Integrand size = 24, antiderivative size = 101 \[ \int (e x)^m \left (a+b x^2\right )^p \left (c+d x^2\right )^q \, dx=\frac {(e x)^{1+m} \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \left (c+d x^2\right )^q \left (1+\frac {d x^2}{c}\right )^{-q} \operatorname {AppellF1}\left (\frac {1+m}{2},-p,-q,\frac {3+m}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )}{e (1+m)} \]
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Time = 0.05 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {525, 524} \[ \int (e x)^m \left (a+b x^2\right )^p \left (c+d x^2\right )^q \, dx=\frac {(e x)^{m+1} \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \left (c+d x^2\right )^q \left (\frac {d x^2}{c}+1\right )^{-q} \operatorname {AppellF1}\left (\frac {m+1}{2},-p,-q,\frac {m+3}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )}{e (m+1)} \]
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Rule 524
Rule 525
Rubi steps \begin{align*} \text {integral}& = \left (\left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p}\right ) \int (e x)^m \left (1+\frac {b x^2}{a}\right )^p \left (c+d x^2\right )^q \, 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 (e x)^m \left (1+\frac {b x^2}{a}\right )^p \left (1+\frac {d x^2}{c}\right )^q \, dx \\ & = \frac {(e x)^{1+m} \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+m}{2};-p,-q;\frac {3+m}{2};-\frac {b x^2}{a},-\frac {d x^2}{c}\right )}{e (1+m)} \\ \end{align*}
Time = 0.17 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.96 \[ \int (e x)^m \left (a+b x^2\right )^p \left (c+d x^2\right )^q \, dx=\frac {x (e x)^m \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \left (c+d x^2\right )^q \left (1+\frac {d x^2}{c}\right )^{-q} \operatorname {AppellF1}\left (\frac {1+m}{2},-p,-q,\frac {3+m}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )}{1+m} \]
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\[\int \left (e x \right )^{m} \left (b \,x^{2}+a \right )^{p} \left (d \,x^{2}+c \right )^{q}d x\]
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\[ \int (e x)^m \left (a+b x^2\right )^p \left (c+d x^2\right )^q \, dx=\int { {\left (b x^{2} + a\right )}^{p} {\left (d x^{2} + c\right )}^{q} \left (e x\right )^{m} \,d x } \]
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Timed out. \[ \int (e x)^m \left (a+b x^2\right )^p \left (c+d x^2\right )^q \, dx=\text {Timed out} \]
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\[ \int (e x)^m \left (a+b x^2\right )^p \left (c+d x^2\right )^q \, dx=\int { {\left (b x^{2} + a\right )}^{p} {\left (d x^{2} + c\right )}^{q} \left (e x\right )^{m} \,d x } \]
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\[ \int (e x)^m \left (a+b x^2\right )^p \left (c+d x^2\right )^q \, dx=\int { {\left (b x^{2} + a\right )}^{p} {\left (d x^{2} + c\right )}^{q} \left (e x\right )^{m} \,d x } \]
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Timed out. \[ \int (e x)^m \left (a+b x^2\right )^p \left (c+d x^2\right )^q \, dx=\int {\left (e\,x\right )}^m\,{\left (b\,x^2+a\right )}^p\,{\left (d\,x^2+c\right )}^q \,d x \]
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