Optimal. Leaf size=90 \[ \frac{d \, _2F_1\left (1,-\frac{1}{n};-\frac{1-n}{n};-\frac{d x^n}{c}\right )}{c x (b c-a d)}-\frac{b \, _2F_1\left (1,-\frac{1}{n};-\frac{1-n}{n};-\frac{b x^n}{a}\right )}{a x (b c-a d)} \]
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Rubi [A] time = 0.0430339, antiderivative size = 90, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {508, 364} \[ \frac{d \, _2F_1\left (1,-\frac{1}{n};-\frac{1-n}{n};-\frac{d x^n}{c}\right )}{c x (b c-a d)}-\frac{b \, _2F_1\left (1,-\frac{1}{n};-\frac{1-n}{n};-\frac{b x^n}{a}\right )}{a x (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 508
Rule 364
Rubi steps
\begin{align*} \int \frac{1}{x^2 \left (a+b x^n\right ) \left (c+d x^n\right )} \, dx &=\frac{b \int \frac{1}{x^2 \left (a+b x^n\right )} \, dx}{b c-a d}-\frac{d \int \frac{1}{x^2 \left (c+d x^n\right )} \, dx}{b c-a d}\\ &=-\frac{b \, _2F_1\left (1,-\frac{1}{n};-\frac{1-n}{n};-\frac{b x^n}{a}\right )}{a (b c-a d) x}+\frac{d \, _2F_1\left (1,-\frac{1}{n};-\frac{1-n}{n};-\frac{d x^n}{c}\right )}{c (b c-a d) x}\\ \end{align*}
Mathematica [A] time = 0.0530143, size = 74, normalized size = 0.82 \[ \frac{b c \, _2F_1\left (1,-\frac{1}{n};\frac{n-1}{n};-\frac{b x^n}{a}\right )-a d \, _2F_1\left (1,-\frac{1}{n};\frac{n-1}{n};-\frac{d x^n}{c}\right )}{a c x (a d-b c)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.073, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2} \left ( a+b{x}^{n} \right ) \left ( c+d{x}^{n} \right ) }}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x^{n} + a\right )}{\left (d x^{n} + c\right )} x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{1}{b d x^{2} x^{2 \, n} +{\left (b c + a d\right )} x^{2} x^{n} + a c x^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x^{2} \left (a + b x^{n}\right ) \left (c + d x^{n}\right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x^{n} + a\right )}{\left (d x^{n} + c\right )} x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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