Optimal. Leaf size=89 \[ \frac{2 \sqrt{e x} \left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} \left (c+d x^2\right )^q \left (\frac{d x^2}{c}+1\right )^{-q} F_1\left (\frac{1}{4};-p,-q;\frac{5}{4};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )}{e} \]
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Rubi [A] time = 0.0720401, antiderivative size = 89, 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} \[ \frac{2 \sqrt{e x} \left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} \left (c+d x^2\right )^q \left (\frac{d x^2}{c}+1\right )^{-q} F_1\left (\frac{1}{4};-p,-q;\frac{5}{4};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )}{e} \]
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}{\sqrt{e x}} \, 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}{\sqrt{e x}} \, 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}{\sqrt{e x}} \, dx\\ &=\frac{2 \sqrt{e x} \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}{4};-p,-q;\frac{5}{4};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )}{e}\\ \end{align*}
Mathematica [A] time = 0.0495254, size = 89, normalized size = 1. \[ \frac{2 x \left (a+b x^2\right )^p \left (\frac{a+b x^2}{a}\right )^{-p} \left (c+d x^2\right )^q \left (\frac{c+d x^2}{c}\right )^{-q} F_1\left (\frac{1}{4};-p,-q;\frac{5}{4};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )}{\sqrt{e x}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.045, size = 0, normalized size = 0. \begin{align*} \int{ \left ( b{x}^{2}+a \right ) ^{p} \left ( d{x}^{2}+c \right ) ^{q}{\frac{1}{\sqrt{ex}}}}\, 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}}{\sqrt{e x}}\,{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 x}, 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}}{\sqrt{e x}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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