Integrand size = 25, antiderivative size = 89 \[ \int (d-e x)^m (d+e x)^m \left (a+c x^2\right )^p \, dx=x (d-e x)^m (d+e x)^m \left (a+c x^2\right )^p \left (1+\frac {c x^2}{a}\right )^{-p} \left (1-\frac {e^2 x^2}{d^2}\right )^{-m} \operatorname {AppellF1}\left (\frac {1}{2},-p,-m,\frac {3}{2},-\frac {c x^2}{a},\frac {e^2 x^2}{d^2}\right ) \]
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Time = 0.05 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {533, 441, 440} \[ \int (d-e x)^m (d+e x)^m \left (a+c x^2\right )^p \, dx=x \left (a+c x^2\right )^p \left (\frac {c x^2}{a}+1\right )^{-p} (d-e x)^m (d+e x)^m \left (1-\frac {e^2 x^2}{d^2}\right )^{-m} \operatorname {AppellF1}\left (\frac {1}{2},-p,-m,\frac {3}{2},-\frac {c x^2}{a},\frac {e^2 x^2}{d^2}\right ) \]
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Rule 440
Rule 441
Rule 533
Rubi steps \begin{align*} \text {integral}& = \left ((d-e x)^m (d+e x)^m \left (d^2-e^2 x^2\right )^{-m}\right ) \int \left (a+c x^2\right )^p \left (d^2-e^2 x^2\right )^m \, dx \\ & = \left ((d-e x)^m (d+e x)^m \left (a+c x^2\right )^p \left (1+\frac {c x^2}{a}\right )^{-p} \left (d^2-e^2 x^2\right )^{-m}\right ) \int \left (1+\frac {c x^2}{a}\right )^p \left (d^2-e^2 x^2\right )^m \, dx \\ & = \left ((d-e x)^m (d+e x)^m \left (a+c x^2\right )^p \left (1+\frac {c x^2}{a}\right )^{-p} \left (1-\frac {e^2 x^2}{d^2}\right )^{-m}\right ) \int \left (1+\frac {c x^2}{a}\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^m \, dx \\ & = x (d-e x)^m (d+e x)^m \left (a+c x^2\right )^p \left (1+\frac {c x^2}{a}\right )^{-p} \left (1-\frac {e^2 x^2}{d^2}\right )^{-m} F_1\left (\frac {1}{2};-p,-m;\frac {3}{2};-\frac {c x^2}{a},\frac {e^2 x^2}{d^2}\right ) \\ \end{align*}
\[ \int (d-e x)^m (d+e x)^m \left (a+c x^2\right )^p \, dx=\int (d-e x)^m (d+e x)^m \left (a+c x^2\right )^p \, dx \]
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\[\int \left (-e x +d \right )^{m} \left (e x +d \right )^{m} \left (c \,x^{2}+a \right )^{p}d x\]
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\[ \int (d-e x)^m (d+e x)^m \left (a+c x^2\right )^p \, dx=\int { {\left (c x^{2} + a\right )}^{p} {\left (e x + d\right )}^{m} {\left (-e x + d\right )}^{m} \,d x } \]
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Timed out. \[ \int (d-e x)^m (d+e x)^m \left (a+c x^2\right )^p \, dx=\text {Timed out} \]
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\[ \int (d-e x)^m (d+e x)^m \left (a+c x^2\right )^p \, dx=\int { {\left (c x^{2} + a\right )}^{p} {\left (e x + d\right )}^{m} {\left (-e x + d\right )}^{m} \,d x } \]
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\[ \int (d-e x)^m (d+e x)^m \left (a+c x^2\right )^p \, dx=\int { {\left (c x^{2} + a\right )}^{p} {\left (e x + d\right )}^{m} {\left (-e x + d\right )}^{m} \,d x } \]
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Timed out. \[ \int (d-e x)^m (d+e x)^m \left (a+c x^2\right )^p \, dx=\int {\left (c\,x^2+a\right )}^p\,{\left (d+e\,x\right )}^m\,{\left (d-e\,x\right )}^m \,d x \]
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