Optimal. Leaf size=101 \[ \frac{(e x)^{m+1} \left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} \left (c+d x^2\right )^q \left (\frac{d x^2}{c}+1\right )^{-q} F_1\left (\frac{m+1}{2};-p,-q;\frac{m+3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )}{e (m+1)} \]
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Rubi [A] time = 0.0790094, antiderivative size = 101, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {511, 510} \[ \frac{(e x)^{m+1} \left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} \left (c+d x^2\right )^q \left (\frac{d x^2}{c}+1\right )^{-q} F_1\left (\frac{m+1}{2};-p,-q;\frac{m+3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )}{e (m+1)} \]
Antiderivative was successfully verified.
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Rule 511
Rule 510
Rubi steps
\begin{align*} \int (e x)^m \left (a+b x^2\right )^p \left (c+d x^2\right )^q \, dx &=\left (\left (a+b x^2\right )^p \left (1+\frac{b x^2}{a}\right )^{-p}\right ) \int (e x)^m \left (1+\frac{b x^2}{a}\right )^p \left (c+d x^2\right )^q \, dx\\ &=\left (\left (a+b x^2\right )^p \left (1+\frac{b x^2}{a}\right )^{-p} \left (c+d x^2\right )^q \left (1+\frac{d x^2}{c}\right )^{-q}\right ) \int (e x)^m \left (1+\frac{b x^2}{a}\right )^p \left (1+\frac{d x^2}{c}\right )^q \, dx\\ &=\frac{(e x)^{1+m} \left (a+b x^2\right )^p \left (1+\frac{b x^2}{a}\right )^{-p} \left (c+d x^2\right )^q \left (1+\frac{d x^2}{c}\right )^{-q} F_1\left (\frac{1+m}{2};-p,-q;\frac{3+m}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )}{e (1+m)}\\ \end{align*}
Mathematica [A] time = 0.113158, size = 97, normalized size = 0.96 \[ \frac{x (e x)^m \left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} \left (c+d x^2\right )^q \left (\frac{d x^2}{c}+1\right )^{-q} F_1\left (\frac{m+1}{2};-p,-q;\frac{m+3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )}{m+1} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.128, size = 0, normalized size = 0. \begin{align*} \int \left ( ex \right ) ^{m} \left ( b{x}^{2}+a \right ) ^{p} \left ( d{x}^{2}+c \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 (b x^{2} + a\right )}^{p}{\left (d x^{2} + c\right )}^{q} \left (e x\right )^{m}\,{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 (b x^{2} + a\right )}^{p}{\left (d x^{2} + c\right )}^{q} \left (e x\right )^{m}, 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 (b x^{2} + a\right )}^{p}{\left (d x^{2} + c\right )}^{q} \left (e x\right )^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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