Integrand size = 27, antiderivative size = 96 \[ \int (g x)^m (d+e x)^n \left (d^2-e^2 x^2\right )^p \, dx=\frac {(g x)^{1+m} (d+e x)^n \left (1-\frac {e x}{d}\right )^{-p} \left (1+\frac {e x}{d}\right )^{-n-p} \left (d^2-e^2 x^2\right )^p \operatorname {AppellF1}\left (1+m,-p,-n-p,2+m,\frac {e x}{d},-\frac {e x}{d}\right )}{g (1+m)} \]
[Out]
Time = 0.06 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {906, 140, 138} \[ \int (g x)^m (d+e x)^n \left (d^2-e^2 x^2\right )^p \, dx=\frac {(g x)^{m+1} (d+e x)^n \left (1-\frac {e x}{d}\right )^{-p} \left (d^2-e^2 x^2\right )^p \left (\frac {e x}{d}+1\right )^{-n-p} \operatorname {AppellF1}\left (m+1,-p,-n-p,m+2,\frac {e x}{d},-\frac {e x}{d}\right )}{g (m+1)} \]
[In]
[Out]
Rule 138
Rule 140
Rule 906
Rubi steps \begin{align*} \text {integral}& = \left ((d-e x)^{-p} (d+e x)^{-p} \left (d^2-e^2 x^2\right )^p\right ) \int (g x)^m (d-e x)^p (d+e x)^{n+p} \, dx \\ & = \left ((d+e x)^{-p} \left (1-\frac {e x}{d}\right )^{-p} \left (d^2-e^2 x^2\right )^p\right ) \int (g x)^m (d+e x)^{n+p} \left (1-\frac {e x}{d}\right )^p \, dx \\ & = \left ((d+e x)^n \left (1-\frac {e x}{d}\right )^{-p} \left (1+\frac {e x}{d}\right )^{-n-p} \left (d^2-e^2 x^2\right )^p\right ) \int (g x)^m \left (1-\frac {e x}{d}\right )^p \left (1+\frac {e x}{d}\right )^{n+p} \, dx \\ & = \frac {(g x)^{1+m} (d+e x)^n \left (1-\frac {e x}{d}\right )^{-p} \left (1+\frac {e x}{d}\right )^{-n-p} \left (d^2-e^2 x^2\right )^p F_1\left (1+m;-p,-n-p;2+m;\frac {e x}{d},-\frac {e x}{d}\right )}{g (1+m)} \\ \end{align*}
Time = 0.24 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.94 \[ \int (g x)^m (d+e x)^n \left (d^2-e^2 x^2\right )^p \, dx=\frac {x (g x)^m (d-e x)^p \left (\frac {d-e x}{d}\right )^{-p} (d+e x)^{n+p} \left (\frac {d+e x}{d}\right )^{-n-p} \operatorname {AppellF1}\left (1+m,-p,-n-p,2+m,\frac {e x}{d},-\frac {e x}{d}\right )}{1+m} \]
[In]
[Out]
\[\int \left (g x \right )^{m} \left (e x +d \right )^{n} \left (-e^{2} x^{2}+d^{2}\right )^{p}d x\]
[In]
[Out]
\[ \int (g x)^m (d+e x)^n \left (d^2-e^2 x^2\right )^p \, dx=\int { {\left (-e^{2} x^{2} + d^{2}\right )}^{p} {\left (e x + d\right )}^{n} \left (g x\right )^{m} \,d x } \]
[In]
[Out]
\[ \int (g x)^m (d+e x)^n \left (d^2-e^2 x^2\right )^p \, dx=\int \left (g x\right )^{m} \left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{p} \left (d + e x\right )^{n}\, dx \]
[In]
[Out]
\[ \int (g x)^m (d+e x)^n \left (d^2-e^2 x^2\right )^p \, dx=\int { {\left (-e^{2} x^{2} + d^{2}\right )}^{p} {\left (e x + d\right )}^{n} \left (g x\right )^{m} \,d x } \]
[In]
[Out]
\[ \int (g x)^m (d+e x)^n \left (d^2-e^2 x^2\right )^p \, dx=\int { {\left (-e^{2} x^{2} + d^{2}\right )}^{p} {\left (e x + d\right )}^{n} \left (g x\right )^{m} \,d x } \]
[In]
[Out]
Timed out. \[ \int (g x)^m (d+e x)^n \left (d^2-e^2 x^2\right )^p \, dx=\int {\left (d^2-e^2\,x^2\right )}^p\,{\left (g\,x\right )}^m\,{\left (d+e\,x\right )}^n \,d x \]
[In]
[Out]