Integrand size = 22, antiderivative size = 238 \[ \int \frac {(g x)^m \left (a+c x^2\right )^p}{(d+e x)^2} \, dx=\frac {x (g x)^m \left (a+c x^2\right )^p \left (1+\frac {c x^2}{a}\right )^{-p} \operatorname {AppellF1}\left (\frac {1+m}{2},-p,2,\frac {3+m}{2},-\frac {c x^2}{a},\frac {e^2 x^2}{d^2}\right )}{d^2 (1+m)}-\frac {2 e x^2 (g x)^m \left (a+c x^2\right )^p \left (1+\frac {c x^2}{a}\right )^{-p} \operatorname {AppellF1}\left (\frac {2+m}{2},-p,2,\frac {4+m}{2},-\frac {c x^2}{a},\frac {e^2 x^2}{d^2}\right )}{d^3 (2+m)}+\frac {e^2 x^3 (g x)^m \left (a+c x^2\right )^p \left (1+\frac {c x^2}{a}\right )^{-p} \operatorname {AppellF1}\left (\frac {3+m}{2},-p,2,\frac {5+m}{2},-\frac {c x^2}{a},\frac {e^2 x^2}{d^2}\right )}{d^4 (3+m)} \]
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Time = 0.18 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {976, 525, 524} \[ \int \frac {(g x)^m \left (a+c x^2\right )^p}{(d+e x)^2} \, dx=\frac {x (g x)^m \left (a+c x^2\right )^p \left (\frac {c x^2}{a}+1\right )^{-p} \operatorname {AppellF1}\left (\frac {m+1}{2},-p,2,\frac {m+3}{2},-\frac {c x^2}{a},\frac {e^2 x^2}{d^2}\right )}{d^2 (m+1)}+\frac {e^2 x^3 (g x)^m \left (a+c x^2\right )^p \left (\frac {c x^2}{a}+1\right )^{-p} \operatorname {AppellF1}\left (\frac {m+3}{2},-p,2,\frac {m+5}{2},-\frac {c x^2}{a},\frac {e^2 x^2}{d^2}\right )}{d^4 (m+3)}-\frac {2 e x^2 (g x)^m \left (a+c x^2\right )^p \left (\frac {c x^2}{a}+1\right )^{-p} \operatorname {AppellF1}\left (\frac {m+2}{2},-p,2,\frac {m+4}{2},-\frac {c x^2}{a},\frac {e^2 x^2}{d^2}\right )}{d^3 (m+2)} \]
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Rule 524
Rule 525
Rule 976
Rubi steps \begin{align*} \text {integral}& = \left (x^{-m} (g x)^m\right ) \int \left (\frac {d^2 x^m \left (a+c x^2\right )^p}{\left (d^2-e^2 x^2\right )^2}-\frac {2 d e x^{1+m} \left (a+c x^2\right )^p}{\left (d^2-e^2 x^2\right )^2}+\frac {e^2 x^{2+m} \left (a+c x^2\right )^p}{\left (-d^2+e^2 x^2\right )^2}\right ) \, dx \\ & = \left (d^2 x^{-m} (g x)^m\right ) \int \frac {x^m \left (a+c x^2\right )^p}{\left (d^2-e^2 x^2\right )^2} \, dx-\left (2 d e x^{-m} (g x)^m\right ) \int \frac {x^{1+m} \left (a+c x^2\right )^p}{\left (d^2-e^2 x^2\right )^2} \, dx+\left (e^2 x^{-m} (g x)^m\right ) \int \frac {x^{2+m} \left (a+c x^2\right )^p}{\left (-d^2+e^2 x^2\right )^2} \, dx \\ & = \left (d^2 x^{-m} (g x)^m \left (a+c x^2\right )^p \left (1+\frac {c x^2}{a}\right )^{-p}\right ) \int \frac {x^m \left (1+\frac {c x^2}{a}\right )^p}{\left (d^2-e^2 x^2\right )^2} \, dx-\left (2 d e x^{-m} (g x)^m \left (a+c x^2\right )^p \left (1+\frac {c x^2}{a}\right )^{-p}\right ) \int \frac {x^{1+m} \left (1+\frac {c x^2}{a}\right )^p}{\left (d^2-e^2 x^2\right )^2} \, dx+\left (e^2 x^{-m} (g x)^m \left (a+c x^2\right )^p \left (1+\frac {c x^2}{a}\right )^{-p}\right ) \int \frac {x^{2+m} \left (1+\frac {c x^2}{a}\right )^p}{\left (-d^2+e^2 x^2\right )^2} \, dx \\ & = \frac {x (g x)^m \left (a+c x^2\right )^p \left (1+\frac {c x^2}{a}\right )^{-p} F_1\left (\frac {1+m}{2};-p,2;\frac {3+m}{2};-\frac {c x^2}{a},\frac {e^2 x^2}{d^2}\right )}{d^2 (1+m)}-\frac {2 e x^2 (g x)^m \left (a+c x^2\right )^p \left (1+\frac {c x^2}{a}\right )^{-p} F_1\left (\frac {2+m}{2};-p,2;\frac {4+m}{2};-\frac {c x^2}{a},\frac {e^2 x^2}{d^2}\right )}{d^3 (2+m)}+\frac {e^2 x^3 (g x)^m \left (a+c x^2\right )^p \left (1+\frac {c x^2}{a}\right )^{-p} F_1\left (\frac {3+m}{2};-p,2;\frac {5+m}{2};-\frac {c x^2}{a},\frac {e^2 x^2}{d^2}\right )}{d^4 (3+m)} \\ \end{align*}
\[ \int \frac {(g x)^m \left (a+c x^2\right )^p}{(d+e x)^2} \, dx=\int \frac {(g x)^m \left (a+c x^2\right )^p}{(d+e x)^2} \, dx \]
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\[\int \frac {\left (g x \right )^{m} \left (c \,x^{2}+a \right )^{p}}{\left (e x +d \right )^{2}}d x\]
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\[ \int \frac {(g x)^m \left (a+c x^2\right )^p}{(d+e x)^2} \, dx=\int { \frac {{\left (c x^{2} + a\right )}^{p} \left (g x\right )^{m}}{{\left (e x + d\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {(g x)^m \left (a+c x^2\right )^p}{(d+e x)^2} \, dx=\text {Timed out} \]
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\[ \int \frac {(g x)^m \left (a+c x^2\right )^p}{(d+e x)^2} \, dx=\int { \frac {{\left (c x^{2} + a\right )}^{p} \left (g x\right )^{m}}{{\left (e x + d\right )}^{2}} \,d x } \]
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\[ \int \frac {(g x)^m \left (a+c x^2\right )^p}{(d+e x)^2} \, dx=\int { \frac {{\left (c x^{2} + a\right )}^{p} \left (g x\right )^{m}}{{\left (e x + d\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {(g x)^m \left (a+c x^2\right )^p}{(d+e x)^2} \, dx=\int \frac {{\left (g\,x\right )}^m\,{\left (c\,x^2+a\right )}^p}{{\left (d+e\,x\right )}^2} \,d x \]
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