(3)/S(2))), x, S(4), - S(3)*b*arctan(sqrt( - a + b*x)/sqrt(a))/a**(S(5)/S(2)) + ( - S(2))/(a*x*sqrt( - a + b*x)) - S(3)*sqrt( - a + b*x)/(a**S(2)*x)],
[x**S(3)*(a + b*x)**(S(1)/S(3)), x, S(2), - S(3)/S(4)*a**S(3)*(a + b*x)**(S(4)/S(3))/b**S(4) + S(9)/S(7)*a**S(2)*(a + b*x)**(S(7)/S(3))/b**S(4) - S(9)/S(10)*a*(a + b*x)**(S(10)/S(3))/b**S(4) + S(3)/S(13)*(a + b*x)**(S(13)/S(3))/b**S(4)],
[x**S(2)*(a + b*x)**(S(2)/S(3)), x, S(2), S(3)/S(5)*a**S(2)*(a + b*x)**(S(5)/S(3))/b**S(3) - S(3)/S(4)*a*(a + b*x)**(S(8)/S(3))/b**S(3) + S(3)/S(11)*(a + b*x)**(S(11)/S(3))/b**S(3)],
[x**S(2)/(a + b*x)**(S(1)/S(3)), x, S(2), S(3)/S(2)*a**S(2)*(a + b*x)**(S(2)/S(3))/b**S(3) - S(6)/S(5)*a*(a + b*x)**(S(5)/S(3))/b**S(3) + S(3)/S(8)*(a + b*x)**(S(8)/S(3))/b**S(3)],
[x**S(3)/( - a + b*x)**(S(1)/S(3)), x, S(2), S(3)/S(2)*a**S(3)*( - a + b*x)**(S(2)/S(3))/b**S(4) + S(9)/S(5)*a**S(2)*( - a + b*x)**(S(5)/S(3))/b**S(4) + S(9)/S(8)*a*( - a + b*x)**(S(8)/S(3))/b**S(4) + S(3)/S(11)*( - a + b*x)**(S(11)/S(3))/b**S(4)],
]
for i in test:
r = rubi_integrate(i[0], i[1])
if len(i) == 5:
assert rubi_test(r, i[1], i[3], expand=True, _diff=True) or rubi_test(r, i[1], i[4], expand=True, _diff=True)
else:
assert rubi_test(r, i[1], i[3], expand=True, _diff=True)
def test_1():
'''
Tests for Rubi Algebraic 1.2 rules. Parsed from Maple syntax
All tests: http://www.apmaths.uwo.ca/~arich/IntegrationProblems/MapleSyntaxFiles/MapleSyntaxFiles.html
Note: Some tests are commented since they depend rules other than Algebraic1.2.
'''
test = [
[(a + b*x)**S(2)*(e + f*x)*sqrt(c + d*x)/x, x, S(5), S(2)/S(7)*f*(a + b*x)**S(2)*(c + d*x)**(S(3)/S(2))/d + S(2)/S(105)*(c + d*x)**(S(3)/S(2))*(S(2)*(S(10)*a**S(2)*d**S(2)*f - b**S(2)*c*(S(7)*d*e - S(4)*c*f) + S(7)*a*b*d*(S(5)*d*e - S(2)*c*f)) + S(3)*b*d*(S(7)*b*d*e - S(4)*b*c*f + S(4)*a*d*f)*x)/d**S(3) - S(2)*a**S(2)*e*arctanh(sqrt(c + d*x)/sqrt(c))*sqrt(c) + S(2)*a**S(2)*e*sqrt(c + d*x)],
[(a + b*x)*(e + f*x)*sqrt(c + d*x)/x, x, S(4), - S(2)/S(15)*(c + d*x)**(S(3)/S(2))*(S(2)*b*c*f - S(5)*d*(b*e + a*f) - S(3)*b*d*f*x)/d**S(2) - S(2)*a*e*arctanh(sqrt(c + d*x)/sqrt(c))*sqrt(c) + S(2)*a*e*sqrt(c + d*x)],
[(c + d*x)**S(2)*(e + f*x)*sqrt(a + b*x)/x, x, S(5), S(2)/S(7)*f*(a + b*x)**(S(3)/S(2))*(c + d*x)**S(2)/b + S(2)/S(105)*(a + b*x)**(S(3)/S(2))*(S(2)*(S(4)*a**S(2)*d**S(2)*f - S(7)*a*b*d*(d*e + S(2)*c*f) + S(5)*b**S(2)*c*(S(7)*d*e + S(2)*c*f)) + S(3)*b*d*(S(7)*b*d*e + S(4)*b*c*f - S(4)*a*d*f)*x)/b**S(3) - S(2)*c**S(2)*e*arctanh(sqrt(a + b*x)/sqrt(a))*sqrt(a) + S(2)*c**S(2)*e*sqrt(a + b*x)],
[(c + d*x)*(e + f*x)*sqrt(a + b*x)/x, x, S(4), - S(2)/S(15)*(a + b*x)**(S(3)/S(2))*(S(2)*a*d*f - S(5)*b*(d*e + c*f) - S(3)*b*d*f*x)/b**S(2) - S(2)*c*e*arctanh(sqrt(a + b*x)/sqrt(a))*sqrt(a) + S(2)*c*e*sqrt(a + b*x)],
[x**S(4)*(e + f*x)**n/((a + b*x)*(c + d*x)), x, S(8), e**S(2)*(e + f*x)**(S(1) + n)/(b*d*f**S(3)*(S(1) + n)) + (b*c + a*d)*e*(e + f*x)**(S(1) + n)/(b**S(2)*d**S(2)*f**S(2)*(S(1) + n)) + (b**S(2)*c**S(2) + a*b*c*d + a**S(2)*d**S(2))*(e + f*x)**(S(1) + n)/(b**S(3)*d**S(3)*f*(S(1) + n)) - S(2)*e*(e + f*x)**(S(2) + n)/(b*d*f**S(3)*(S(2) + n)) - (b*c + a*d)*(e + f*x)**(S(2) + n)/(b**S(2)*d**S(2)*f**S(2)*(S(2) + n)) + (e + f*x)**(S(3) + n)/(b*d*f**S(3)*(S(3) + n)) - a**S(4)*(e + f*x)**(S(1) + n)*hypergeom([S(1), S(1) + n], [S(2) + n], b*(e + f*x)/(b*e - a*f))/(b**S(3)*(b*c - a*d)*(b*e - a*f)*(S(1) + n)) + c**S(4)*(e + f*x)**(S(1) + n)*hypergeom([S(1), S(1) + n], [S(2) + n], d*(e + f*x)/(d*e - c*f))/(d**S(3)*(b*c - a*d)*(d*e - c*f)*(S(1) + n))],
[(a + b*x)*(c + d*x)*(e + f*x)*(g + h*x), x, S(2), a*c*e*g*x + S(1)/S(2)*(b*c*e*g + a*(d*e*g + c*f*g + c*e*h))*x**S(2) + S(1)/S(3)*(b*(d*e*g + c*f*g + c*e*h) + a*(d*f*g + d*e*h + c*f*h))*x**S(3) + S(1)/S(4)*(a*d*f*h + b*(d*f*g + d*e*h + c*f*h))*x**S(4) + S(1)/S(5)*b*d*f*h*x**S(5)],
[(a + b*x)*(c + d*x)*(e + f*x)/(g + h*x), x, S(2), (b*(d*g - c*h)*(f*g - e*h) - a*h*(d*f*g - d*e*h - c*f*h))*x/h**S(3) + S(1)/S(2)*(a*d*f*h - b*(d*f*g - d*e*h - c*f*h))*x**S(2)/h**S(2) + S(1)/S(3)*b*d*f*x**S(3)/h - (b*g - a*h)*(d*g - c*h)*(f*g - e*h)*log(g + h*x)/h**S(4)],
[(a + b*x)**m*(c + d*x)*(e + f*x)*(g + h*x), x, S(2), (b*c - a*d)*(b*e - a*f)*(b*g - a*h)*(a + b*x)**(S(1) + m)/(b**S(4)*(S(1) + m)) + (S(3)*a**S(2)*d*f*h + b**S(2)*(d*e*g + c*f*g + c*e*h) - S(2)*a*b*(d*f*g + d*e*h + c*f*h))*(a + b*x)**(S(2) + m)/(b**S(4)*(S(2) + m)) - (S(3)*a*d*f*h - b*(d*f*g + d*e*h + c*f*h))*(a + b*x)**(S(3) + m)/(b**S(4)*(S(3) + m)) + d*f*h*(a + b*x)**(S(4) + m)/(b**S(4)*(S(4) + m))],
[(c + d*x)**( - S(4) - m)*(e + f*x)**m*(g + h*x), x, S(3), - (d*g - c*h)*(c + d*x)**( - S(3) - m)*(e + f*x)**(S(1) + m)/(d*(d*e - c*f)*(S(3) + m)) + (c*f*h*(S(1) + m) + d*(S(2)*f*g - e*h*(S(3) + m)))*(c + d*x)**( - S(2) - m)*(e + f*x)**(S(1) + m)/(d*(d*e - c*f)**S(2)*(S(2) + m)*(S(3) + m)) - f*(c*f*h*(S(1) + m) + d*(S(2)*f*g - e*h*(S(3) + m)))*(c + d*x)**( - S(1) - m)*(e + f*x)**(S(1) + m)/(d*(d*e - c*f)**S(3)*(S(1) + m)*(S(2) + m)*(S(3) + m))],
]
for i in test:
r = rubi_integrate(i[0], i[1])
if len(i) == 5:
assert rubi_test(r, i[1], i[3], expand=True, _diff=True) or rubi_test(r, i[1], i[4], expand=True, _diff=True)
else:
assert rubi_test(r, i[1], i[3], expand=True, _diff=True)
def test_7():
test = [
# [(a + b*x)**(S(3)/S(2))*(c + d*x)**(S(1)/S(3)), x, S(5), S(12)/S(187)*(b*c - a*d)*(a + b*x)**(S(3)/S(2))*(c + d*x)**(S(1)/S(3))/(b*d) + S(6)/S(17)*(a + b*x)**(S(5)/S(2))*(c + d*x)**(S(1)/S(3))/b - S(108)/S(935)*(b*c - a*d)**S(2)*(c + d*x)**(S(1)/S(3))*sqrt(a + b*x)/(b*d**S(2)) - S(108)/S(935)*S(3)**(S(3)/S(4))*(b*c - a*d)**S(3)*((b*c - a*d)**(S(1)/S(3)) - b**(S(1)/S(3))*(c + d*x)**(S(1)/S(3)))*EllipticF(( - b**(S(1)/S(3))*(c + d*x)**(S(1)/S(3)) + (b*c - a*d)**(S(1)/S(3))*(S(1) + sqrt(S(3))))/( - b**(S(1)/S(3))*(c + d*x)**(S(1)/S(3)) + (b*c - a*d)**(S(1)/S(3))*(S(1) - sqrt(S(3)))), sqrt( - S(7) + S(4)*sqrt(S(3))))*sqrt(((b*c - a*d)**(S(2)/S(3)) + b**(S(1)/S(3))*(b*c - a*d)**(S(1)/S(3))*(c + d*x)**(S(1)/S(3)) + b**(S(2)/S(3))*(c + d*x)**(S(2)/S(3)))/( - b**(S(1)/S(3))*(c + d*x)**(S(1)/S(3)) + (b*c - a*d)**(S(1)/S(3))*(S(1) - sqrt(S(3))))**S(2))*sqrt(S(2) - sqrt(S(3)))/(b**(S(4)/S(3))*d**S(3)*sqrt(a - b*c/d + b*(c + d*x)/d)*sqrt( - (b*c - a*d)**(S(1)/S(3))*((b*c - a*d)**(S(1)/S(3)) - b**(S(1)/S(3))*(c + d*x)**(S(1)/S(3)))/( - b**(S(1)/S(3))*(c + d*x)**(S(1)/S(3)) + (b*c - a*d)**(S(1)/S(3))*(S(1) - sqrt(S(3))))**S(2)))],
# [(a + b*x)**(S(3)/S(2))/(c + d*x)**(S(1)/S(3)), x, S(6), S(6)/S(13)*(a + b*x)**(S(3)/S(2))*(c + d*x)**(S(2)/S(3))/d - S(54)/S(91)*(b*c - a*d)*(c + d*x)**(S(2)/S(3))*sqrt(a + b*x)/d**S(2) - S(162)/S(91)*(b*c - a*d)**S(2)*sqrt(a - b*c/d + b*(c + d*x)/d)/(b**(S(2)/S(3))*d**S(2)*( - b**(S(1)/S(3))*(c + d*x)**(S(1)/S(3)) + (b*c - a*d)**(S(1)/S(3))*(S(1) - sqrt(S(3))))) - S(54)/S(91)*S(3)**(S(3)/S(4))*(b*c - a*d)**(S(7)/S(3))*((b*c - a*d)**(S(1)/S(3)) - b**(S(1)/S(3))*(c + d*x)**(S(1)/S(3)))*EllipticF(( - b**(S(1)/S(3))*(c + d*x)**(S(1)/S(3)) + (b*c - a*d)**(S(1)/S(3))*(S(1) + sqrt(S(3))))/( - b**(S(1)/S(3))*(c + d*x)**(S(1)/S(3)) + (b*c - a*d)**(S(1)/S(3))*(S(1) - sqrt(S(3)))), sqrt( - S(7) + S(4)*sqrt(S(3))))*sqrt(S(2))*sqrt(((b*c - a*d)**(S(2)/S(3)) + b**(S(1)/S(3))*(b*c - a*d)**(S(1)/S(3))*(c + d*x)**(S(1)/S(3)) + b**(S(2)/S(3))*(c + d*x)**(S(2)/S(3)))/( - b**(S(1)/S(3))*(c + d*x)**(S(1)/S(3)) + (b*c - a*d)**(S(1)/S(3))*(S(1) - sqrt(S(3))))**S(2))/(b**(S(2)/S(3))*d**S(3)*sqrt(a - b*c/d + b*(c + d*x)/d)*sqrt( - (b*c - a*d)**(S(1)/S(3))*((b*c - a*d)**(S(1)/S(3)) - b**(S(1)/S(3))*(c + d*x)**(S(1)/S(3)))/( - b**(S(1)/S(3))*(c + d*x)**(S(1)/S(3)) + (b*c - a*d)**(S(1)/S(3))*(S(1) - sqrt(S(3))))**S(2))) + S(81)/S(91)*S(3)**(S(1)/S(4))*(b*c - a*d)**(S(7)/S(3))*((b*c - a*d)**(S(1)/S(3)) - b**(S(1)/S(3))*(c + d*x)**(S(1)/S(3)))*EllipticE(( - b**(S(1)/S(3))*(c + d*x)**(S(1)/S(3)) + (b*c - a*d)**(S(1)/S(3))*(S(1) + sqrt(S(3))))/( - b**(S(1)/S(3))*(c + d*x)**(S(1)/S(3)) + (b*c - a*d)**(S(1)/S(3))*(S(1) - sqrt(S(3)))), sqrt( - S(7) + S(4)*sqrt(S(3))))*sqrt(((b*c - a*d)**(S(2)/S(3)) + b**(S(1)/S(3))*(b*c - a*d)**(S(1)/S(3))*(c + d*x)**(S(1)/S(3)) + b**(S(2)/S(3))*(c + d*x)**(S(2)/S(3)))/( - b**(S(1)/S(3))*(c + d*x)**(S(1)/S(3)) + (b*c - a*d)**(S(1)/S(3))*(S(1) - sqrt(S(3))))**S(2))*sqrt(S(2) + sqrt(S(3)))/(b**(S(2)/S(3))*d**S(3)*sqrt(a - b*c/d + b*(c + d*x)/d)*sqrt( - (b*c - a*d)**(S(1)/S(3))*((b*c - a*d)**(S(1)/S(3)) - b**(S(1)/S(3))*(c + d*x)**(S(1)/S(3)))/( - b**(S(1)/S(3))*(c + d*x)**(S(1)/S(3)) + (b*c - a*d)**(S(1)/S(3))*(S(1) - sqrt(S(3))))**S(2)))],
[
(a + b * x) ** (S(2) / S(3)) * (c + d * x) ** (S(1) / S(3)),
x,
S(3),
S(1)
/ S(6)
* (b * c - a * d)
* (a + b * x) ** (S(2) / S(3))
* (c + d * x) ** (S(1) / S(3))
/ (b * d)
+ S(1)
/ S(2)
* (a + b * x) ** (S(5) / S(3))
* (c + d * x) ** (S(1) / S(3))
/ b
+ S(1)
/ S(18)
* (b * c - a * d) ** S(2)
* log(c + d * x)
/ (b ** (S(4) / S(3)) * d ** (S(5) / S(3)))
+ S(1)
/ S(6)
* (b * c - a * d) ** S(2)
* log(
-S(1)
+ d ** (S(1) / S(3))
* (a + b * x) ** (S(1) / S(3))
/ (b ** (S(1) / S(3)) * (c + d * x) ** (S(1) / S(3)))
)
/ (b ** (S(4) / S(3)) * d ** (S(5) / S(3)))
+ S(1)
/ S(3)
* (b * c - a * d) ** S(2)
* arctan(
S(1) / sqrt(S(3))
+ S(2)
* d ** (S(1) / S(3))
* (a + b * x) ** (S(1) / S(3))
/ (b ** (S(1) / S(3)) * (c + d * x) ** (S(1) / S(3)) * sqrt(S(3)))
)
/ (b ** (S(4) / S(3)) * d ** (S(5) / S(3)) * sqrt(S(3))),
],
[
(a + b * x) ** (S(4) / S(3)) / (c + d * x) ** (S(1) / S(3)),
x,
S(3),
-S(2)
/ S(3)
* (b * c - a * d)
* (a + b * x) ** (S(1) / S(3))
* (c + d * x) ** (S(2) / S(3))
/ d ** S(2)
+ S(1)
/ S(2)
* (a + b * x) ** (S(4) / S(3))
* (c + d * x) ** (S(2) / S(3))
/ d
- S(1)
/ S(9)
* (b * c - a * d) ** S(2)
* log(a + b * x)
/ (b ** (S(2) / S(3)) * d ** (S(7) / S(3)))
- S(1)
/ S(3)
* (b * c - a * d) ** S(2)
* log(
-S(1)
+ b ** (S(1) / S(3))
* (c + d * x) ** (S(1) / S(3))
/ (d ** (S(1) / S(3)) * (a + b * x) ** (S(1) / S(3)))
)
/ (b ** (S(2) / S(3)) * d ** (S(7) / S(3)))
- S(2)
/ S(3)
* (b * c - a * d) ** S(2)
* arctan(
S(1) / sqrt(S(3))
+ S(2)
* b ** (S(1) / S(3))
* (c + d * x) ** (S(1) / S(3))
/ (d ** (S(1) / S(3)) * (a + b * x) ** (S(1) / S(3)) * sqrt(S(3)))
)
/ (b ** (S(2) / S(3)) * d ** (S(7) / S(3)) * sqrt(S(3))),
],
# [(a + b*x)**(S(5)/S(2))/(c + d*x)**(S(1)/S(4)), x, S(10), - S(40)/S(117)*(b*c - a*d)*(a + b*x)**(S(3)/S(2))*(c + d*x)**(S(3)/S(4))/d**S(2) + S(4)/S(13)*(a + b*x)**(S(5)/S(2))*(c + d*x)**(S(3)/S(4))/d + S(16)/S(39)*(b*c - a*d)**S(2)*(c + d*x)**(S(3)/S(4))*sqrt(a + b*x)/d**S(3) - S(32)/S(39)*(b*c - a*d)**(S(15)/S(4))*EllipticE(b**(S(1)/S(4))*(c + d*x)**(S(1)/S(4))/(b*c - a*d)**(S(1)/S(4)), I)*sqrt(S(1) - b*(c + d*x)/(b*c - a*d))/(b**(S(3)/S(4))*d**S(4)*sqrt(a - b*c/d + b*(c + d*x)/d)) + S(32)/S(39)*(b*c - a*d)**(S(15)/S(4))*EllipticF(b**(S(1)/S(4))*(c + d*x)**(S(1)/S(4))/(b*c - a*d)**(S(1)/S(4)), I)*sqrt(S(1) - b*(c + d*x)/(b*c - a*d))/(b**(S(3)/S(4))*d**S(4)*sqrt(a - b*c/d + b*(c + d*x)/d))],
[
(c + d * x) ** (S(5) / S(4)) / (a + b * x) ** (S(25) / S(4)),
x,
S(4),
-S(4)
/ S(21)
* (c + d * x) ** (S(9) / S(4))
/ ((b * c - a * d) * (a + b * x) ** (S(21) / S(4)))
+ S(16)
/ S(119)
* d
* (c + d * x) ** (S(9) / S(4))
/ ((b * c - a * d) ** S(2) * (a + b * x) ** (S(17) / S(4)))
- S(128)
/ S(1547)
* d ** S(2)
* (c + d * x) ** (S(9) / S(4))
/ ((b * c - a * d) ** S(3) * (a + b * x) ** (S(13) / S(4)))
+ S(512)
/ S(13923)
* d ** S(3)
* (c + d * x) ** (S(9) / S(4))
/ ((b * c - a * d) ** S(4) * (a + b * x) ** (S(9) / S(4))),
],
[
(a + b * x) ** (S(5) / S(4)) / (c + d * x) ** (S(1) / S(4)),
x,
S(6),
-S(5)
/ S(8)
* (b * c - a * d)
* (a + b * x) ** (S(1) / S(4))
* (c + d * x) ** (S(3) / S(4))
/ d ** S(2)
+ S(1)
/ S(2)
* (a + b * x) ** (S(5) / S(4))
* (c + d * x) ** (S(3) / S(4))
/ d
+ S(5)
/ S(16)
* (b * c - a * d) ** S(2)
* arctan(
d ** (S(1) / S(4))
* (a + b * x) ** (S(1) / S(4))
/ (b ** (S(1) / S(4)) * (c + d * x) ** (S(1) / S(4)))
)
/ (b ** (S(3) / S(4)) * d ** (S(9) / S(4)))
+ S(5)
/ S(16)
* (b * c - a * d) ** S(2)
* arctanh(
d ** (S(1) / S(4))
* (a + b * x) ** (S(1) / S(4))
/ (b ** (S(1) / S(4)) * (c + d * x) ** (S(1) / S(4)))
)
/ (b ** (S(3) / S(4)) * d ** (S(9) / S(4))),
],
[
S(1) / ((a + b * x) ** (S(3) / S(4)) * (c + d * x) ** (S(1) / S(4))),
x,
S(4),
S(2)
* arctan(
d ** (S(1) / S(4))
* (a + b * x) ** (S(1) / S(4))
/ (b ** (S(1) / S(4)) * (c + d * x) ** (S(1) / S(4)))
)
/ (b ** (S(3) / S(4)) * d ** (S(1) / S(4)))
+ S(2)
* arctanh(
d ** (S(1) / S(4))
* (a + b * x) ** (S(1) / S(4))
/ (b ** (S(1) / S(4)) * (c + d * x) ** (S(1) / S(4)))
)
/ (b ** (S(3) / S(4)) * d ** (S(1) / S(4))),
],
# [(a + b*x)**(S(3)/S(2))/(c + d*x)**(S(1)/S(5)), x, S(2), S(2)/S(5)*(a + b*x)**(S(5)/S(2))*(b*(c + d*x)/(b*c - a*d))**(S(1)/S(5))*hypergeom([S(1)/S(5), S(5)/S(2)], [S(7)/S(2)], - d*(a + b*x)/(b*c - a*d))/(b*(c + d*x)**(S(1)/S(5)))],
# [(a + b*x)**(S(5)/S(2))/(c + d*x)**(S(1)/S(6)), x, S(7), - S(9)/S(28)*(b*c - a*d)*(a + b*x)**(S(3)/S(2))*(c + d*x)**(S(5)/S(6))/d**S(2) + S(3)/S(10)*(a + b*x)**(S(5)/S(2))*(c + d*x)**(S(5)/S(6))/d + S(81)/S(224)*(b*c - a*d)**S(2)*(c + d*x)**(S(5)/S(6))*sqrt(a + b*x)/d**S(3) + S(243)/S(448)*(b*c - a*d)**S(3)*(c + d*x)**(S(1)/S(6))*(S(1) + sqrt(S(3)))*sqrt(a - b*c/d + b*(c + d*x)/d)/(b**(S(2)/S(3))*d**S(3)*((b*c - a*d)**(S(1)/S(3)) - b**(S(1)/S(3))*(c + d*x)**(S(1)/S(3))*(S(1) + sqrt(S(3))))) + S(243)/S(448)*S(3)**(S(1)/S(4))*(b*c - a*d)**(S(10)/S(3))*(c + d*x)**(S(1)/S(6))*((b*c - a*d)**(S(1)/S(3)) - b**(S(1)/S(3))*(c + d*x)**(S(1)/S(3)))*sqrt(cos(arccos(((b*c - a*d)**(S(1)/S(3)) - b**(S(1)/S(3))*(c + d*x)**(S(1)/S(3))*(S(1) - sqrt(S(3))))/((b*c - a*d)**(S(1)/S(3)) - b**(S(1)/S(3))*(c + d*x)**(S(1)/S(3))*(S(1) + sqrt(S(3))))))**S(2))/cos(arccos(((b*c - a*d)**(S(1)/S(3)) - b**(S(1)/S(3))*(c + d*x)**(S(1)/S(3))*(S(1) - sqrt(S(3))))/((b*c - a*d)**(S(1)/S(3)) - b**(S(1)/S(3))*(c + d*x)**(S(1)/S(3))*(S(1) + sqrt(S(3))))))*EllipticE(sin(arccos(((b*c - a*d)**(S(1)/S(3)) - b**(S(1)/S(3))*(c + d*x)**(S(1)/S(3))*(S(1) - sqrt(S(3))))/((b*c - a*d)**(S(1)/S(3)) - b**(S(1)/S(3))*(c + d*x)**(S(1)/S(3))*(S(1) + sqrt(S(3)))))), sqrt(S(1)/S(4)*(S(2) + sqrt(S(3)))))*sqrt(((b*c - a*d)**(S(2)/S(3)) + b**(S(1)/S(3))*(b*c - a*d)**(S(1)/S(3))*(c + d*x)**(S(1)/S(3)) + b**(S(2)/S(3))*(c + d*x)**(S(2)/S(3)))/((b*c - a*d)**(S(1)/S(3)) - b**(S(1)/S(3))*(c + d*x)**(S(1)/S(3))*(S(1) + sqrt(S(3))))**S(2))/(b**(S(2)/S(3))*d**S(4)*sqrt(a - b*c/d + b*(c + d*x)/d)*sqrt( - b**(S(1)/S(3))*(c + d*x)**(S(1)/S(3))*((b*c - a*d)**(S(1)/S(3)) - b**(S(1)/S(3))*(c + d*x)**(S(1)/S(3)))/((b*c - a*d)**(S(1)/S(3)) - b**(S(1)/S(3))*(c + d*x)**(S(1)/S(3))*(S(1) + sqrt(S(3))))**S(2))) + S(81)/S(896)*S(3)**(S(3)/S(4))*(b*c - a*d)**(S(10)/S(3))*(c + d*x)**(S(1)/S(6))*((b*c - a*d)**(S(1)/S(3)) - b**(S(1)/S(3))*(c + d*x)**(S(1)/S(3)))*sqrt(cos(arccos(((b*c - a*d)**(S(1)/S(3)) - b**(S(1)/S(3))*(c + d*x)**(S(1)/S(3))*(S(1) - sqrt(S(3))))/((b*c - a*d)**(S(1)/S(3)) - b**(S(1)/S(3))*(c + d*x)**(S(1)/S(3))*(S(1) + sqrt(S(3))))))**S(2))/cos(arccos(((b*c - a*d)**(S(1)/S(3)) - b**(S(1)/S(3))*(c + d*x)**(S(1)/S(3))*(S(1) - sqrt(S(3))))/((b*c - a*d)**(S(1)/S(3)) - b**(S(1)/S(3))*(c + d*x)**(S(1)/S(3))*(S(1) + sqrt(S(3))))))*EllipticF(sin(arccos(((b*c - a*d)**(S(1)/S(3)) - b**(S(1)/S(3))*(c + d*x)**(S(1)/S(3))*(S(1) - sqrt(S(3))))/((b*c - a*d)**(S(1)/S(3)) - b**(S(1)/S(3))*(c + d*x)**(S(1)/S(3))*(S(1) + sqrt(S(3)))))), sqrt(S(1)/S(4)*(S(2) + sqrt(S(3)))))*(S(1) - sqrt(S(3)))*sqrt(((b*c - a*d)**(S(2)/S(3)) + b**(S(1)/S(3))*(b*c - a*d)**(S(1)/S(3))*(c + d*x)**(S(1)/S(3)) + b**(S(2)/S(3))*(c + d*x)**(S(2)/S(3)))/((b*c - a*d)**(S(1)/S(3)) - b**(S(1)/S(3))*(c + d*x)**(S(1)/S(3))*(S(1) + sqrt(S(3))))**S(2))/(b**(S(2)/S(3))*d**S(4)*sqrt(a - b*c/d + b*(c + d*x)/d)*sqrt( - b**(S(1)/S(3))*(c + d*x)**(S(1)/S(3))*((b*c - a*d)**(S(1)/S(3)) - b**(S(1)/S(3))*(c + d*x)**(S(1)/S(3)))/((b*c - a*d)**(S(1)/S(3)) - b**(S(1)/S(3))*(c + d*x)**(S(1)/S(3))*(S(1) + sqrt(S(3))))**S(2)))],
# [(a + b*x)**m*(c + d*x)**n, x, S(2), - (a + b*x)**(S(1) + m)*(c + d*x)**(S(1) + n)*hypergeom([S(1), S(2) + m + n], [S(2) + n], b*(c + d*x)/(b*c - a*d))/((b*c - a*d)*(S(1) + n)), (a + b*x)**(S(1) + m)*(c + d*x)**n*hypergeom([S(1) + m, - n], [S(2) + m], - d*(a + b*x)/(b*c - a*d))/(b*(S(1) + m)*(b*(c + d*x)/(b*c - a*d))**n)],
]
for i in test:
r = rubi_integrate(i[0], i[1])
if len(i) == 5:
assert rubi_test(r, i[1], i[3], expand=True, _diff=True) or rubi_test(
r, i[1], i[4], expand=True, _diff=True
)
else:
assert rubi_test(r, i[1], i[3], expand=True, _diff=True)
def test_5():
test = [
[
(S(1) + a * x) ** (S(3) / S(2)) / sqrt(S(1) - a * x),
x,
S(4),
S(3) / S(2) * arcsin(a * x) / a
- S(1) / S(2) * (S(1) + a * x) ** (S(3) / S(2)) * sqrt(S(1) - a * x) / a
- S(3) / S(2) * sqrt(S(1) - a * x) * sqrt(S(1) + a * x) / a,
],
[
(S(1) - x) ** (S(1) / S(2)) / (S(1) + x) ** (S(1) / S(2)),
x,
S(3),
arcsin(x) + sqrt(S(1) - x) * sqrt(S(1) + x),
],
[
S(1) / ((S(1) - x) ** (S(1) / S(2)) * (S(1) + x) ** (S(3) / S(2))),
x,
S(1),
-sqrt(S(1) - x) / sqrt(S(1) + x),
],
[
(a + a * x) ** (S(5) / S(2)) * (c - c * x) ** (S(5) / S(2)),
x,
S(5),
S(5)
/ S(24)
* a
* c
* x
* (a + a * x) ** (S(3) / S(2))
* (c - c * x) ** (S(3) / S(2))
+ S(1)
/ S(6)
* x
* (a + a * x) ** (S(5) / S(2))
* (c - c * x) ** (S(5) / S(2))
+ S(5)
/ S(8)
* a ** (S(5) / S(2))
* c ** (S(5) / S(2))
* arctan(sqrt(c) * sqrt(a + a * x) / (sqrt(a) * sqrt(c - c * x)))
+ S(5)
/ S(16)
* a ** S(2)
* c ** S(2)
* x
* sqrt(a + a * x)
* sqrt(c - c * x),
],
[
S(1) / ((a + a * x) ** (S(5) / S(2)) * (c - c * x) ** (S(5) / S(2))),
x,
S(2),
S(1)
/ S(3)
* x
/ (a * c * (a + a * x) ** (S(3) / S(2)) * (c - c * x) ** (S(3) / S(2)))
+ S(2)
/ S(3)
* x
/ (a ** S(2) * c ** S(2) * sqrt(a + a * x) * sqrt(c - c * x)),
],
[
(S(3) - x) ** (S(1) / S(2)) * (-S(2) + x) ** (S(1) / S(2)),
x,
S(5),
-S(1) / S(8) * arcsin(S(5) - S(2) * x)
- S(1) / S(2) * (S(3) - x) ** (S(3) / S(2)) * sqrt(-S(2) + x)
+ S(1) / S(4) * sqrt(S(3) - x) * sqrt(-S(2) + x),
],
[
S(1) / (sqrt(a + b * x) * sqrt(-a * d + b * d * x)),
x,
S(2),
S(2)
* arctanh(sqrt(d) * sqrt(a + b * x) / sqrt(-a * d + b * d * x))
/ (b * sqrt(d)),
],
[
S(1)
/ ((a - I * a * x) ** (S(7) / S(4)) * (a + I * a * x) ** (S(1) / S(4))),
x,
S(1),
-S(2)
/ S(3)
* I
* (a + I * a * x) ** (S(3) / S(4))
/ (a ** S(2) * (a - I * a * x) ** (S(3) / S(4))),
],
[
(a + b * x) ** S(2) * (a * c - b * c * x) ** n,
x,
S(2),
-S(4) * a ** S(2) * (a * c - b * c * x) ** (S(1) + n) / (b * c * (S(1) + n))
+ S(4)
* a
* (a * c - b * c * x) ** (S(2) + n)
/ (b * c ** S(2) * (S(2) + n))
- (a * c - b * c * x) ** (S(3) + n) / (b * c ** S(3) * (S(3) + n)),
],
[
(a + b * x) ** S(4) * (c + d * x),
x,
S(2),
S(1) / S(5) * (b * c - a * d) * (a + b * x) ** S(5) / b ** S(2)
+ S(1) / S(6) * d * (a + b * x) ** S(6) / b ** S(2),
],
[
(a + b * x) * (c + d * x),
x,
S(2),
a * c * x
+ S(1) / S(2) * (b * c + a * d) * x ** S(2)
+ S(1) / S(3) * b * d * x ** S(3),
],
[
(a + b * x) ** S(5) / (c + d * x),
x,
S(2),
b * (b * c - a * d) ** S(4) * x / d ** S(5)
- S(1) / S(2) * (b * c - a * d) ** S(3) * (a + b * x) ** S(2) / d ** S(4)
+ S(1) / S(3) * (b * c - a * d) ** S(2) * (a + b * x) ** S(3) / d ** S(3)
- S(1) / S(4) * (b * c - a * d) * (a + b * x) ** S(4) / d ** S(2)
+ S(1) / S(5) * (a + b * x) ** S(5) / d
- (b * c - a * d) ** S(5) * log(c + d * x) / d ** S(6),
],
[
(a + b * x) / (c + d * x) ** S(3),
x,
S(1),
S(1) / S(2) * (a + b * x) ** S(2) / ((b * c - a * d) * (c + d * x) ** S(2)),
],
[
(a + b * x) ** S(5) * (c + d * x) ** (S(1) / S(2)),
x,
S(2),
-S(2)
/ S(3)
* (b * c - a * d) ** S(5)
* (c + d * x) ** (S(3) / S(2))
/ d ** S(6)
+ S(2)
* b
* (b * c - a * d) ** S(4)
* (c + d * x) ** (S(5) / S(2))
/ d ** S(6)
- S(20)
/ S(7)
* b ** S(2)
* (b * c - a * d) ** S(3)
* (c + d * x) ** (S(7) / S(2))
/ d ** S(6)
+ S(20)
/ S(9)
* b ** S(3)
* (b * c - a * d) ** S(2)
* (c + d * x) ** (S(9) / S(2))
/ d ** S(6)
- S(10)
/ S(11)
* b ** S(4)
* (b * c - a * d)
* (c + d * x) ** (S(11) / S(2))
/ d ** S(6)
+ S(2) / S(13) * b ** S(5) * (c + d * x) ** (S(13) / S(2)) / d ** S(6),
],
[
(c + d * x) ** (S(1) / S(2)) / (a + b * x) ** S(2),
x,
S(3),
-d
* arctanh(sqrt(b) * sqrt(c + d * x) / sqrt(b * c - a * d))
/ (b ** (S(3) / S(2)) * sqrt(b * c - a * d))
- sqrt(c + d * x) / (b * (a + b * x)),
],
]
for i in test:
r = rubi_integrate(i[0], i[1])
if len(i) == 5:
assert rubi_test(r, i[1], i[3], expand=True, _diff=True) or rubi_test(
r, i[1], i[4], expand=True, _diff=True
)
else:
assert rubi_test(r, i[1], i[3], expand=True, _diff=True)
def test_4():
test = [
[
x ** m / (a + b * x) ** S(2),
x,
S(1),
x ** (S(1) + m)
* hypergeom([S(2), S(1) + m], [S(2) + m], -b * x / a)
/ (a ** S(2) * (S(1) + m)),
],
[
x ** m / sqrt(S(2) + S(3) * x),
x,
S(1),
x ** (S(1) + m)
* hypergeom([S(1) / S(2), S(1) + m], [S(2) + m], -S(3) / S(2) * x)
/ ((S(1) + m) * sqrt(S(2))),
],
[
x ** m * (a + b * x) ** n,
x,
S(2),
x ** (S(1) + m)
* (a + b * x) ** n
* hypergeom([S(1) + m, -n], [S(2) + m], -b * x / a)
/ ((S(1) + m) * (S(1) + b * x / a) ** n),
],
[
x ** (-S(1) + n) / (a + b * x) ** n,
x,
S(2),
x ** n
* (S(1) + b * x / a) ** n
* hypergeom([n, n], [S(1) + n], -b * x / a)
/ (n * (a + b * x) ** n),
],
[
(c + d * (a + b * x)) ** (S(5) / S(2)),
x,
S(2),
S(2) / S(7) * (c + d * (a + b * x)) ** (S(7) / S(2)) / (b * d),
],
[
(c + d * (a + b * x)) ** (S(3) / S(2)),
x,
S(2),
S(2) / S(5) * (c + d * (a + b * x)) ** (S(5) / S(2)) / (b * d),
],
[
(a + b * x) ** S(3) / (a * d / b + d * x) ** S(3),
x,
S(2),
b ** S(3) * x / d ** S(3),
],
[
(a + b * x) * (a * c - b * c * x) ** S(3),
x,
S(2),
-S(1) / S(2) * a * c ** S(3) * (a - b * x) ** S(4) / b
+ S(1) / S(5) * c ** S(3) * (a - b * x) ** S(5) / b,
],
[
(a * c - b * c * x) ** S(3) / (a + b * x),
x,
S(2),
-S(4) * a ** S(2) * c ** S(3) * x
+ a * c ** S(3) * (a - b * x) ** S(2) / b
+ S(1) / S(3) * c ** S(3) * (a - b * x) ** S(3) / b
+ S(8) * a ** S(3) * c ** S(3) * log(a + b * x) / b,
],
[
S(1) / ((a + b * x) ** S(2) * (a * c - b * c * x)),
x,
S(3),
(-S(1) / S(2)) / (a * b * c * (a + b * x))
+ S(1) / S(2) * arctanh(b * x / a) / (a ** S(2) * b * c),
],
[
(S(1) + x) ** (S(1) / S(2)) / (S(1) - x) ** (S(9) / S(2)),
x,
S(3),
S(1) / S(7) * (S(1) + x) ** (S(3) / S(2)) / (S(1) - x) ** (S(7) / S(2))
+ S(2) / S(35) * (S(1) + x) ** (S(3) / S(2)) / (S(1) - x) ** (S(5) / S(2))
+ S(2) / S(105) * (S(1) + x) ** (S(3) / S(2)) / (S(1) - x) ** (S(3) / S(2)),
],
[
(S(1) + x) ** (S(5) / S(2)) / (S(1) - x) ** (S(1) / S(2)),
x,
S(5),
S(5) / S(2) * arcsin(x)
- S(5) / S(6) * (S(1) + x) ** (S(3) / S(2)) * sqrt(S(1) - x)
- S(1) / S(3) * (S(1) + x) ** (S(5) / S(2)) * sqrt(S(1) - x)
- S(5) / S(2) * sqrt(S(1) - x) * sqrt(S(1) + x),
],
]
for i in test:
r = rubi_integrate(i[0], i[1])
if len(i) == 5:
assert rubi_test(r, i[1], i[3], expand=True, _diff=True) or rubi_test(
r, i[1], i[4], expand=True, _diff=True
)
else:
assert rubi_test(r, i[1], i[3], expand=True, _diff=True)
def test_7():
test = [
#[(a + b*x)**(S(3)/S(2))*(c + d*x)**(S(1)/S(3)), x, S(5), S(12)/S(187)*(b*c - a*d)*(a + b*x)**(S(3)/S(2))*(c + d*x)**(S(1)/S(3))/(b*d) + S(6)/S(17)*(a + b*x)**(S(5)/S(2))*(c + d*x)**(S(1)/S(3))/b - S(108)/S(935)*(b*c - a*d)**S(2)*(c + d*x)**(S(1)/S(3))*sqrt(a + b*x)/(b*d**S(2)) - S(108)/S(935)*S(3)**(S(3)/S(4))*(b*c - a*d)**S(3)*((b*c - a*d)**(S(1)/S(3)) - b**(S(1)/S(3))*(c + d*x)**(S(1)/S(3)))*EllipticF(( - b**(S(1)/S(3))*(c + d*x)**(S(1)/S(3)) + (b*c - a*d)**(S(1)/S(3))*(S(1) + sqrt(S(3))))/( - b**(S(1)/S(3))*(c + d*x)**(S(1)/S(3)) + (b*c - a*d)**(S(1)/S(3))*(S(1) - sqrt(S(3)))), sqrt( - S(7) + S(4)*sqrt(S(3))))*sqrt(((b*c - a*d)**(S(2)/S(3)) + b**(S(1)/S(3))*(b*c - a*d)**(S(1)/S(3))*(c + d*x)**(S(1)/S(3)) + b**(S(2)/S(3))*(c + d*x)**(S(2)/S(3)))/( - b**(S(1)/S(3))*(c + d*x)**(S(1)/S(3)) + (b*c - a*d)**(S(1)/S(3))*(S(1) - sqrt(S(3))))**S(2))*sqrt(S(2) - sqrt(S(3)))/(b**(S(4)/S(3))*d**S(3)*sqrt(a - b*c/d + b*(c + d*x)/d)*sqrt( - (b*c - a*d)**(S(1)/S(3))*((b*c - a*d)**(S(1)/S(3)) - b**(S(1)/S(3))*(c + d*x)**(S(1)/S(3)))/( - b**(S(1)/S(3))*(c + d*x)**(S(1)/S(3)) + (b*c - a*d)**(S(1)/S(3))*(S(1) - sqrt(S(3))))**S(2)))],
#[(a + b*x)**(S(3)/S(2))/(c + d*x)**(S(1)/S(3)), x, S(6), S(6)/S(13)*(a + b*x)**(S(3)/S(2))*(c + d*x)**(S(2)/S(3))/d - S(54)/S(91)*(b*c - a*d)*(c + d*x)**(S(2)/S(3))*sqrt(a + b*x)/d**S(2) - S(162)/S(91)*(b*c - a*d)**S(2)*sqrt(a - b*c/d + b*(c + d*x)/d)/(b**(S(2)/S(3))*d**S(2)*( - b**(S(1)/S(3))*(c + d*x)**(S(1)/S(3)) + (b*c - a*d)**(S(1)/S(3))*(S(1) - sqrt(S(3))))) - S(54)/S(91)*S(3)**(S(3)/S(4))*(b*c - a*d)**(S(7)/S(3))*((b*c - a*d)**(S(1)/S(3)) - b**(S(1)/S(3))*(c + d*x)**(S(1)/S(3)))*EllipticF(( - b**(S(1)/S(3))*(c + d*x)**(S(1)/S(3)) + (b*c - a*d)**(S(1)/S(3))*(S(1) + sqrt(S(3))))/( - b**(S(1)/S(3))*(c + d*x)**(S(1)/S(3)) + (b*c - a*d)**(S(1)/S(3))*(S(1) - sqrt(S(3)))), sqrt( - S(7) + S(4)*sqrt(S(3))))*sqrt(S(2))*sqrt(((b*c - a*d)**(S(2)/S(3)) + b**(S(1)/S(3))*(b*c - a*d)**(S(1)/S(3))*(c + d*x)**(S(1)/S(3)) + b**(S(2)/S(3))*(c + d*x)**(S(2)/S(3)))/( - b**(S(1)/S(3))*(c + d*x)**(S(1)/S(3)) + (b*c - a*d)**(S(1)/S(3))*(S(1) - sqrt(S(3))))**S(2))/(b**(S(2)/S(3))*d**S(3)*sqrt(a - b*c/d + b*(c + d*x)/d)*sqrt( - (b*c - a*d)**(S(1)/S(3))*((b*c - a*d)**(S(1)/S(3)) - b**(S(1)/S(3))*(c + d*x)**(S(1)/S(3)))/( - b**(S(1)/S(3))*(c + d*x)**(S(1)/S(3)) + (b*c - a*d)**(S(1)/S(3))*(S(1) - sqrt(S(3))))**S(2))) + S(81)/S(91)*S(3)**(S(1)/S(4))*(b*c - a*d)**(S(7)/S(3))*((b*c - a*d)**(S(1)/S(3)) - b**(S(1)/S(3))*(c + d*x)**(S(1)/S(3)))*EllipticE(( - b**(S(1)/S(3))*(c + d*x)**(S(1)/S(3)) + (b*c - a*d)**(S(1)/S(3))*(S(1) + sqrt(S(3))))/( - b**(S(1)/S(3))*(c + d*x)**(S(1)/S(3)) + (b*c - a*d)**(S(1)/S(3))*(S(1) - sqrt(S(3)))), sqrt( - S(7) + S(4)*sqrt(S(3))))*sqrt(((b*c - a*d)**(S(2)/S(3)) + b**(S(1)/S(3))*(b*c - a*d)**(S(1)/S(3))*(c + d*x)**(S(1)/S(3)) + b**(S(2)/S(3))*(c + d*x)**(S(2)/S(3)))/( - b**(S(1)/S(3))*(c + d*x)**(S(1)/S(3)) + (b*c - a*d)**(S(1)/S(3))*(S(1) - sqrt(S(3))))**S(2))*sqrt(S(2) + sqrt(S(3)))/(b**(S(2)/S(3))*d**S(3)*sqrt(a - b*c/d + b*(c + d*x)/d)*sqrt( - (b*c - a*d)**(S(1)/S(3))*((b*c - a*d)**(S(1)/S(3)) - b**(S(1)/S(3))*(c + d*x)**(S(1)/S(3)))/( - b**(S(1)/S(3))*(c + d*x)**(S(1)/S(3)) + (b*c - a*d)**(S(1)/S(3))*(S(1) - sqrt(S(3))))**S(2)))],
[(a + b*x)**(S(2)/S(3))*(c + d*x)**(S(1)/S(3)), x, S(3), S(1)/S(6)*(b*c - a*d)*(a + b*x)**(S(2)/S(3))*(c + d*x)**(S(1)/S(3))/(b*d) + S(1)/S(2)*(a + b*x)**(S(5)/S(3))*(c + d*x)**(S(1)/S(3))/b + S(1)/S(18)*(b*c - a*d)**S(2)*log(c + d*x)/(b**(S(4)/S(3))*d**(S(5)/S(3))) + S(1)/S(6)*(b*c - a*d)**S(2)*log( - S(1) + d**(S(1)/S(3))*(a + b*x)**(S(1)/S(3))/(b**(S(1)/S(3))*(c + d*x)**(S(1)/S(3))))/(b**(S(4)/S(3))*d**(S(5)/S(3))) + S(1)/S(3)*(b*c - a*d)**S(2)*arctan(S(1)/sqrt(S(3)) + S(2)*d**(S(1)/S(3))*(a + b*x)**(S(1)/S(3))/(b**(S(1)/S(3))*(c + d*x)**(S(1)/S(3))*sqrt(S(3))))/(b**(S(4)/S(3))*d**(S(5)/S(3))*sqrt(S(3)))],
[(a + b*x)**(S(4)/S(3))/(c + d*x)**(S(1)/S(3)), x, S(3), - S(2)/S(3)*(b*c - a*d)*(a + b*x)**(S(1)/S(3))*(c + d*x)**(S(2)/S(3))/d**S(2) + S(1)/S(2)*(a + b*x)**(S(4)/S(3))*(c + d*x)**(S(2)/S(3))/d - S(1)/S(9)*(b*c - a*d)**S(2)*log(a + b*x)/(b**(S(2)/S(3))*d**(S(7)/S(3))) - S(1)/S(3)*(b*c - a*d)**S(2)*log( - S(1) + b**(S(1)/S(3))*(c + d*x)**(S(1)/S(3))/(d**(S(1)/S(3))*(a + b*x)**(S(1)/S(3))))/(b**(S(2)/S(3))*d**(S(7)/S(3))) - S(2)/S(3)*(b*c - a*d)**S(2)*arctan(S(1)/sqrt(S(3)) + S(2)*b**(S(1)/S(3))*(c + d*x)**(S(1)/S(3))/(d**(S(1)/S(3))*(a + b*x)**(S(1)/S(3))*sqrt(S(3))))/(b**(S(2)/S(3))*d**(S(7)/S(3))*sqrt(S(3)))],
#[(a + b*x)**(S(5)/S(2))/(c + d*x)**(S(1)/S(4)), x, S(10), - S(40)/S(117)*(b*c - a*d)*(a + b*x)**(S(3)/S(2))*(c + d*x)**(S(3)/S(4))/d**S(2) + S(4)/S(13)*(a + b*x)**(S(5)/S(2))*(c + d*x)**(S(3)/S(4))/d + S(16)/S(39)*(b*c - a*d)**S(2)*(c + d*x)**(S(3)/S(4))*sqrt(a + b*x)/d**S(3) - S(32)/S(39)*(b*c - a*d)**(S(15)/S(4))*EllipticE(b**(S(1)/S(4))*(c + d*x)**(S(1)/S(4))/(b*c - a*d)**(S(1)/S(4)), I)*sqrt(S(1) - b*(c + d*x)/(b*c - a*d))/(b**(S(3)/S(4))*d**S(4)*sqrt(a - b*c/d + b*(c + d*x)/d)) + S(32)/S(39)*(b*c - a*d)**(S(15)/S(4))*EllipticF(b**(S(1)/S(4))*(c + d*x)**(S(1)/S(4))/(b*c - a*d)**(S(1)/S(4)), I)*sqrt(S(1) - b*(c + d*x)/(b*c - a*d))/(b**(S(3)/S(4))*d**S(4)*sqrt(a - b*c/d + b*(c + d*x)/d))],
[(c + d*x)**(S(5)/S(4))/(a + b*x)**(S(25)/S(4)), x, S(4), - S(4)/S(21)*(c + d*x)**(S(9)/S(4))/((b*c - a*d)*(a + b*x)**(S(21)/S(4))) + S(16)/S(119)*d*(c + d*x)**(S(9)/S(4))/((b*c - a*d)**S(2)*(a + b*x)**(S(17)/S(4))) - S(128)/S(1547)*d**S(2)*(c + d*x)**(S(9)/S(4))/((b*c - a*d)**S(3)*(a + b*x)**(S(13)/S(4))) + S(512)/S(13923)*d**S(3)*(c + d*x)**(S(9)/S(4))/((b*c - a*d)**S(4)*(a + b*x)**(S(9)/S(4)))],
[(a + b*x)**(S(5)/S(4))/(c + d*x)**(S(1)/S(4)), x, S(6), - S(5)/S(8)*(b*c - a*d)*(a + b*x)**(S(1)/S(4))*(c + d*x)**(S(3)/S(4))/d**S(2) + S(1)/S(2)*(a + b*x)**(S(5)/S(4))*(c + d*x)**(S(3)/S(4))/d + S(5)/S(16)*(b*c - a*d)**S(2)*arctan(d**(S(1)/S(4))*(a + b*x)**(S(1)/S(4))/(b**(S(1)/S(4))*(c + d*x)**(S(1)/S(4))))/(b**(S(3)/S(4))*d**(S(9)/S(4))) + S(5)/S(16)*(b*c - a*d)**S(2)*arctanh(d**(S(1)/S(4))*(a + b*x)**(S(1)/S(4))/(b**(S(1)/S(4))*(c + d*x)**(S(1)/S(4))))/(b**(S(3)/S(4))*d**(S(9)/S(4)))],
[S(1)/((a + b*x)**(S(3)/S(4))*(c + d*x)**(S(1)/S(4))), x, S(4), S(2)*arctan(d**(S(1)/S(4))*(a + b*x)**(S(1)/S(4))/(b**(S(1)/S(4))*(c + d*x)**(S(1)/S(4))))/(b**(S(3)/S(4))*d**(S(1)/S(4))) + S(2)*arctanh(d**(S(1)/S(4))*(a + b*x)**(S(1)/S(4))/(b**(S(1)/S(4))*(c + d*x)**(S(1)/S(4))))/(b**(S(3)/S(4))*d**(S(1)/S(4)))],
#[(a + b*x)**(S(3)/S(2))/(c + d*x)**(S(1)/S(5)), x, S(2), S(2)/S(5)*(a + b*x)**(S(5)/S(2))*(b*(c + d*x)/(b*c - a*d))**(S(1)/S(5))*hypergeom([S(1)/S(5), S(5)/S(2)], [S(7)/S(2)], - d*(a + b*x)/(b*c - a*d))/(b*(c + d*x)**(S(1)/S(5)))],
#[(a + b*x)**(S(5)/S(2))/(c + d*x)**(S(1)/S(6)), x, S(7), - S(9)/S(28)*(b*c - a*d)*(a + b*x)**(S(3)/S(2))*(c + d*x)**(S(5)/S(6))/d**S(2) + S(3)/S(10)*(a + b*x)**(S(5)/S(2))*(c + d*x)**(S(5)/S(6))/d + S(81)/S(224)*(b*c - a*d)**S(2)*(c + d*x)**(S(5)/S(6))*sqrt(a + b*x)/d**S(3) + S(243)/S(448)*(b*c - a*d)**S(3)*(c + d*x)**(S(1)/S(6))*(S(1) + sqrt(S(3)))*sqrt(a - b*c/d + b*(c + d*x)/d)/(b**(S(2)/S(3))*d**S(3)*((b*c - a*d)**(S(1)/S(3)) - b**(S(1)/S(3))*(c + d*x)**(S(1)/S(3))*(S(1) + sqrt(S(3))))) + S(243)/S(448)*S(3)**(S(1)/S(4))*(b*c - a*d)**(S(10)/S(3))*(c + d*x)**(S(1)/S(6))*((b*c - a*d)**(S(1)/S(3)) - b**(S(1)/S(3))*(c + d*x)**(S(1)/S(3)))*sqrt(cos(arccos(((b*c - a*d)**(S(1)/S(3)) - b**(S(1)/S(3))*(c + d*x)**(S(1)/S(3))*(S(1) - sqrt(S(3))))/((b*c - a*d)**(S(1)/S(3)) - b**(S(1)/S(3))*(c + d*x)**(S(1)/S(3))*(S(1) + sqrt(S(3))))))**S(2))/cos(arccos(((b*c - a*d)**(S(1)/S(3)) - b**(S(1)/S(3))*(c + d*x)**(S(1)/S(3))*(S(1) - sqrt(S(3))))/((b*c - a*d)**(S(1)/S(3)) - b**(S(1)/S(3))*(c + d*x)**(S(1)/S(3))*(S(1) + sqrt(S(3))))))*EllipticE(sin(arccos(((b*c - a*d)**(S(1)/S(3)) - b**(S(1)/S(3))*(c + d*x)**(S(1)/S(3))*(S(1) - sqrt(S(3))))/((b*c - a*d)**(S(1)/S(3)) - b**(S(1)/S(3))*(c + d*x)**(S(1)/S(3))*(S(1) + sqrt(S(3)))))), sqrt(S(1)/S(4)*(S(2) + sqrt(S(3)))))*sqrt(((b*c - a*d)**(S(2)/S(3)) + b**(S(1)/S(3))*(b*c - a*d)**(S(1)/S(3))*(c + d*x)**(S(1)/S(3)) + b**(S(2)/S(3))*(c + d*x)**(S(2)/S(3)))/((b*c - a*d)**(S(1)/S(3)) - b**(S(1)/S(3))*(c + d*x)**(S(1)/S(3))*(S(1) + sqrt(S(3))))**S(2))/(b**(S(2)/S(3))*d**S(4)*sqrt(a - b*c/d + b*(c + d*x)/d)*sqrt( - b**(S(1)/S(3))*(c + d*x)**(S(1)/S(3))*((b*c - a*d)**(S(1)/S(3)) - b**(S(1)/S(3))*(c + d*x)**(S(1)/S(3)))/((b*c - a*d)**(S(1)/S(3)) - b**(S(1)/S(3))*(c + d*x)**(S(1)/S(3))*(S(1) + sqrt(S(3))))**S(2))) + S(81)/S(896)*S(3)**(S(3)/S(4))*(b*c - a*d)**(S(10)/S(3))*(c + d*x)**(S(1)/S(6))*((b*c - a*d)**(S(1)/S(3)) - b**(S(1)/S(3))*(c + d*x)**(S(1)/S(3)))*sqrt(cos(arccos(((b*c - a*d)**(S(1)/S(3)) - b**(S(1)/S(3))*(c + d*x)**(S(1)/S(3))*(S(1) - sqrt(S(3))))/((b*c - a*d)**(S(1)/S(3)) - b**(S(1)/S(3))*(c + d*x)**(S(1)/S(3))*(S(1) + sqrt(S(3))))))**S(2))/cos(arccos(((b*c - a*d)**(S(1)/S(3)) - b**(S(1)/S(3))*(c + d*x)**(S(1)/S(3))*(S(1) - sqrt(S(3))))/((b*c - a*d)**(S(1)/S(3)) - b**(S(1)/S(3))*(c + d*x)**(S(1)/S(3))*(S(1) + sqrt(S(3))))))*EllipticF(sin(arccos(((b*c - a*d)**(S(1)/S(3)) - b**(S(1)/S(3))*(c + d*x)**(S(1)/S(3))*(S(1) - sqrt(S(3))))/((b*c - a*d)**(S(1)/S(3)) - b**(S(1)/S(3))*(c + d*x)**(S(1)/S(3))*(S(1) + sqrt(S(3)))))), sqrt(S(1)/S(4)*(S(2) + sqrt(S(3)))))*(S(1) - sqrt(S(3)))*sqrt(((b*c - a*d)**(S(2)/S(3)) + b**(S(1)/S(3))*(b*c - a*d)**(S(1)/S(3))*(c + d*x)**(S(1)/S(3)) + b**(S(2)/S(3))*(c + d*x)**(S(2)/S(3)))/((b*c - a*d)**(S(1)/S(3)) - b**(S(1)/S(3))*(c + d*x)**(S(1)/S(3))*(S(1) + sqrt(S(3))))**S(2))/(b**(S(2)/S(3))*d**S(4)*sqrt(a - b*c/d + b*(c + d*x)/d)*sqrt( - b**(S(1)/S(3))*(c + d*x)**(S(1)/S(3))*((b*c - a*d)**(S(1)/S(3)) - b**(S(1)/S(3))*(c + d*x)**(S(1)/S(3)))/((b*c - a*d)**(S(1)/S(3)) - b**(S(1)/S(3))*(c + d*x)**(S(1)/S(3))*(S(1) + sqrt(S(3))))**S(2)))],
#[(a + b*x)**m*(c + d*x)**n, x, S(2), - (a + b*x)**(S(1) + m)*(c + d*x)**(S(1) + n)*hypergeom([S(1), S(2) + m + n], [S(2) + n], b*(c + d*x)/(b*c - a*d))/((b*c - a*d)*(S(1) + n)), (a + b*x)**(S(1) + m)*(c + d*x)**n*hypergeom([S(1) + m, - n], [S(2) + m], - d*(a + b*x)/(b*c - a*d))/(b*(S(1) + m)*(b*(c + d*x)/(b*c - a*d))**n)],
]
for i in test:
r = rubi_integrate(i[0], i[1])
if len(i) == 5:
assert rubi_test(r, i[1], i[3], expand=True, _diff=True) or rubi_test(r, i[1], i[4], expand=True, _diff=True)
else:
assert rubi_test(r, i[1], i[3], expand=True, _diff=True)
def test_4():
test = [
[x**m/(a + b*x)**S(2), x, S(1), x**(S(1) + m)*hypergeom([S(2), S(1) + m], [S(2) + m], - b*x/a)/(a**S(2)*(S(1) + m))],
[x**m/sqrt(S(2) + S(3)*x), x, S(1), x**(S(1) + m)*hypergeom([S(1)/S(2), S(1) + m], [S(2) + m], - S(3)/S(2)*x)/((S(1) + m)*sqrt(S(2)))],
[x**m*(a + b*x)**n, x, S(2), x**(S(1) + m)*(a + b*x)**n*hypergeom([S(1) + m, - n], [S(2) + m], - b*x/a)/((S(1) + m)*(S(1) + b*x/a)**n)],
[x**( - S(1) + n)/(a + b*x)**n, x, S(2), x**n*(S(1) + b*x/a)**n*hypergeom([n, n], [S(1) + n], - b*x/a)/(n*(a + b*x)**n)],
[(c + d*(a + b*x))**(S(5)/S(2)), x, S(2), S(2)/S(7)*(c + d*(a + b*x))**(S(7)/S(2))/(b*d)],
[(c + d*(a + b*x))**(S(3)/S(2)), x, S(2), S(2)/S(5)*(c + d*(a + b*x))**(S(5)/S(2))/(b*d)],
[(a + b*x)**S(3)/(a*d/b + d*x)**S(3), x, S(2), b**S(3)*x/d**S(3)],
[(a + b*x)*(a*c - b*c*x)**S(3), x, S(2), - S(1)/S(2)*a*c**S(3)*(a - b*x)**S(4)/b + S(1)/S(5)*c**S(3)*(a - b*x)**S(5)/b],
[(a*c - b*c*x)**S(3)/(a + b*x), x, S(2), - S(4)*a**S(2)*c**S(3)*x + a*c**S(3)*(a - b*x)**S(2)/b + S(1)/S(3)*c**S(3)*(a - b*x)**S(3)/b + S(8)*a**S(3)*c**S(3)*log(a + b*x)/b],
[S(1)/((a + b*x)**S(2)*(a*c - b*c*x)), x, S(3), ( - S(1)/S(2))/(a*b*c*(a + b*x)) + S(1)/S(2)*arctanh(b*x/a)/(a**S(2)*b*c)],
[(S(1) + x)**(S(1)/S(2))/(S(1) - x)**(S(9)/S(2)), x, S(3), S(1)/S(7)*(S(1) + x)**(S(3)/S(2))/(S(1) - x)**(S(7)/S(2)) + S(2)/S(35)*(S(1) + x)**(S(3)/S(2))/(S(1) - x)**(S(5)/S(2)) + S(2)/S(105)*(S(1) + x)**(S(3)/S(2))/(S(1) - x)**(S(3)/S(2))],
[(S(1) + x)**(S(5)/S(2))/(S(1) - x)**(S(1)/S(2)), x, S(5), S(5)/S(2)*arcsin(x) - S(5)/S(6)*(S(1) + x)**(S(3)/S(2))*sqrt(S(1) - x) - S(1)/S(3)*(S(1) + x)**(S(5)/S(2))*sqrt(S(1) - x) - S(5)/S(2)*sqrt(S(1) - x)*sqrt(S(1) + x)],
]
for i in test:
r = rubi_integrate(i[0], i[1])
if len(i) == 5:
assert rubi_test(r, i[1], i[3], expand=True, _diff=True) or rubi_test(r, i[1], i[4], expand=True, _diff=True)
else:
assert rubi_test(r, i[1], i[3], expand=True, _diff=True)
def test_1():
'''
Tests for Rubi Algebraic 1.2 rules. Parsed from Maple syntax
All tests: http://www.apmaths.uwo.ca/~arich/IntegrationProblems/MapleSyntaxFiles/MapleSyntaxFiles.html
Note: Some tests are commented since they depend rules other than Algebraic1.2.
'''
test = [
[(a + b*x)**S(2)*(e + f*x)*sqrt(c + d*x)/x, x, S(5), S(2)/S(7)*f*(a + b*x)**S(2)*(c + d*x)**(S(3)/S(2))/d + S(2)/S(105)*(c + d*x)**(S(3)/S(2))*(S(2)*(S(10)*a**S(2)*d**S(2)*f - b**S(2)*c*(S(7)*d*e - S(4)*c*f) + S(7)*a*b*d*(S(5)*d*e - S(2)*c*f)) + S(3)*b*d*(S(7)*b*d*e - S(4)*b*c*f + S(4)*a*d*f)*x)/d**S(3) - S(2)*a**S(2)*e*arctanh(sqrt(c + d*x)/sqrt(c))*sqrt(c) + S(2)*a**S(2)*e*sqrt(c + d*x)],
[(a + b*x)*(e + f*x)*sqrt(c + d*x)/x, x, S(4), - S(2)/S(15)*(c + d*x)**(S(3)/S(2))*(S(2)*b*c*f - S(5)*d*(b*e + a*f) - S(3)*b*d*f*x)/d**S(2) - S(2)*a*e*arctanh(sqrt(c + d*x)/sqrt(c))*sqrt(c) + S(2)*a*e*sqrt(c + d*x)],
[(c + d*x)**S(2)*(e + f*x)*sqrt(a + b*x)/x, x, S(5), S(2)/S(7)*f*(a + b*x)**(S(3)/S(2))*(c + d*x)**S(2)/b + S(2)/S(105)*(a + b*x)**(S(3)/S(2))*(S(2)*(S(4)*a**S(2)*d**S(2)*f - S(7)*a*b*d*(d*e + S(2)*c*f) + S(5)*b**S(2)*c*(S(7)*d*e + S(2)*c*f)) + S(3)*b*d*(S(7)*b*d*e + S(4)*b*c*f - S(4)*a*d*f)*x)/b**S(3) - S(2)*c**S(2)*e*arctanh(sqrt(a + b*x)/sqrt(a))*sqrt(a) + S(2)*c**S(2)*e*sqrt(a + b*x)],
[(c + d*x)*(e + f*x)*sqrt(a + b*x)/x, x, S(4), - S(2)/S(15)*(a + b*x)**(S(3)/S(2))*(S(2)*a*d*f - S(5)*b*(d*e + c*f) - S(3)*b*d*f*x)/b**S(2) - S(2)*c*e*arctanh(sqrt(a + b*x)/sqrt(a))*sqrt(a) + S(2)*c*e*sqrt(a + b*x)],
[x**S(4)*(e + f*x)**n/((a + b*x)*(c + d*x)), x, S(8), e**S(2)*(e + f*x)**(S(1) + n)/(b*d*f**S(3)*(S(1) + n)) + (b*c + a*d)*e*(e + f*x)**(S(1) + n)/(b**S(2)*d**S(2)*f**S(2)*(S(1) + n)) + (b**S(2)*c**S(2) + a*b*c*d + a**S(2)*d**S(2))*(e + f*x)**(S(1) + n)/(b**S(3)*d**S(3)*f*(S(1) + n)) - S(2)*e*(e + f*x)**(S(2) + n)/(b*d*f**S(3)*(S(2) + n)) - (b*c + a*d)*(e + f*x)**(S(2) + n)/(b**S(2)*d**S(2)*f**S(2)*(S(2) + n)) + (e + f*x)**(S(3) + n)/(b*d*f**S(3)*(S(3) + n)) - a**S(4)*(e + f*x)**(S(1) + n)*hypergeom([S(1), S(1) + n], [S(2) + n], b*(e + f*x)/(b*e - a*f))/(b**S(3)*(b*c - a*d)*(b*e - a*f)*(S(1) + n)) + c**S(4)*(e + f*x)**(S(1) + n)*hypergeom([S(1), S(1) + n], [S(2) + n], d*(e + f*x)/(d*e - c*f))/(d**S(3)*(b*c - a*d)*(d*e - c*f)*(S(1) + n))],
[(a + b*x)*(c + d*x)*(e + f*x)*(g + h*x), x, S(2), a*c*e*g*x + S(1)/S(2)*(b*c*e*g + a*(d*e*g + c*f*g + c*e*h))*x**S(2) + S(1)/S(3)*(b*(d*e*g + c*f*g + c*e*h) + a*(d*f*g + d*e*h + c*f*h))*x**S(3) + S(1)/S(4)*(a*d*f*h + b*(d*f*g + d*e*h + c*f*h))*x**S(4) + S(1)/S(5)*b*d*f*h*x**S(5)],
[(a + b*x)*(c + d*x)*(e + f*x)/(g + h*x), x, S(2), (b*(d*g - c*h)*(f*g - e*h) - a*h*(d*f*g - d*e*h - c*f*h))*x/h**S(3) + S(1)/S(2)*(a*d*f*h - b*(d*f*g - d*e*h - c*f*h))*x**S(2)/h**S(2) + S(1)/S(3)*b*d*f*x**S(3)/h - (b*g - a*h)*(d*g - c*h)*(f*g - e*h)*log(g + h*x)/h**S(4)],
[(a + b*x)**m*(c + d*x)*(e + f*x)*(g + h*x), x, S(2), (b*c - a*d)*(b*e - a*f)*(b*g - a*h)*(a + b*x)**(S(1) + m)/(b**S(4)*(S(1) + m)) + (S(3)*a**S(2)*d*f*h + b**S(2)*(d*e*g + c*f*g + c*e*h) - S(2)*a*b*(d*f*g + d*e*h + c*f*h))*(a + b*x)**(S(2) + m)/(b**S(4)*(S(2) + m)) - (S(3)*a*d*f*h - b*(d*f*g + d*e*h + c*f*h))*(a + b*x)**(S(3) + m)/(b**S(4)*(S(3) + m)) + d*f*h*(a + b*x)**(S(4) + m)/(b**S(4)*(S(4) + m))],
[(c + d*x)**( - S(4) - m)*(e + f*x)**m*(g + h*x), x, S(3), - (d*g - c*h)*(c + d*x)**( - S(3) - m)*(e + f*x)**(S(1) + m)/(d*(d*e - c*f)*(S(3) + m)) + (c*f*h*(S(1) + m) + d*(S(2)*f*g - e*h*(S(3) + m)))*(c + d*x)**( - S(2) - m)*(e + f*x)**(S(1) + m)/(d*(d*e - c*f)**S(2)*(S(2) + m)*(S(3) + m)) - f*(c*f*h*(S(1) + m) + d*(S(2)*f*g - e*h*(S(3) + m)))*(c + d*x)**( - S(1) - m)*(e + f*x)**(S(1) + m)/(d*(d*e - c*f)**S(3)*(S(1) + m)*(S(2) + m)*(S(3) + m))],
]
for i in test:
r = rubi_integrate(i[0], i[1])
if len(i) == 5:
assert rubi_test(r, i[1], i[3], expand=True, _diff=True) or rubi_test(r, i[1], i[4], expand=True, _diff=True)
else:
assert rubi_test(r, i[1], i[3], expand=True, _diff=True)