3.204 \(\int \frac{1}{\left (a+b x^n\right ) \left (c+d x^n\right )^2} \, dx\)

Optimal. Leaf size=123 \[ \frac{b^2 x \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )}{a (b c-a d)^2}+\frac{d x (b c (1-2 n)-a d (1-n)) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{d x^n}{c}\right )}{c^2 n (b c-a d)^2}-\frac{d x}{c n (b c-a d) \left (c+d x^n\right )} \]

[Out]

-((d*x)/(c*(b*c - a*d)*n*(c + d*x^n))) + (b^2*x*Hypergeometric2F1[1, n^(-1), 1 +
n^(-1), -((b*x^n)/a)])/(a*(b*c - a*d)^2) + (d*(b*c*(1 - 2*n) - a*d*(1 - n))*x*H
ypergeometric2F1[1, n^(-1), 1 + n^(-1), -((d*x^n)/c)])/(c^2*(b*c - a*d)^2*n)

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Rubi [A]  time = 0.359577, antiderivative size = 123, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ \frac{b^2 x \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )}{a (b c-a d)^2}-\frac{d x (a d (1-n)-b (c-2 c n)) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{d x^n}{c}\right )}{c^2 n (b c-a d)^2}-\frac{d x}{c n (b c-a d) \left (c+d x^n\right )} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b*x^n)*(c + d*x^n)^2),x]

[Out]

-((d*x)/(c*(b*c - a*d)*n*(c + d*x^n))) + (b^2*x*Hypergeometric2F1[1, n^(-1), 1 +
n^(-1), -((b*x^n)/a)])/(a*(b*c - a*d)^2) - (d*(a*d*(1 - n) - b*(c - 2*c*n))*x*H
ypergeometric2F1[1, n^(-1), 1 + n^(-1), -((d*x^n)/c)])/(c^2*(b*c - a*d)^2*n)

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Rubi in Sympy [A]  time = 48.4865, size = 100, normalized size = 0.81 \[ \frac{d x}{c n \left (c + d x^{n}\right ) \left (a d - b c\right )} - \frac{d x \left (- a d n + a d + 2 b c n - b c\right ){{}_{2}F_{1}\left (\begin{matrix} 1, \frac{1}{n} \\ 1 + \frac{1}{n} \end{matrix}\middle |{- \frac{d x^{n}}{c}} \right )}}{c^{2} n \left (a d - b c\right )^{2}} + \frac{b^{2} x{{}_{2}F_{1}\left (\begin{matrix} 1, \frac{1}{n} \\ 1 + \frac{1}{n} \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )}}{a \left (a d - b c\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a+b*x**n)/(c+d*x**n)**2,x)

[Out]

d*x/(c*n*(c + d*x**n)*(a*d - b*c)) - d*x*(-a*d*n + a*d + 2*b*c*n - b*c)*hyper((1
, 1/n), (1 + 1/n,), -d*x**n/c)/(c**2*n*(a*d - b*c)**2) + b**2*x*hyper((1, 1/n),
(1 + 1/n,), -b*x**n/a)/(a*(a*d - b*c)**2)

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Mathematica [A]  time = 0.217176, size = 121, normalized size = 0.98 \[ \frac{x \left (b^2 c^2 n \left (c+d x^n\right ) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )+a d \left (\left (c+d x^n\right ) (a d (n-1)+b (c-2 c n)) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{d x^n}{c}\right )+c (a d-b c)\right )\right )}{a c^2 n (b c-a d)^2 \left (c+d x^n\right )} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b*x^n)*(c + d*x^n)^2),x]

[Out]

(x*(b^2*c^2*n*(c + d*x^n)*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), -((b*x^n)/a)]
+ a*d*(c*(-(b*c) + a*d) + (a*d*(-1 + n) + b*(c - 2*c*n))*(c + d*x^n)*Hypergeome
tric2F1[1, n^(-1), 1 + n^(-1), -((d*x^n)/c)])))/(a*c^2*(b*c - a*d)^2*n*(c + d*x^
n))

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Maple [F]  time = 0.165, size = 0, normalized size = 0. \[ \int{\frac{1}{ \left ( a+b{x}^{n} \right ) \left ( c+d{x}^{n} \right ) ^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b*x^n)/(c+d*x^n)^2,x)

[Out]

int(1/(a+b*x^n)/(c+d*x^n)^2,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ b^{2} \int \frac{1}{a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2} +{\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} x^{n}}\,{d x} -{\left (b c d{\left (2 \, n - 1\right )} - a d^{2}{\left (n - 1\right )}\right )} \int \frac{1}{b^{2} c^{4} n - 2 \, a b c^{3} d n + a^{2} c^{2} d^{2} n +{\left (b^{2} c^{3} d n - 2 \, a b c^{2} d^{2} n + a^{2} c d^{3} n\right )} x^{n}}\,{d x} - \frac{d x}{b c^{3} n - a c^{2} d n +{\left (b c^{2} d n - a c d^{2} n\right )} x^{n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^n + a)*(d*x^n + c)^2),x, algorithm="maxima")

[Out]

b^2*integrate(1/(a*b^2*c^2 - 2*a^2*b*c*d + a^3*d^2 + (b^3*c^2 - 2*a*b^2*c*d + a^
2*b*d^2)*x^n), x) - (b*c*d*(2*n - 1) - a*d^2*(n - 1))*integrate(1/(b^2*c^4*n - 2
*a*b*c^3*d*n + a^2*c^2*d^2*n + (b^2*c^3*d*n - 2*a*b*c^2*d^2*n + a^2*c*d^3*n)*x^n
), x) - d*x/(b*c^3*n - a*c^2*d*n + (b*c^2*d*n - a*c*d^2*n)*x^n)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{b d^{2} x^{3 \, n} + a c^{2} +{\left (2 \, b c d + a d^{2}\right )} x^{2 \, n} +{\left (b c^{2} + 2 \, a c d\right )} x^{n}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^n + a)*(d*x^n + c)^2),x, algorithm="fricas")

[Out]

integral(1/(b*d^2*x^(3*n) + a*c^2 + (2*b*c*d + a*d^2)*x^(2*n) + (b*c^2 + 2*a*c*d
)*x^n), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a+b*x**n)/(c+d*x**n)**2,x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{n} + a\right )}{\left (d x^{n} + c\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^n + a)*(d*x^n + c)^2),x, algorithm="giac")

[Out]

integrate(1/((b*x^n + a)*(d*x^n + c)^2), x)