Optimal. Leaf size=238 \[ \frac{x (g x)^m \left (a+c x^2\right )^p \left (\frac{c x^2}{a}+1\right )^{-p} F_1\left (\frac{m+1}{2};-p,2;\frac{m+3}{2};-\frac{c x^2}{a},\frac{e^2 x^2}{d^2}\right )}{d^2 (m+1)}-\frac{2 e x^2 (g x)^m \left (a+c x^2\right )^p \left (\frac{c x^2}{a}+1\right )^{-p} F_1\left (\frac{m+2}{2};-p,2;\frac{m+4}{2};-\frac{c x^2}{a},\frac{e^2 x^2}{d^2}\right )}{d^3 (m+2)}+\frac{e^2 x^3 (g x)^m \left (a+c x^2\right )^p \left (\frac{c x^2}{a}+1\right )^{-p} F_1\left (\frac{m+3}{2};-p,2;\frac{m+5}{2};-\frac{c x^2}{a},\frac{e^2 x^2}{d^2}\right )}{d^4 (m+3)} \]
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Rubi [A] time = 0.278811, antiderivative size = 238, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {962, 511, 510} \[ \frac{x (g x)^m \left (a+c x^2\right )^p \left (\frac{c x^2}{a}+1\right )^{-p} F_1\left (\frac{m+1}{2};-p,2;\frac{m+3}{2};-\frac{c x^2}{a},\frac{e^2 x^2}{d^2}\right )}{d^2 (m+1)}-\frac{2 e x^2 (g x)^m \left (a+c x^2\right )^p \left (\frac{c x^2}{a}+1\right )^{-p} F_1\left (\frac{m+2}{2};-p,2;\frac{m+4}{2};-\frac{c x^2}{a},\frac{e^2 x^2}{d^2}\right )}{d^3 (m+2)}+\frac{e^2 x^3 (g x)^m \left (a+c x^2\right )^p \left (\frac{c x^2}{a}+1\right )^{-p} F_1\left (\frac{m+3}{2};-p,2;\frac{m+5}{2};-\frac{c x^2}{a},\frac{e^2 x^2}{d^2}\right )}{d^4 (m+3)} \]
Antiderivative was successfully verified.
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Rule 962
Rule 511
Rule 510
Rubi steps
\begin{align*} \int \frac{(g x)^m \left (a+c x^2\right )^p}{(d+e x)^2} \, dx &=\left (x^{-m} (g x)^m\right ) \int \left (\frac{d^2 x^m \left (a+c x^2\right )^p}{\left (d^2-e^2 x^2\right )^2}-\frac{2 d e x^{1+m} \left (a+c x^2\right )^p}{\left (d^2-e^2 x^2\right )^2}+\frac{e^2 x^{2+m} \left (a+c x^2\right )^p}{\left (-d^2+e^2 x^2\right )^2}\right ) \, dx\\ &=\left (d^2 x^{-m} (g x)^m\right ) \int \frac{x^m \left (a+c x^2\right )^p}{\left (d^2-e^2 x^2\right )^2} \, dx-\left (2 d e x^{-m} (g x)^m\right ) \int \frac{x^{1+m} \left (a+c x^2\right )^p}{\left (d^2-e^2 x^2\right )^2} \, dx+\left (e^2 x^{-m} (g x)^m\right ) \int \frac{x^{2+m} \left (a+c x^2\right )^p}{\left (-d^2+e^2 x^2\right )^2} \, dx\\ &=\left (d^2 x^{-m} (g x)^m \left (a+c x^2\right )^p \left (1+\frac{c x^2}{a}\right )^{-p}\right ) \int \frac{x^m \left (1+\frac{c x^2}{a}\right )^p}{\left (d^2-e^2 x^2\right )^2} \, dx-\left (2 d e x^{-m} (g x)^m \left (a+c x^2\right )^p \left (1+\frac{c x^2}{a}\right )^{-p}\right ) \int \frac{x^{1+m} \left (1+\frac{c x^2}{a}\right )^p}{\left (d^2-e^2 x^2\right )^2} \, dx+\left (e^2 x^{-m} (g x)^m \left (a+c x^2\right )^p \left (1+\frac{c x^2}{a}\right )^{-p}\right ) \int \frac{x^{2+m} \left (1+\frac{c x^2}{a}\right )^p}{\left (-d^2+e^2 x^2\right )^2} \, dx\\ &=\frac{x (g x)^m \left (a+c x^2\right )^p \left (1+\frac{c x^2}{a}\right )^{-p} F_1\left (\frac{1+m}{2};-p,2;\frac{3+m}{2};-\frac{c x^2}{a},\frac{e^2 x^2}{d^2}\right )}{d^2 (1+m)}-\frac{2 e x^2 (g x)^m \left (a+c x^2\right )^p \left (1+\frac{c x^2}{a}\right )^{-p} F_1\left (\frac{2+m}{2};-p,2;\frac{4+m}{2};-\frac{c x^2}{a},\frac{e^2 x^2}{d^2}\right )}{d^3 (2+m)}+\frac{e^2 x^3 (g x)^m \left (a+c x^2\right )^p \left (1+\frac{c x^2}{a}\right )^{-p} F_1\left (\frac{3+m}{2};-p,2;\frac{5+m}{2};-\frac{c x^2}{a},\frac{e^2 x^2}{d^2}\right )}{d^4 (3+m)}\\ \end{align*}
Mathematica [F] time = 0.0936035, size = 0, normalized size = 0. \[ \int \frac{(g x)^m \left (a+c x^2\right )^p}{(d+e x)^2} \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 0.638, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( gx \right ) ^{m} \left ( c{x}^{2}+a \right ) ^{p}}{ \left ( ex+d \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} + a\right )}^{p} \left (g x\right )^{m}}{{\left (e x + 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} + a\right )}^{p} \left (g x\right )^{m}}{e^{2} x^{2} + 2 \, d e x + 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} + a\right )}^{p} \left (g x\right )^{m}}{{\left (e x + d\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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