Optimal. Leaf size=357 \[ -\frac{3 e x^{n+1} \left (a+c x^{2 n}\right )^p \left (\frac{c x^{2 n}}{a}+1\right )^{-p} F_1\left (\frac{n+1}{2 n};-p,3;\frac{1}{2} \left (3+\frac{1}{n}\right );-\frac{c x^{2 n}}{a},\frac{e^2 x^{2 n}}{d^2}\right )}{d^4 (n+1)}+\frac{3 e^2 x^{2 n+1} \left (a+c x^{2 n}\right )^p \left (\frac{c x^{2 n}}{a}+1\right )^{-p} F_1\left (\frac{1}{2} \left (2+\frac{1}{n}\right );-p,3;\frac{1}{2} \left (4+\frac{1}{n}\right );-\frac{c x^{2 n}}{a},\frac{e^2 x^{2 n}}{d^2}\right )}{d^5 (2 n+1)}-\frac{e^3 x^{3 n+1} \left (a+c x^{2 n}\right )^p \left (\frac{c x^{2 n}}{a}+1\right )^{-p} F_1\left (\frac{1}{2} \left (3+\frac{1}{n}\right );-p,3;\frac{1}{2} \left (5+\frac{1}{n}\right );-\frac{c x^{2 n}}{a},\frac{e^2 x^{2 n}}{d^2}\right )}{d^6 (3 n+1)}+\frac{x \left (a+c x^{2 n}\right )^p \left (\frac{c x^{2 n}}{a}+1\right )^{-p} F_1\left (\frac{1}{2 n};-p,3;\frac{1}{2} \left (2+\frac{1}{n}\right );-\frac{c x^{2 n}}{a},\frac{e^2 x^{2 n}}{d^2}\right )}{d^3} \]
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Rubi [A] time = 0.338036, antiderivative size = 357, normalized size of antiderivative = 1., number of steps used = 10, 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} \[ -\frac{3 e x^{n+1} \left (a+c x^{2 n}\right )^p \left (\frac{c x^{2 n}}{a}+1\right )^{-p} F_1\left (\frac{n+1}{2 n};-p,3;\frac{1}{2} \left (3+\frac{1}{n}\right );-\frac{c x^{2 n}}{a},\frac{e^2 x^{2 n}}{d^2}\right )}{d^4 (n+1)}+\frac{3 e^2 x^{2 n+1} \left (a+c x^{2 n}\right )^p \left (\frac{c x^{2 n}}{a}+1\right )^{-p} F_1\left (\frac{1}{2} \left (2+\frac{1}{n}\right );-p,3;\frac{1}{2} \left (4+\frac{1}{n}\right );-\frac{c x^{2 n}}{a},\frac{e^2 x^{2 n}}{d^2}\right )}{d^5 (2 n+1)}-\frac{e^3 x^{3 n+1} \left (a+c x^{2 n}\right )^p \left (\frac{c x^{2 n}}{a}+1\right )^{-p} F_1\left (\frac{1}{2} \left (3+\frac{1}{n}\right );-p,3;\frac{1}{2} \left (5+\frac{1}{n}\right );-\frac{c x^{2 n}}{a},\frac{e^2 x^{2 n}}{d^2}\right )}{d^6 (3 n+1)}+\frac{x \left (a+c x^{2 n}\right )^p \left (\frac{c x^{2 n}}{a}+1\right )^{-p} F_1\left (\frac{1}{2 n};-p,3;\frac{1}{2} \left (2+\frac{1}{n}\right );-\frac{c x^{2 n}}{a},\frac{e^2 x^{2 n}}{d^2}\right )}{d^3} \]
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}{\left (d+e x^n\right )^3} \, dx &=\int \left (\frac{d^3 \left (a+c x^{2 n}\right )^p}{\left (d^2-e^2 x^{2 n}\right )^3}+\frac{3 d^2 e x^n \left (a+c x^{2 n}\right )^p}{\left (-d^2+e^2 x^{2 n}\right )^3}-\frac{3 d e^2 x^{2 n} \left (a+c x^{2 n}\right )^p}{\left (-d^2+e^2 x^{2 n}\right )^3}+\frac{e^3 x^{3 n} \left (a+c x^{2 n}\right )^p}{\left (-d^2+e^2 x^{2 n}\right )^3}\right ) \, dx\\ &=d^3 \int \frac{\left (a+c x^{2 n}\right )^p}{\left (d^2-e^2 x^{2 n}\right )^3} \, dx+\left (3 d^2 e\right ) \int \frac{x^n \left (a+c x^{2 n}\right )^p}{\left (-d^2+e^2 x^{2 n}\right )^3} \, dx-\left (3 d e^2\right ) \int \frac{x^{2 n} \left (a+c x^{2 n}\right )^p}{\left (-d^2+e^2 x^{2 n}\right )^3} \, dx+e^3 \int \frac{x^{3 n} \left (a+c x^{2 n}\right )^p}{\left (-d^2+e^2 x^{2 n}\right )^3} \, dx\\ &=\left (d^3 \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}{\left (d^2-e^2 x^{2 n}\right )^3} \, dx+\left (3 d^2 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}{\left (-d^2+e^2 x^{2 n}\right )^3} \, dx-\left (3 d e^2 \left (a+c x^{2 n}\right )^p \left (1+\frac{c x^{2 n}}{a}\right )^{-p}\right ) \int \frac{x^{2 n} \left (1+\frac{c x^{2 n}}{a}\right )^p}{\left (-d^2+e^2 x^{2 n}\right )^3} \, dx+\left (e^3 \left (a+c x^{2 n}\right )^p \left (1+\frac{c x^{2 n}}{a}\right )^{-p}\right ) \int \frac{x^{3 n} \left (1+\frac{c x^{2 n}}{a}\right )^p}{\left (-d^2+e^2 x^{2 n}\right )^3} \, dx\\ &=\frac{3 e^2 x^{1+2 n} \left (a+c x^{2 n}\right )^p \left (1+\frac{c x^{2 n}}{a}\right )^{-p} F_1\left (\frac{1}{2} \left (2+\frac{1}{n}\right );-p,3;\frac{1}{2} \left (4+\frac{1}{n}\right );-\frac{c x^{2 n}}{a},\frac{e^2 x^{2 n}}{d^2}\right )}{d^5 (1+2 n)}-\frac{e^3 x^{1+3 n} \left (a+c x^{2 n}\right )^p \left (1+\frac{c x^{2 n}}{a}\right )^{-p} F_1\left (\frac{1}{2} \left (3+\frac{1}{n}\right );-p,3;\frac{1}{2} \left (5+\frac{1}{n}\right );-\frac{c x^{2 n}}{a},\frac{e^2 x^{2 n}}{d^2}\right )}{d^6 (1+3 n)}+\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,3;\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^3}-\frac{3 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,3;\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^4 (1+n)}\\ \end{align*}
Mathematica [F] time = 0.294008, size = 0, normalized size = 0. \[ \int \frac{\left (a+c x^{2 n}\right )^p}{\left (d+e x^n\right )^3} \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 0.086, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+c{x}^{2\,n} \right ) ^{p}}{ \left ( d+e{x}^{n} \right ) ^{3}}}\, 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}}{{\left (e x^{n} + d\right )}^{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{{\left (c x^{2 \, n} + a\right )}^{p}}{e^{3} x^{3 \, n} + 3 \, d e^{2} x^{2 \, n} + 3 \, d^{2} e x^{n} + d^{3}}, 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}}{{\left (e x^{n} + d\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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