Optimal. Leaf size=167 \[ \frac{x \left (a+c x^{2 n}\right )^p \left (\frac{c x^{2 n}}{a}+1\right )^{-p} F_1\left (\frac{1}{2 n};-p,1;\frac{1}{2} \left (2+\frac{1}{n}\right );-\frac{c x^{2 n}}{a},\frac{e^2 x^{2 n}}{d^2}\right )}{d}-\frac{e x^{n+1} \left (a+c x^{2 n}\right )^p \left (\frac{c x^{2 n}}{a}+1\right )^{-p} F_1\left (\frac{n+1}{2 n};-p,1;\frac{1}{2} \left (3+\frac{1}{n}\right );-\frac{c x^{2 n}}{a},\frac{e^2 x^{2 n}}{d^2}\right )}{d^2 (n+1)} \]
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Rubi [A] time = 0.141518, antiderivative size = 167, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {1438, 430, 429, 511, 510} \[ \frac{x \left (a+c x^{2 n}\right )^p \left (\frac{c x^{2 n}}{a}+1\right )^{-p} F_1\left (\frac{1}{2 n};-p,1;\frac{1}{2} \left (2+\frac{1}{n}\right );-\frac{c x^{2 n}}{a},\frac{e^2 x^{2 n}}{d^2}\right )}{d}-\frac{e x^{n+1} \left (a+c x^{2 n}\right )^p \left (\frac{c x^{2 n}}{a}+1\right )^{-p} F_1\left (\frac{n+1}{2 n};-p,1;\frac{1}{2} \left (3+\frac{1}{n}\right );-\frac{c x^{2 n}}{a},\frac{e^2 x^{2 n}}{d^2}\right )}{d^2 (n+1)} \]
Antiderivative was successfully verified.
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Rule 1438
Rule 430
Rule 429
Rule 511
Rule 510
Rubi steps
\begin{align*} \int \frac{\left (a+c x^{2 n}\right )^p}{d+e x^n} \, dx &=\int \left (\frac{d \left (a+c x^{2 n}\right )^p}{d^2-e^2 x^{2 n}}+\frac{e x^n \left (a+c x^{2 n}\right )^p}{-d^2+e^2 x^{2 n}}\right ) \, dx\\ &=d \int \frac{\left (a+c x^{2 n}\right )^p}{d^2-e^2 x^{2 n}} \, dx+e \int \frac{x^n \left (a+c x^{2 n}\right )^p}{-d^2+e^2 x^{2 n}} \, dx\\ &=\left (d \left (a+c x^{2 n}\right )^p \left (1+\frac{c x^{2 n}}{a}\right )^{-p}\right ) \int \frac{\left (1+\frac{c x^{2 n}}{a}\right )^p}{d^2-e^2 x^{2 n}} \, dx+\left (e \left (a+c x^{2 n}\right )^p \left (1+\frac{c x^{2 n}}{a}\right )^{-p}\right ) \int \frac{x^n \left (1+\frac{c x^{2 n}}{a}\right )^p}{-d^2+e^2 x^{2 n}} \, dx\\ &=\frac{x \left (a+c x^{2 n}\right )^p \left (1+\frac{c x^{2 n}}{a}\right )^{-p} F_1\left (\frac{1}{2 n};-p,1;\frac{1}{2} \left (2+\frac{1}{n}\right );-\frac{c x^{2 n}}{a},\frac{e^2 x^{2 n}}{d^2}\right )}{d}-\frac{e x^{1+n} \left (a+c x^{2 n}\right )^p \left (1+\frac{c x^{2 n}}{a}\right )^{-p} F_1\left (\frac{1+n}{2 n};-p,1;\frac{1}{2} \left (3+\frac{1}{n}\right );-\frac{c x^{2 n}}{a},\frac{e^2 x^{2 n}}{d^2}\right )}{d^2 (1+n)}\\ \end{align*}
Mathematica [F] time = 0.080604, size = 0, normalized size = 0. \[ \int \frac{\left (a+c x^{2 n}\right )^p}{d+e x^n} \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 0.092, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+c{x}^{2\,n} \right ) ^{p}}{d+e{x}^{n}}}\, 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{{\left (c x^{2 \, n} + a\right )}^{p}}{e x^{n} + d}\,{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{{\left (c x^{2 \, n} + a\right )}^{p}}{e x^{n} + d}, 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 \frac{{\left (c x^{2 \, n} + a\right )}^{p}}{e x^{n} + d}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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