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_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_2():
test = [
[x**m*(e + f*x)**n/((a + b*x)*(c + d*x)), x, S(6), b*x**(S(1) + m)*(e + f*x)**n*AppellF1(S(1) + m, - n, S(1), S(2) + m, - f*x/e, - b*x/a)/(a*(b*c - a*d)*(S(1) + m)*(S(1) + f*x/e)**n) - d*x**(S(1) + m)*(e + f*x)**n*AppellF1(S(1) + m, - n, S(1), S(2) + m, - f*x/e, - d*x/c)/(c*(b*c - a*d)*(S(1) + m)*(S(1) + f*x/e)**n)],
[(a + b*x)**m*(c + d*x)**n*(e + f*x)*(g + h*x), x, S(3), - (a + b*x)**(S(1) + m)*(c + d*x)**(S(1) + n)*(b*c*f*h*(S(2) + m) + a*d*f*h*(S(2) + n) - b*d*(f*g + e*h)*(S(3) + m + n) - b*d*f*h*(S(2) + m + n)*x)/(b**S(2)*d**S(2)*(S(2) + m + n)*(S(3) + m + n)) + (a**S(2)*d**S(2)*f*h*(S(1) + n)*(S(2) + n) + a*b*d*(S(1) + n)*(S(2)*c*f*h*(S(1) + m) - d*(f*g + e*h)*(S(3) + m + n)) + b**S(2)*(c**S(2)*f*h*(S(1) + m)*(S(2) + m) - c*d*(f*g + e*h)*(S(1) + m)*(S(3) + m + n) + d**S(2)*e*g*(S(2) + m + n)*(S(3) + m + 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(3)*d**S(2)*(S(1) + m)*(S(2) + m + n)*(S(3) + m + n)*(b*(c + d*x)/(b*c - a*d))**n)],
[(a + b*x)**m*(A + B*x)*(c + d*x)**n*(e + f*x)**p, x, S(7), (A*b - a*B)*(a + b*x)**(S(1) + m)*(c + d*x)**n*(e + f*x)**p*AppellF1(S(1) + m, - n, - p, S(2) + m, - d*(a + b*x)/(b*c - a*d), - f*(a + b*x)/(b*e - a*f))/(b**S(2)*(S(1) + m)*(b*(c + d*x)/(b*c - a*d))**n*(b*(e + f*x)/(b*e - a*f))**p) + B*(a + b*x)**(S(2) + m)*(c + d*x)**n*(e + f*x)**p*AppellF1(S(2) + m, - n, - p, S(3) + m, - d*(a + b*x)/(b*c - a*d), - f*(a + b*x)/(b*e - a*f))/(b**S(2)*(S(2) + m)*(b*(c + d*x)/(b*c - a*d))**n*(b*(e + f*x)/(b*e - a*f))**p)],
[(A + B*x)*(c + d*x)**n*(e + f*x)**p/(a + b*x), x, S(5), - (A*b - a*B)*(c + d*x)**(S(1) + n)*(e + f*x)**p*AppellF1(S(1) + n, S(1), - p, S(2) + n, b*(c + d*x)/(b*c - a*d), - f*(c + d*x)/(d*e - c*f))/(b*(b*c - a*d)*(S(1) + n)*(d*(e + f*x)/(d*e - c*f))**p) - B*(c + d*x)**(S(1) + n)*(e + f*x)**(S(1) + p)*hypergeom([S(1), S(2) + n + p], [S(2) + p], d*(e + f*x)/(d*e - c*f))/(b*(d*e - c*f)*(S(1) + p)), - (A*b - a*B)*(c + d*x)**(S(1) + n)*(e + f*x)**p*AppellF1(S(1) + n, - p, S(1), S(2) + n, - f*(c + d*x)/(d*e - c*f), b*(c + d*x)/(b*c - a*d))/(b*(b*c - a*d)*(S(1) + n)*(d*(e + f*x)/(d*e - c*f))**p) + B*(c + d*x)**(S(1) + n)*(e + f*x)**p*hypergeom([S(1) + n, - p], [S(2) + n], - f*(c + d*x)/(d*e - c*f))/(b*d*(S(1) + n)*(d*(e + f*x)/(d*e - c*f))**p)],
[(c*i + d*i*x)/(sqrt(c + d*x)*sqrt(e + f*x)*sqrt(g + h*x)), x, S(3), S(2)*i*EllipticE(sqrt(h)*sqrt(e + f*x)/sqrt( - f*g + e*h), sqrt( - d*(f*g - e*h)/((d*e - c*f)*h)))*sqrt( - f*g + e*h)*sqrt(c + d*x)*sqrt(f*(g + h*x)/(f*g - e*h))/(f*sqrt(h)*sqrt( - f*(c + d*x)/(d*e - c*f))*sqrt(g + h*x))],
[(a + b*x)**m*(c + d*x)**n*(e + f*x)**p, x, S(3), (a + b*x)**(S(1) + m)*(c + d*x)**n*(e + f*x)**p*AppellF1(S(1) + m, - n, - p, S(2) + m, - d*(a + b*x)/(b*c - a*d), - f*(a + b*x)/(b*e - a*f))/(b*(S(1) + m)*(b*(c + d*x)/(b*c - a*d))**n*(b*(e + f*x)/(b*e - a*f))**p)],
[(a + b*x)**m*(c + d*x)**n*(e + f*x)**p/(g + h*x), x, S(0), Integrate((a + b*x)**m*(c + d*x)**n*(e + f*x)**p/(g + h*x), x)]
[x**S(3)*(S(1) + a*x)/(sqrt(a*x)*sqrt(S(1) - a*x)), x, S(8), - S(75)/S(128)*arcsin(S(1) - S(2)*a*x)/a**S(4) - S(25)/S(32)*(a*x)**(S(3)/S(2))*sqrt(S(1) - a*x)/a**S(4) - S(5)/S(8)*(a*x)**(S(5)/S(2))*sqrt(S(1) - a*x)/a**S(4) - S(1)/S(4)*(a*x)**(S(7)/S(2))*sqrt(S(1) - a*x)/a**S(4) - S(75)/S(64)*sqrt(a*x)*sqrt(S(1) - a*x)/a**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_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_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 = [
[x**m*(e + f*x)**n/((a + b*x)*(c + d*x)), x, S(6), b*x**(S(1) + m)*(e + f*x)**n*AppellF1(S(1) + m, - n, S(1), S(2) + m, - f*x/e, - b*x/a)/(a*(b*c - a*d)*(S(1) + m)*(S(1) + f*x/e)**n) - d*x**(S(1) + m)*(e + f*x)**n*AppellF1(S(1) + m, - n, S(1), S(2) + m, - f*x/e, - d*x/c)/(c*(b*c - a*d)*(S(1) + m)*(S(1) + f*x/e)**n)],
[(a + b*x)**m*(c + d*x)**n*(e + f*x)*(g + h*x), x, S(3), - (a + b*x)**(S(1) + m)*(c + d*x)**(S(1) + n)*(b*c*f*h*(S(2) + m) + a*d*f*h*(S(2) + n) - b*d*(f*g + e*h)*(S(3) + m + n) - b*d*f*h*(S(2) + m + n)*x)/(b**S(2)*d**S(2)*(S(2) + m + n)*(S(3) + m + n)) + (a**S(2)*d**S(2)*f*h*(S(1) + n)*(S(2) + n) + a*b*d*(S(1) + n)*(S(2)*c*f*h*(S(1) + m) - d*(f*g + e*h)*(S(3) + m + n)) + b**S(2)*(c**S(2)*f*h*(S(1) + m)*(S(2) + m) - c*d*(f*g + e*h)*(S(1) + m)*(S(3) + m + n) + d**S(2)*e*g*(S(2) + m + n)*(S(3) + m + 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(3)*d**S(2)*(S(1) + m)*(S(2) + m + n)*(S(3) + m + n)*(b*(c + d*x)/(b*c - a*d))**n)],
[(a + b*x)**m*(A + B*x)*(c + d*x)**n*(e + f*x)**p, x, S(7), (A*b - a*B)*(a + b*x)**(S(1) + m)*(c + d*x)**n*(e + f*x)**p*AppellF1(S(1) + m, - n, - p, S(2) + m, - d*(a + b*x)/(b*c - a*d), - f*(a + b*x)/(b*e - a*f))/(b**S(2)*(S(1) + m)*(b*(c + d*x)/(b*c - a*d))**n*(b*(e + f*x)/(b*e - a*f))**p) + B*(a + b*x)**(S(2) + m)*(c + d*x)**n*(e + f*x)**p*AppellF1(S(2) + m, - n, - p, S(3) + m, - d*(a + b*x)/(b*c - a*d), - f*(a + b*x)/(b*e - a*f))/(b**S(2)*(S(2) + m)*(b*(c + d*x)/(b*c - a*d))**n*(b*(e + f*x)/(b*e - a*f))**p)],
[(A + B*x)*(c + d*x)**n*(e + f*x)**p/(a + b*x), x, S(5), - (A*b - a*B)*(c + d*x)**(S(1) + n)*(e + f*x)**p*AppellF1(S(1) + n, S(1), - p, S(2) + n, b*(c + d*x)/(b*c - a*d), - f*(c + d*x)/(d*e - c*f))/(b*(b*c - a*d)*(S(1) + n)*(d*(e + f*x)/(d*e - c*f))**p) - B*(c + d*x)**(S(1) + n)*(e + f*x)**(S(1) + p)*hypergeom([S(1), S(2) + n + p], [S(2) + p], d*(e + f*x)/(d*e - c*f))/(b*(d*e - c*f)*(S(1) + p)), - (A*b - a*B)*(c + d*x)**(S(1) + n)*(e + f*x)**p*AppellF1(S(1) + n, - p, S(1), S(2) + n, - f*(c + d*x)/(d*e - c*f), b*(c + d*x)/(b*c - a*d))/(b*(b*c - a*d)*(S(1) + n)*(d*(e + f*x)/(d*e - c*f))**p) + B*(c + d*x)**(S(1) + n)*(e + f*x)**p*hypergeom([S(1) + n, - p], [S(2) + n], - f*(c + d*x)/(d*e - c*f))/(b*d*(S(1) + n)*(d*(e + f*x)/(d*e - c*f))**p)],
[(c*i + d*i*x)/(sqrt(c + d*x)*sqrt(e + f*x)*sqrt(g + h*x)), x, S(3), S(2)*i*EllipticE(sqrt(h)*sqrt(e + f*x)/sqrt( - f*g + e*h), sqrt( - d*(f*g - e*h)/((d*e - c*f)*h)))*sqrt( - f*g + e*h)*sqrt(c + d*x)*sqrt(f*(g + h*x)/(f*g - e*h))/(f*sqrt(h)*sqrt( - f*(c + d*x)/(d*e - c*f))*sqrt(g + h*x))],
[(a + b*x)**m*(c + d*x)**n*(e + f*x)**p, x, S(3), (a + b*x)**(S(1) + m)*(c + d*x)**n*(e + f*x)**p*AppellF1(S(1) + m, - n, - p, S(2) + m, - d*(a + b*x)/(b*c - a*d), - f*(a + b*x)/(b*e - a*f))/(b*(S(1) + m)*(b*(c + d*x)/(b*c - a*d))**n*(b*(e + f*x)/(b*e - a*f))**p)],
[(a + b*x)**m*(c + d*x)**n*(e + f*x)**p/(g + h*x), x, S(0), Integrate((a + b*x)**m*(c + d*x)**n*(e + f*x)**p/(g + h*x), x)]
[x**S(3)*(S(1) + a*x)/(sqrt(a*x)*sqrt(S(1) - a*x)), x, S(8), - S(75)/S(128)*arcsin(S(1) - S(2)*a*x)/a**S(4) - S(25)/S(32)*(a*x)**(S(3)/S(2))*sqrt(S(1) - a*x)/a**S(4) - S(5)/S(8)*(a*x)**(S(5)/S(2))*sqrt(S(1) - a*x)/a**S(4) - S(1)/S(4)*(a*x)**(S(7)/S(2))*sqrt(S(1) - a*x)/a**S(4) - S(75)/S(64)*sqrt(a*x)*sqrt(S(1) - a*x)/a**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)