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

Optimal. Leaf size=72 \[ \frac{b x \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )}{a (b c-a d)}-\frac{d x \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{d x^n}{c}\right )}{c (b c-a d)} \]

[Out]

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

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Rubi [A]  time = 0.0794435, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{b x \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )}{a (b c-a d)}-\frac{d x \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{d x^n}{c}\right )}{c (b c-a d)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 10.9658, size = 53, normalized size = 0.74 \[ \frac{d x{{}_{2}F_{1}\left (\begin{matrix} 1, \frac{1}{n} \\ 1 + \frac{1}{n} \end{matrix}\middle |{- \frac{d x^{n}}{c}} \right )}}{c \left (a d - b c\right )} - \frac{b x{{}_{2}F_{1}\left (\begin{matrix} 1, \frac{1}{n} \\ 1 + \frac{1}{n} \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )}}{a \left (a d - b c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0545357, size = 64, normalized size = 0.89 \[ \frac{x \left (a d \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{d x^n}{c}\right )-b c \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )\right )}{a c (a d-b c)} \]

Antiderivative was successfully verified.

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

[Out]

(x*(-(b*c*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), -((b*x^n)/a)]) + a*d*Hypergeo
metric2F1[1, n^(-1), 1 + n^(-1), -((d*x^n)/c)]))/(a*c*(-(b*c) + a*d))

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Maple [F]  time = 0.091, size = 0, normalized size = 0. \[ \int{\frac{1}{ \left ( a+b{x}^{n} \right ) \left ( c+d{x}^{n} \right ) }}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b*x^n)/(c+d*x^n),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/((a + b*x**n)*(c + d*x**n)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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