Optimal. Leaf size=89 \[ -\frac {2 \left (a+\frac {b}{x^2}\right )^p \left (\frac {b}{a x^2}+1\right )^{-p} \left (c+\frac {d}{x^2}\right )^q \left (\frac {d}{c x^2}+1\right )^{-q} F_1\left (\frac {1}{4};-p,-q;\frac {5}{4};-\frac {b}{a x^2},-\frac {d}{c x^2}\right )}{e \sqrt {e x}} \]
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Rubi [A] time = 0.09, antiderivative size = 89, normalized size of antiderivative = 1.00, 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, 430, 429} \[ -\frac {2 \left (a+\frac {b}{x^2}\right )^p \left (\frac {b}{a x^2}+1\right )^{-p} \left (c+\frac {d}{x^2}\right )^q \left (\frac {d}{c x^2}+1\right )^{-q} F_1\left (\frac {1}{4};-p,-q;\frac {5}{4};-\frac {b}{a x^2},-\frac {d}{c x^2}\right )}{e \sqrt {e x}} \]
Antiderivative was successfully verified.
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Rule 429
Rule 430
Rule 496
Rubi steps
\begin {align*} \int \frac {\left (a+\frac {b}{x^2}\right )^p \left (c+\frac {d}{x^2}\right )^q}{(e x)^{3/2}} \, dx &=-\frac {2 \operatorname {Subst}\left (\int \left (a+b e^2 x^4\right )^p \left (c+d e^2 x^4\right )^q \, 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 \left (1+\frac {b e^2 x^4}{a}\right )^p \left (c+d e^2 x^4\right )^q \, 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 \left (1+\frac {b e^2 x^4}{a}\right )^p \left (1+\frac {d e^2 x^4}{c}\right )^q \, 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} F_1\left (\frac {1}{4};-p,-q;\frac {5}{4};-\frac {b}{a x^2},-\frac {d}{c x^2}\right )}{e \sqrt {e x}}\\ \end {align*}
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Mathematica [A] time = 0.17, size = 111, normalized size = 1.25 \[ -\frac {2 x \left (a+\frac {b}{x^2}\right )^p \left (\frac {a x^2}{b}+1\right )^{-p} \left (c+\frac {d}{x^2}\right )^q \left (\frac {c x^2}{d}+1\right )^{-q} F_1\left (-p-q-\frac {1}{4};-p,-q;-p-q+\frac {3}{4};-\frac {a x^2}{b},-\frac {c x^2}{d}\right )}{(e x)^{3/2} (4 p+4 q+1)} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.87, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {e x} \left (\frac {a x^{2} + b}{x^{2}}\right )^{p} \left (\frac {c x^{2} + d}{x^{2}}\right )^{q}}{e^{2} x^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (a + \frac {b}{x^{2}}\right )}^{p} {\left (c + \frac {d}{x^{2}}\right )}^{q}}{\left (e x\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.08, size = 0, normalized size = 0.00 \[ \int \frac {\left (a +\frac {b}{x^{2}}\right )^{p} \left (c +\frac {d}{x^{2}}\right )^{q}}{\left (e x \right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (a + \frac {b}{x^{2}}\right )}^{p} {\left (c + \frac {d}{x^{2}}\right )}^{q}}{\left (e x\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (a+\frac {b}{x^2}\right )}^p\,{\left (c+\frac {d}{x^2}\right )}^q}{{\left (e\,x\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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