def replacement1885(v, x, u):
a = D(u, x)
b = D(v, x)
rubi.append(1885)
return -Dist(a / (-a * v + b * u), Int(S(1) / u, x), x) + Dist(
b / (-a * v + b * u), Int(S(1) / v, x), x)
def With1899(u, b, a, n, x):
c = D(u, x)
rubi.append(1899)
return -Dist(c * n / b,
Int(u**(n + S(-1)) *
(a + b * x) * log(a + b * x), x), x) - Int(
u**n, x) + Simp(
u**n * (a + b * x) * log(a + b * x) / b, x)
def With1903(a, b, m, n, u, x):
c = D(u, x)
return -Dist(
c * n / (b * (m + S(1))),
Int(u**(n + S(-1)) *
(a + b * x)**(m + S(1)) * log(a + b * x), x), x) - Dist(
S(1) / (m + S(1)), Int(u**n * (a + b * x)**m, x), x) + Simp(
u**n * (a + b * x)**(m + S(1)) * log(a + b * x) /
(b * (m + S(1))), x)
def With1900(u, m, b, a, n, x):
c = D(u, x)
rubi.append(1900)
return -Dist(
c * n / (b * (m + S(1))),
Int(u**(n + S(-1)) *
(a + b * x)**(m + S(1)) * log(a + b * x), x), x) - Dist(
S(1) /
(m + S(1)), Int(u**n * (a + b * x)**m, x), x) + Simp(
u**n * (a + b * x)**(m + S(1)) * log(a + b * x) /
(b * (m + S(1))), x)
def replacement1883(v, x, u):
a = D(u, x)
b = D(v, x)
rubi.append(1883)
return -Dist(
(-a * v + b * u) / a, Int(S(1) / u, x), x) + Simp(b * x / a, x)
def replacement36(p, u, Pq, Qr, x):
gcd = PolyGCD(Pq, Qr, x)
rubi.append(36)
return Int(
gcd**(p + S(1)) * u * PolynomialQuotient(Pq, gcd, x) *
PolynomialQuotient(Qr, gcd, x)**p, x)
def replacement35(p, u, m, Pq, Qr, x):
gcd = PolyGCD(Pq, Qr, x)
rubi.append(35)
return Int(
gcd**(m + p) * u * PolynomialQuotient(Pq, gcd, x)**m *
PolynomialQuotient(Qr, gcd, x)**p, x)
def replacement34(p, Pm, b, n2, c, a, n, Qm, x):
m = Expon(Pm, x)
rubi.append(34)
return Dist(
Coeff(Qm, x, m + S(-1)) / (m * Coeff(Pm, x, m)),
Subst(Int((a + b * x**n + c * x**(S(2) * n))**p, x), x, Pm), x)
def replacement33(p, Pm, b, a, n, Qm, x):
m = Expon(Pm, x)
rubi.append(33)
return Dist(
Coeff(Qm, x, m + S(-1)) / (m * Coeff(Pm, x, m)),
Subst(Int((a + b * x**n)**p, x), x, Pm), x)
def replacement22(B, v, C, u, m, b, a, x, A):
rubi.append(22)
return Dist(
b**(S(-2)),
Int(
u * (a + b * v)**(m + S(1)) *
Simp(B * b - C * a + C * b * v, x), x), x)
def replacement1896(m, n, u, v, x):
a = D(u, x)
b = D(v, x)
return -Dist(b * n / (a *
(m + S(1))), Int(u**(m + S(1)) * v**(n + S(-1)), x),
x) + Simp(u**(m + S(1)) * v**n / (a * (m + S(1))), x)
def replacement19(a, b, m, n, u, v, x):
return Dist(
a**(-IntPart(n)) * b**IntPart(n) * (a * v)**(-FracPart(n)) *
(b * v)**FracPart(n),
Int(u * (a * v)**(m + n), x),
x,
)
def replacement24(A, B, C, a, b, m, u, v, x):
return Dist(
b**(S(-2)),
Int(u * (a + b * v)**(m + S(1)) * Simp(B * b - C * a + C * b * v, x),
x),
x,
)
def replacement37(Pq, Qr, m, p, u, x):
gcd = PolyGCD(Pq, Qr, x)
return Int(
gcd**(m + p) * u * PolynomialQuotient(Pq, gcd, x)**m *
PolynomialQuotient(Qr, gcd, x)**p,
x,
)
def replacement36(Pm, Qm, a, b, c, n, n2, p, x):
m = Expon(Pm, x)
return Dist(
Coeff(Qm, x, m + S(-1)) / (m * Coeff(Pm, x, m)),
Subst(Int((a + b * x**n + c * x**(S(2) * n))**p, x), x, Pm),
x,
)
def replacement35(Pm, Qm, a, b, n, p, x):
m = Expon(Pm, x)
return Dist(
Coeff(Qm, x, m + S(-1)) / (m * Coeff(Pm, x, m)),
Subst(Int((a + b * x**n)**p, x), x, Pm),
x,
)
def replacement1884(v, x, n, u):
a = D(u, x)
b = D(v, x)
rubi.append(1884)
return -Dist(
(-a * v + b * u) / a, Int(v**(n + S(-1)) / u, x), x) + Simp(
v**n / (a * n), x)
def replacement1898(m, n, u, v, x):
a = D(u, x)
b = D(v, x)
return -Dist(n * (-a * v + b * u) /
(a *
(m + n + S(1))), Int(u**m * v**(n + S(-1)), x), x) + Simp(
u**(m + S(1)) * v**n / (a * (m + n + S(1))), x)
def replacement1895(v, u, m, n, x):
a = D(u, x)
b = D(v, x)
rubi.append(1895)
return -Dist(n * (-a * v + b * u) /
(a * (m + n + S(1))), Int(u**m * v**(n + S(-1)), x),
x) + Simp(u**(m + S(1)) * v**n / (a * (m + n + S(1))), x)
def replacement1893(v, u, m, n, x):
a = D(u, x)
b = D(v, x)
rubi.append(1893)
return -Dist(b * n /
(a * (m + S(1))), Int(u**(m + S(1)) * v**(n + S(-1)), x),
x) + Simp(u**(m + S(1)) * v**n / (a * (m + S(1))), x)
def replacement1888(v, x, n, u):
a = D(u, x)
b = D(v, x)
rubi.append(1888)
return -Dist(a /
(-a * v + b * u), Int(v**(n + S(1)) / u, x), x) + Simp(
v**(n + S(1)) / ((n + S(1)) * (-a * v + b * u)), x)
def replacement38(Pq, Qr, p, u, x):
gcd = PolyGCD(Pq, Qr, x)
return Int(
gcd**(p + S(1)) * u * PolynomialQuotient(Pq, gcd, x) *
PolynomialQuotient(Qr, gcd, x)**p,
x,
)
def replacement1900(m, n, u, v, x):
a = D(u, x)
b = D(v, x)
return Dist(b * (m + n + S(2)) /
((m + S(1)) *
(-a * v + b * u)), Int(u**(m + S(1)) * v**n, x), x) - Simp(
u**(m + S(1)) * v**(n + S(1)) / ((m + S(1)) *
(-a * v + b * u)), x)
def replacement1897(v, u, m, n, x):
a = D(u, x)
b = D(v, x)
rubi.append(1897)
return Dist(b * (m + n + S(2)) / (
(m + S(1)) *
(-a * v + b * u)), Int(u**(m + S(1)) * v**n, x), x) - Simp(
u**(m + S(1)) * v**(n + S(1)) / ((m + S(1)) *
(-a * v + b * u)), x)
def With19(x, n, b, u, a, m):
c = D(u, x)
return -Int(u**n * (a + b * x)**m, x) / (m + S(1)) - c * n * Int(
u**(n + S(-1)) * (a + b * x)**
(m + S(1)) * log(a + b * x), x) / (b * (m + S(1))) + u**n * (
a + b * x)**(m + S(1)) * log(a + b * x) / (b * (m + S(1)))
def With18(x, n, b, u, a):
c = D(u, x)
return -Int(u**n, x) - c * n * Int(
u**(n + S(-1)) * (a + b * x) * log(a + b * x),
x) / b + u**n * (a + b * x) * log(a + b * x) / b
def With16(x, n, v, u, m):
a = D(u, x)
b = D(v, x)
return b * (m + n + S(2)) * Int(u**(m + S(1)) * v**n, x) / (
(m + S(1)) * (-a * v + b * u)) - u**(m + S(1)) * v**(n + S(1)) / (
(m + S(1)) * (-a * v + b * u))
def With14(x, n, v, u, m):
a = D(u, x)
b = D(v, x)
return -n * (-a * v + b * u) * Int(u**m * v**(n + S(-1)), x) / (
a * (m + n + S(1))) + u**(m + S(1)) * v**n / (a * (m + n + S(1)))
def With12(x, n, v, u, m):
a = D(u, x)
b = D(v, x)
return -b * n * Int(u**(m + S(1)) * v**(n + S(-1)), x) / (
a * (m + S(1))) + u**(m + S(1)) * v**n / (a * (m + S(1)))
def With7(x, n, v, u):
a = D(u, x)
b = D(v, x)
return -a * Int(v**(n + S(1)) / u, x) / (-a * v + b * u) + v**(
n + S(1)) / ((n + S(1)) * (-a * v + b * u))