Integrand size = 27, antiderivative size = 214 \[ \int \frac {(g x)^m \left (d^2-e^2 x^2\right )^p}{(d+e x)^2} \, dx=\frac {(g x)^{1+m} \left (d^2-e^2 x^2\right )^{-1+p}}{g (1-m-2 p)}-\frac {2 (m+p) (g x)^{1+m} \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1+m}{2},2-p,\frac {3+m}{2},\frac {e^2 x^2}{d^2}\right )}{d^2 g (1+m) (1-m-2 p)}-\frac {2 e (g x)^{2+m} \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {2+m}{2},2-p,\frac {4+m}{2},\frac {e^2 x^2}{d^2}\right )}{d^3 g^2 (2+m)} \]
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Time = 0.16 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {866, 1823, 822, 372, 371} \[ \int \frac {(g x)^m \left (d^2-e^2 x^2\right )^p}{(d+e x)^2} \, dx=-\frac {2 (m+p) (g x)^{m+1} \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {m+1}{2},2-p,\frac {m+3}{2},\frac {e^2 x^2}{d^2}\right )}{d^2 g (m+1) (-m-2 p+1)}+\frac {(g x)^{m+1} \left (d^2-e^2 x^2\right )^{p-1}}{g (-m-2 p+1)}-\frac {2 e (g x)^{m+2} \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {m+2}{2},2-p,\frac {m+4}{2},\frac {e^2 x^2}{d^2}\right )}{d^3 g^2 (m+2)} \]
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Rule 371
Rule 372
Rule 822
Rule 866
Rule 1823
Rubi steps \begin{align*} \text {integral}& = \int (g x)^m (d-e x)^2 \left (d^2-e^2 x^2\right )^{-2+p} \, dx \\ & = \frac {(g x)^{1+m} \left (d^2-e^2 x^2\right )^{-1+p}}{g (1-m-2 p)}+\frac {\int (g x)^m \left (-2 d^2 e^2 (m+p)-2 d e^3 (1-m-2 p) x\right ) \left (d^2-e^2 x^2\right )^{-2+p} \, dx}{e^2 (1-m-2 p)} \\ & = \frac {(g x)^{1+m} \left (d^2-e^2 x^2\right )^{-1+p}}{g (1-m-2 p)}-\frac {(2 d e) \int (g x)^{1+m} \left (d^2-e^2 x^2\right )^{-2+p} \, dx}{g}-\frac {\left (2 d^2 (m+p)\right ) \int (g x)^m \left (d^2-e^2 x^2\right )^{-2+p} \, dx}{1-m-2 p} \\ & = \frac {(g x)^{1+m} \left (d^2-e^2 x^2\right )^{-1+p}}{g (1-m-2 p)}-\frac {\left (2 e \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p}\right ) \int (g x)^{1+m} \left (1-\frac {e^2 x^2}{d^2}\right )^{-2+p} \, dx}{d^3 g}-\frac {\left (2 (m+p) \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p}\right ) \int (g x)^m \left (1-\frac {e^2 x^2}{d^2}\right )^{-2+p} \, dx}{d^2 (1-m-2 p)} \\ & = \frac {(g x)^{1+m} \left (d^2-e^2 x^2\right )^{-1+p}}{g (1-m-2 p)}-\frac {2 (m+p) (g x)^{1+m} \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac {1+m}{2},2-p;\frac {3+m}{2};\frac {e^2 x^2}{d^2}\right )}{d^2 g (1+m) (1-m-2 p)}-\frac {2 e (g x)^{2+m} \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac {2+m}{2},2-p;\frac {4+m}{2};\frac {e^2 x^2}{d^2}\right )}{d^3 g^2 (2+m)} \\ \end{align*}
Time = 0.16 (sec) , antiderivative size = 180, normalized size of antiderivative = 0.84 \[ \int \frac {(g x)^m \left (d^2-e^2 x^2\right )^p}{(d+e x)^2} \, dx=\frac {x (g x)^m \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \left (d^2 \left (6+5 m+m^2\right ) \operatorname {Hypergeometric2F1}\left (\frac {1+m}{2},2-p,\frac {3+m}{2},\frac {e^2 x^2}{d^2}\right )-e (1+m) x \left (2 d (3+m) \operatorname {Hypergeometric2F1}\left (\frac {2+m}{2},2-p,\frac {4+m}{2},\frac {e^2 x^2}{d^2}\right )-e (2+m) x \operatorname {Hypergeometric2F1}\left (\frac {3+m}{2},2-p,\frac {5+m}{2},\frac {e^2 x^2}{d^2}\right )\right )\right )}{d^4 (1+m) (2+m) (3+m)} \]
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\[\int \frac {\left (g x \right )^{m} \left (-e^{2} x^{2}+d^{2}\right )^{p}}{\left (e x +d \right )^{2}}d x\]
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\[ \int \frac {(g x)^m \left (d^2-e^2 x^2\right )^p}{(d+e x)^2} \, dx=\int { \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{p} \left (g x\right )^{m}}{{\left (e x + d\right )}^{2}} \,d x } \]
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\[ \int \frac {(g x)^m \left (d^2-e^2 x^2\right )^p}{(d+e x)^2} \, dx=\int \frac {\left (g x\right )^{m} \left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{p}}{\left (d + e x\right )^{2}}\, dx \]
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\[ \int \frac {(g x)^m \left (d^2-e^2 x^2\right )^p}{(d+e x)^2} \, dx=\int { \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{p} \left (g x\right )^{m}}{{\left (e x + d\right )}^{2}} \,d x } \]
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\[ \int \frac {(g x)^m \left (d^2-e^2 x^2\right )^p}{(d+e x)^2} \, dx=\int { \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{p} \left (g x\right )^{m}}{{\left (e x + d\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {(g x)^m \left (d^2-e^2 x^2\right )^p}{(d+e x)^2} \, dx=\int \frac {{\left (d^2-e^2\,x^2\right )}^p\,{\left (g\,x\right )}^m}{{\left (d+e\,x\right )}^2} \,d x \]
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