Optimal. Leaf size=95 \[ \frac{d \, _2F_1\left (1,-\frac{2}{n};-\frac{2-n}{n};-\frac{d x^n}{c}\right )}{2 c x^2 (b c-a d)}-\frac{b \, _2F_1\left (1,-\frac{2}{n};-\frac{2-n}{n};-\frac{b x^n}{a}\right )}{2 a x^2 (b c-a d)} \]
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Rubi [A] time = 0.0428551, antiderivative size = 95, 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} \[ \frac{d \, _2F_1\left (1,-\frac{2}{n};-\frac{2-n}{n};-\frac{d x^n}{c}\right )}{2 c x^2 (b c-a d)}-\frac{b \, _2F_1\left (1,-\frac{2}{n};-\frac{2-n}{n};-\frac{b x^n}{a}\right )}{2 a x^2 (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 508
Rule 364
Rubi steps
\begin{align*} \int \frac{1}{x^3 \left (a+b x^n\right ) \left (c+d x^n\right )} \, dx &=\frac{b \int \frac{1}{x^3 \left (a+b x^n\right )} \, dx}{b c-a d}-\frac{d \int \frac{1}{x^3 \left (c+d x^n\right )} \, dx}{b c-a d}\\ &=-\frac{b \, _2F_1\left (1,-\frac{2}{n};-\frac{2-n}{n};-\frac{b x^n}{a}\right )}{2 a (b c-a d) x^2}+\frac{d \, _2F_1\left (1,-\frac{2}{n};-\frac{2-n}{n};-\frac{d x^n}{c}\right )}{2 c (b c-a d) x^2}\\ \end{align*}
Mathematica [A] time = 0.0519527, size = 77, normalized size = 0.81 \[ \frac{b c \, _2F_1\left (1,-\frac{2}{n};\frac{n-2}{n};-\frac{b x^n}{a}\right )-a d \, _2F_1\left (1,-\frac{2}{n};\frac{n-2}{n};-\frac{d x^n}{c}\right )}{2 a c x^2 (a d-b c)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.079, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{3} \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^{3}}\,{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^{3} x^{2 \, n} +{\left (b c + a d\right )} x^{3} x^{n} + a c x^{3}}, 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^{3} \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^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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