Optimal. Leaf size=113 \[ x \left (a-b x^n\right )^p \left (a+b x^n\right )^p \left (1-\frac{b^2 x^{2 n}}{a^2}\right )^{-p} \left (c+d x^{2 n}\right )^q \left (\frac{d x^{2 n}}{c}+1\right )^{-q} F_1\left (\frac{1}{2 n};-p,-q;\frac{1}{2} \left (2+\frac{1}{n}\right );\frac{b^2 x^{2 n}}{a^2},-\frac{d x^{2 n}}{c}\right ) \]
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Rubi [A] time = 0.0904555, antiderivative size = 113, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.097, Rules used = {519, 430, 429} \[ x \left (a-b x^n\right )^p \left (a+b x^n\right )^p \left (1-\frac{b^2 x^{2 n}}{a^2}\right )^{-p} \left (c+d x^{2 n}\right )^q \left (\frac{d x^{2 n}}{c}+1\right )^{-q} F_1\left (\frac{1}{2 n};-p,-q;\frac{1}{2} \left (2+\frac{1}{n}\right );\frac{b^2 x^{2 n}}{a^2},-\frac{d x^{2 n}}{c}\right ) \]
Antiderivative was successfully verified.
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Rule 519
Rule 430
Rule 429
Rubi steps
\begin{align*} \int \left (a-b x^n\right )^p \left (a+b x^n\right )^p \left (c+d x^{2 n}\right )^q \, dx &=\left (\left (a-b x^n\right )^p \left (a+b x^n\right )^p \left (a^2-b^2 x^{2 n}\right )^{-p}\right ) \int \left (a^2-b^2 x^{2 n}\right )^p \left (c+d x^{2 n}\right )^q \, dx\\ &=\left (\left (a-b x^n\right )^p \left (a+b x^n\right )^p \left (1-\frac{b^2 x^{2 n}}{a^2}\right )^{-p}\right ) \int \left (1-\frac{b^2 x^{2 n}}{a^2}\right )^p \left (c+d x^{2 n}\right )^q \, dx\\ &=\left (\left (a-b x^n\right )^p \left (a+b x^n\right )^p \left (1-\frac{b^2 x^{2 n}}{a^2}\right )^{-p} \left (c+d x^{2 n}\right )^q \left (1+\frac{d x^{2 n}}{c}\right )^{-q}\right ) \int \left (1-\frac{b^2 x^{2 n}}{a^2}\right )^p \left (1+\frac{d x^{2 n}}{c}\right )^q \, dx\\ &=x \left (a-b x^n\right )^p \left (a+b x^n\right )^p \left (1-\frac{b^2 x^{2 n}}{a^2}\right )^{-p} \left (c+d x^{2 n}\right )^q \left (1+\frac{d x^{2 n}}{c}\right )^{-q} F_1\left (\frac{1}{2 n};-p,-q;\frac{1}{2} \left (2+\frac{1}{n}\right );\frac{b^2 x^{2 n}}{a^2},-\frac{d x^{2 n}}{c}\right )\\ \end{align*}
Mathematica [F] time = 0.238882, size = 0, normalized size = 0. \[ \int \left (a-b x^n\right )^p \left (a+b x^n\right )^p \left (c+d x^{2 n}\right )^q \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 1.492, size = 0, normalized size = 0. \begin{align*} \int \left ( a-b{x}^{n} \right ) ^{p} \left ( a+b{x}^{n} \right ) ^{p} \left ( c+d{x}^{2\,n} \right ) ^{q}\, 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{\left (d x^{2 \, n} + c\right )}^{q}{\left (b x^{n} + a\right )}^{p}{\left (-b x^{n} + a\right )}^{p}\,{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 ({\left (d x^{2 \, n} + c\right )}^{q}{\left (b x^{n} + a\right )}^{p}{\left (-b x^{n} + a\right )}^{p}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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{\left (d x^{2 \, n} + c\right )}^{q}{\left (b x^{n} + a\right )}^{p}{\left (-b x^{n} + a\right )}^{p}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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