Optimal. Leaf size=89 \[ \frac{2 \sqrt{e x} \left (a+\frac{b}{x^2}\right )^p \left (\frac{b}{a x^2}+1\right )^{-p} \left (c+\frac{d}{x^2}\right )^q \left (\frac{d}{c x^2}+1\right )^{-q} F_1\left (-\frac{1}{4};-p,-q;\frac{3}{4};-\frac{b}{a x^2},-\frac{d}{c x^2}\right )}{e} \]
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Rubi [A] time = 0.12595, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used = {496, 511, 510} \[ \frac{2 \sqrt{e x} \left (a+\frac{b}{x^2}\right )^p \left (\frac{b}{a x^2}+1\right )^{-p} \left (c+\frac{d}{x^2}\right )^q \left (\frac{d}{c x^2}+1\right )^{-q} F_1\left (-\frac{1}{4};-p,-q;\frac{3}{4};-\frac{b}{a x^2},-\frac{d}{c x^2}\right )}{e} \]
Antiderivative was successfully verified.
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Rule 496
Rule 511
Rule 510
Rubi steps
\begin{align*} \int \frac{\left (a+\frac{b}{x^2}\right )^p \left (c+\frac{d}{x^2}\right )^q}{\sqrt{e x}} \, dx &=-\frac{2 \operatorname{Subst}\left (\int \frac{\left (a+b e^2 x^4\right )^p \left (c+d e^2 x^4\right )^q}{x^2} \, dx,x,\frac{1}{\sqrt{e x}}\right )}{e}\\ &=-\frac{\left (2 \left (a+\frac{b}{x^2}\right )^p \left (1+\frac{b}{a x^2}\right )^{-p}\right ) \operatorname{Subst}\left (\int \frac{\left (1+\frac{b e^2 x^4}{a}\right )^p \left (c+d e^2 x^4\right )^q}{x^2} \, dx,x,\frac{1}{\sqrt{e x}}\right )}{e}\\ &=-\frac{\left (2 \left (a+\frac{b}{x^2}\right )^p \left (1+\frac{b}{a x^2}\right )^{-p} \left (c+\frac{d}{x^2}\right )^q \left (1+\frac{d}{c x^2}\right )^{-q}\right ) \operatorname{Subst}\left (\int \frac{\left (1+\frac{b e^2 x^4}{a}\right )^p \left (1+\frac{d e^2 x^4}{c}\right )^q}{x^2} \, dx,x,\frac{1}{\sqrt{e x}}\right )}{e}\\ &=\frac{2 \left (a+\frac{b}{x^2}\right )^p \left (1+\frac{b}{a x^2}\right )^{-p} \left (c+\frac{d}{x^2}\right )^q \left (1+\frac{d}{c x^2}\right )^{-q} \sqrt{e x} F_1\left (-\frac{1}{4};-p,-q;\frac{3}{4};-\frac{b}{a x^2},-\frac{d}{c x^2}\right )}{e}\\ \end{align*}
Mathematica [A] time = 0.109108, size = 111, normalized size = 1.25 \[ -\frac{2 x \left (a+\frac{b}{x^2}\right )^p \left (\frac{a x^2}{b}+1\right )^{-p} \left (c+\frac{d}{x^2}\right )^q \left (\frac{c x^2}{d}+1\right )^{-q} F_1\left (-p-q+\frac{1}{4};-p,-q;-p-q+\frac{5}{4};-\frac{a x^2}{b},-\frac{c x^2}{d}\right )}{\sqrt{e x} (4 p+4 q-1)} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.026, size = 0, normalized size = 0. \begin{align*} \int{ \left ( a+{\frac{b}{{x}^{2}}} \right ) ^{p} \left ( c+{\frac{d}{{x}^{2}}} \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 (a + \frac{b}{x^{2}}\right )}^{p}{\left (c + \frac{d}{x^{2}}\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 (\frac{a x^{2} + b}{x^{2}}\right )^{p} \left (\frac{c x^{2} + d}{x^{2}}\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 (a + \frac{b}{x^{2}}\right )}^{p}{\left (c + \frac{d}{x^{2}}\right )}^{q}}{\sqrt{e x}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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