Optimal. Leaf size=89 \[ \frac{b x^3 \, _2F_1\left (1,\frac{3}{n};\frac{n+3}{n};-\frac{b x^n}{a}\right )}{3 a (b c-a d)}-\frac{d x^3 \, _2F_1\left (1,\frac{3}{n};\frac{n+3}{n};-\frac{d x^n}{c}\right )}{3 c (b c-a d)} \]
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Rubi [A] time = 0.0418385, antiderivative size = 89, 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} \[ \frac{b x^3 \, _2F_1\left (1,\frac{3}{n};\frac{n+3}{n};-\frac{b x^n}{a}\right )}{3 a (b c-a d)}-\frac{d x^3 \, _2F_1\left (1,\frac{3}{n};\frac{n+3}{n};-\frac{d x^n}{c}\right )}{3 c (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 508
Rule 364
Rubi steps
\begin{align*} \int \frac{x^2}{\left (a+b x^n\right ) \left (c+d x^n\right )} \, dx &=\frac{b \int \frac{x^2}{a+b x^n} \, dx}{b c-a d}-\frac{d \int \frac{x^2}{c+d x^n} \, dx}{b c-a d}\\ &=\frac{b x^3 \, _2F_1\left (1,\frac{3}{n};\frac{3+n}{n};-\frac{b x^n}{a}\right )}{3 a (b c-a d)}-\frac{d x^3 \, _2F_1\left (1,\frac{3}{n};\frac{3+n}{n};-\frac{d x^n}{c}\right )}{3 c (b c-a d)}\\ \end{align*}
Mathematica [A] time = 0.0538311, size = 78, normalized size = 0.88 \[ \frac{b c x^3 \, _2F_1\left (1,\frac{3}{n};\frac{n+3}{n};-\frac{b x^n}{a}\right )-a d x^3 \, _2F_1\left (1,\frac{3}{n};\frac{n+3}{n};-\frac{d x^n}{c}\right )}{3 a b c^2-3 a^2 c d} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.076, size = 0, normalized size = 0. \begin{align*} \int{\frac{{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{x^{2}}{{\left (b x^{n} + a\right )}{\left (d x^{n} + c\right )}}\,{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{x^{2}}{b d x^{2 \, n} + a c +{\left (b c + a d\right )} x^{n}}, 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{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{x^{2}}{{\left (b x^{n} + a\right )}{\left (d x^{n} + c\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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