Optimal. Leaf size=302 \[ \frac{x (f x)^m \left (a+c x^{2 n}\right )^p \left (\frac{c x^{2 n}}{a}+1\right )^{-p} F_1\left (\frac{m+1}{2 n};-p,2;\frac{m+1}{2 n}+1;-\frac{c x^{2 n}}{a},\frac{e^2 x^{2 n}}{d^2}\right )}{d^2 (m+1)}-\frac{2 e x^{n+1} (f x)^m \left (a+c x^{2 n}\right )^p \left (\frac{c x^{2 n}}{a}+1\right )^{-p} F_1\left (\frac{m+n+1}{2 n};-p,2;\frac{m+3 n+1}{2 n};-\frac{c x^{2 n}}{a},\frac{e^2 x^{2 n}}{d^2}\right )}{d^3 (m+n+1)}+\frac{e^2 x^{2 n+1} (f x)^m \left (a+c x^{2 n}\right )^p \left (\frac{c x^{2 n}}{a}+1\right )^{-p} F_1\left (\frac{m+2 n+1}{2 n};-p,2;\frac{m+4 n+1}{2 n};-\frac{c x^{2 n}}{a},\frac{e^2 x^{2 n}}{d^2}\right )}{d^4 (m+2 n+1)} \]
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Rubi [A] time = 0.332553, antiderivative size = 302, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used = {1562, 511, 510} \[ \frac{x (f x)^m \left (a+c x^{2 n}\right )^p \left (\frac{c x^{2 n}}{a}+1\right )^{-p} F_1\left (\frac{m+1}{2 n};-p,2;\frac{m+1}{2 n}+1;-\frac{c x^{2 n}}{a},\frac{e^2 x^{2 n}}{d^2}\right )}{d^2 (m+1)}-\frac{2 e x^{n+1} (f x)^m \left (a+c x^{2 n}\right )^p \left (\frac{c x^{2 n}}{a}+1\right )^{-p} F_1\left (\frac{m+n+1}{2 n};-p,2;\frac{m+3 n+1}{2 n};-\frac{c x^{2 n}}{a},\frac{e^2 x^{2 n}}{d^2}\right )}{d^3 (m+n+1)}+\frac{e^2 x^{2 n+1} (f x)^m \left (a+c x^{2 n}\right )^p \left (\frac{c x^{2 n}}{a}+1\right )^{-p} F_1\left (\frac{m+2 n+1}{2 n};-p,2;\frac{m+4 n+1}{2 n};-\frac{c x^{2 n}}{a},\frac{e^2 x^{2 n}}{d^2}\right )}{d^4 (m+2 n+1)} \]
Antiderivative was successfully verified.
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Rule 1562
Rule 511
Rule 510
Rubi steps
\begin{align*} \int \frac{(f x)^m \left (a+c x^{2 n}\right )^p}{\left (d+e x^n\right )^2} \, dx &=\left (x^{-m} (f x)^m\right ) \int \left (\frac{d^2 x^m \left (a+c x^{2 n}\right )^p}{\left (d^2-e^2 x^{2 n}\right )^2}-\frac{2 d e x^{m+n} \left (a+c x^{2 n}\right )^p}{\left (-d^2+e^2 x^{2 n}\right )^2}+\frac{e^2 x^{m+2 n} \left (a+c x^{2 n}\right )^p}{\left (-d^2+e^2 x^{2 n}\right )^2}\right ) \, dx\\ &=\left (d^2 x^{-m} (f x)^m\right ) \int \frac{x^m \left (a+c x^{2 n}\right )^p}{\left (d^2-e^2 x^{2 n}\right )^2} \, dx-\left (2 d e x^{-m} (f x)^m\right ) \int \frac{x^{m+n} \left (a+c x^{2 n}\right )^p}{\left (-d^2+e^2 x^{2 n}\right )^2} \, dx+\left (e^2 x^{-m} (f x)^m\right ) \int \frac{x^{m+2 n} \left (a+c x^{2 n}\right )^p}{\left (-d^2+e^2 x^{2 n}\right )^2} \, dx\\ &=\left (d^2 x^{-m} (f x)^m \left (a+c x^{2 n}\right )^p \left (1+\frac{c x^{2 n}}{a}\right )^{-p}\right ) \int \frac{x^m \left (1+\frac{c x^{2 n}}{a}\right )^p}{\left (d^2-e^2 x^{2 n}\right )^2} \, dx-\left (2 d e x^{-m} (f x)^m \left (a+c x^{2 n}\right )^p \left (1+\frac{c x^{2 n}}{a}\right )^{-p}\right ) \int \frac{x^{m+n} \left (1+\frac{c x^{2 n}}{a}\right )^p}{\left (-d^2+e^2 x^{2 n}\right )^2} \, dx+\left (e^2 x^{-m} (f x)^m \left (a+c x^{2 n}\right )^p \left (1+\frac{c x^{2 n}}{a}\right )^{-p}\right ) \int \frac{x^{m+2 n} \left (1+\frac{c x^{2 n}}{a}\right )^p}{\left (-d^2+e^2 x^{2 n}\right )^2} \, dx\\ &=\frac{x (f x)^m \left (a+c x^{2 n}\right )^p \left (1+\frac{c x^{2 n}}{a}\right )^{-p} F_1\left (\frac{1+m}{2 n};-p,2;1+\frac{1+m}{2 n};-\frac{c x^{2 n}}{a},\frac{e^2 x^{2 n}}{d^2}\right )}{d^2 (1+m)}-\frac{2 e x^{1+n} (f x)^m \left (a+c x^{2 n}\right )^p \left (1+\frac{c x^{2 n}}{a}\right )^{-p} F_1\left (\frac{1+m+n}{2 n};-p,2;\frac{1+m+3 n}{2 n};-\frac{c x^{2 n}}{a},\frac{e^2 x^{2 n}}{d^2}\right )}{d^3 (1+m+n)}+\frac{e^2 x^{1+2 n} (f x)^m \left (a+c x^{2 n}\right )^p \left (1+\frac{c x^{2 n}}{a}\right )^{-p} F_1\left (\frac{1+m+2 n}{2 n};-p,2;\frac{1+m+4 n}{2 n};-\frac{c x^{2 n}}{a},\frac{e^2 x^{2 n}}{d^2}\right )}{d^4 (1+m+2 n)}\\ \end{align*}
Mathematica [F] time = 0.16251, size = 0, normalized size = 0. \[ \int \frac{(f x)^m \left (a+c x^{2 n}\right )^p}{\left (d+e x^n\right )^2} \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 0.094, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( fx \right ) ^{m} \left ( a+c{x}^{2\,n} \right ) ^{p}}{ \left ( d+e{x}^{n} \right ) ^{2}}}\, 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 (f x\right )^{m}}{{\left (e x^{n} + d\right )}^{2}}\,{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} \left (f x\right )^{m}}{e^{2} x^{2 \, n} + 2 \, d e x^{n} + d^{2}}, 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 (f x\right )^{m}}{{\left (e x^{n} + d\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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