Integrand size = 19, antiderivative size = 79 \[ \int \left (a+\frac {b}{x^2}\right )^p \left (c+\frac {d}{x^2}\right )^q \, dx=\left (a+\frac {b}{x^2}\right )^p \left (1+\frac {b}{a x^2}\right )^{-p} \left (c+\frac {d}{x^2}\right )^q \left (1+\frac {d}{c x^2}\right )^{-q} x \operatorname {AppellF1}\left (-\frac {1}{2},-p,-q,\frac {1}{2},-\frac {b}{a x^2},-\frac {d}{c x^2}\right ) \]
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Time = 0.07 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {382, 525, 524} \[ \int \left (a+\frac {b}{x^2}\right )^p \left (c+\frac {d}{x^2}\right )^q \, dx=x \left (a+\frac {b}{x^2}\right )^p \left (\frac {b}{a x^2}+1\right )^{-p} \left (c+\frac {d}{x^2}\right )^q \left (\frac {d}{c x^2}+1\right )^{-q} \operatorname {AppellF1}\left (-\frac {1}{2},-p,-q,\frac {1}{2},-\frac {b}{a x^2},-\frac {d}{c x^2}\right ) \]
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Rule 382
Rule 524
Rule 525
Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int \frac {\left (a+b x^2\right )^p \left (c+d x^2\right )^q}{x^2} \, dx,x,\frac {1}{x}\right ) \\ & = -\left (\left (\left (a+\frac {b}{x^2}\right )^p \left (1+\frac {b}{a x^2}\right )^{-p}\right ) \text {Subst}\left (\int \frac {\left (1+\frac {b x^2}{a}\right )^p \left (c+d x^2\right )^q}{x^2} \, dx,x,\frac {1}{x}\right )\right ) \\ & = -\left (\left (\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 ) \text {Subst}\left (\int \frac {\left (1+\frac {b x^2}{a}\right )^p \left (1+\frac {d x^2}{c}\right )^q}{x^2} \, dx,x,\frac {1}{x}\right )\right ) \\ & = \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} x F_1\left (-\frac {1}{2};-p,-q;\frac {1}{2};-\frac {b}{a x^2},-\frac {d}{c x^2}\right ) \\ \end{align*}
Time = 0.25 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.32 \[ \int \left (a+\frac {b}{x^2}\right )^p \left (c+\frac {d}{x^2}\right )^q \, dx=-\frac {\left (a+\frac {b}{x^2}\right )^p \left (c+\frac {d}{x^2}\right )^q x \left (1+\frac {a x^2}{b}\right )^{-p} \left (1+\frac {c x^2}{d}\right )^{-q} \operatorname {AppellF1}\left (\frac {1}{2}-p-q,-p,-q,\frac {3}{2}-p-q,-\frac {a x^2}{b},-\frac {c x^2}{d}\right )}{-1+2 p+2 q} \]
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\[\int \left (a +\frac {b}{x^{2}}\right )^{p} \left (c +\frac {d}{x^{2}}\right )^{q}d x\]
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\[ \int \left (a+\frac {b}{x^2}\right )^p \left (c+\frac {d}{x^2}\right )^q \, dx=\int { {\left (a + \frac {b}{x^{2}}\right )}^{p} {\left (c + \frac {d}{x^{2}}\right )}^{q} \,d x } \]
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Timed out. \[ \int \left (a+\frac {b}{x^2}\right )^p \left (c+\frac {d}{x^2}\right )^q \, dx=\text {Timed out} \]
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\[ \int \left (a+\frac {b}{x^2}\right )^p \left (c+\frac {d}{x^2}\right )^q \, dx=\int { {\left (a + \frac {b}{x^{2}}\right )}^{p} {\left (c + \frac {d}{x^{2}}\right )}^{q} \,d x } \]
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\[ \int \left (a+\frac {b}{x^2}\right )^p \left (c+\frac {d}{x^2}\right )^q \, dx=\int { {\left (a + \frac {b}{x^{2}}\right )}^{p} {\left (c + \frac {d}{x^{2}}\right )}^{q} \,d x } \]
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Timed out. \[ \int \left (a+\frac {b}{x^2}\right )^p \left (c+\frac {d}{x^2}\right )^q \, dx=\int {\left (a+\frac {b}{x^2}\right )}^p\,{\left (c+\frac {d}{x^2}\right )}^q \,d x \]
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