Optimal. Leaf size=91 \[ -\frac{2 \left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} \left (c+d x^2\right )^q \left (\frac{d x^2}{c}+1\right )^{-q} F_1\left (-\frac{3}{4};-p,-q;\frac{1}{4};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )}{3 e (e x)^{3/2}} \]
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Rubi [A] time = 0.076363, antiderivative size = 91, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {511, 510} \[ -\frac{2 \left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} \left (c+d x^2\right )^q \left (\frac{d x^2}{c}+1\right )^{-q} F_1\left (-\frac{3}{4};-p,-q;\frac{1}{4};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )}{3 e (e x)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 511
Rule 510
Rubi steps
\begin{align*} \int \frac{\left (a+b x^2\right )^p \left (c+d x^2\right )^q}{(e x)^{5/2}} \, dx &=\left (\left (a+b x^2\right )^p \left (1+\frac{b x^2}{a}\right )^{-p}\right ) \int \frac{\left (1+\frac{b x^2}{a}\right )^p \left (c+d x^2\right )^q}{(e x)^{5/2}} \, 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 \frac{\left (1+\frac{b x^2}{a}\right )^p \left (1+\frac{d x^2}{c}\right )^q}{(e x)^{5/2}} \, dx\\ &=-\frac{2 \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{3}{4};-p,-q;\frac{1}{4};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )}{3 e (e x)^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0579623, size = 91, normalized size = 1. \[ -\frac{2 x \left (a+b x^2\right )^p \left (\frac{a+b x^2}{a}\right )^{-p} \left (c+d x^2\right )^q \left (\frac{c+d x^2}{c}\right )^{-q} F_1\left (-\frac{3}{4};-p,-q;\frac{1}{4};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )}{3 (e x)^{5/2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.044, size = 0, normalized size = 0. \begin{align*} \int{ \left ( b{x}^{2}+a \right ) ^{p} \left ( d{x}^{2}+c \right ) ^{q} \left ( ex \right ) ^{-{\frac{5}{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 (b x^{2} + a\right )}^{p}{\left (d x^{2} + c\right )}^{q}}{\left (e x\right )^{\frac{5}{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{\sqrt{e x}{\left (b x^{2} + a\right )}^{p}{\left (d x^{2} + c\right )}^{q}}{e^{3} x^{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 (b x^{2} + a\right )}^{p}{\left (d x^{2} + c\right )}^{q}}{\left (e x\right )^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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