Integrand size = 17, antiderivative size = 322 \[ \int \frac {\left (a+c x^2\right )^p}{(d+e x)^3} \, dx=-\frac {d^2 e \left (a+c x^2\right )^{1+p}}{4 \left (c d^2+a e^2\right ) \left (d^2-e^2 x^2\right )^2}+\frac {x \left (a+c x^2\right )^p \left (1+\frac {c x^2}{a}\right )^{-p} \operatorname {AppellF1}\left (\frac {1}{2},-p,3,\frac {3}{2},-\frac {c x^2}{a},\frac {e^2 x^2}{d^2}\right )}{d^3}+\frac {e^2 x^3 \left (a+c x^2\right )^p \left (1+\frac {c x^2}{a}\right )^{-p} \operatorname {AppellF1}\left (\frac {3}{2},-p,3,\frac {5}{2},-\frac {c x^2}{a},\frac {e^2 x^2}{d^2}\right )}{d^5}+\frac {c e \left (2 a e^2+c d^2 (1+p)\right ) \left (a+c x^2\right )^{1+p} \operatorname {Hypergeometric2F1}\left (2,1+p,2+p,\frac {e^2 \left (a+c x^2\right )}{c d^2+a e^2}\right )}{4 \left (c d^2+a e^2\right )^3 (1+p)}-\frac {3 c^2 d^2 e \left (a+c x^2\right )^{1+p} \operatorname {Hypergeometric2F1}\left (3,1+p,2+p,\frac {e^2 \left (a+c x^2\right )}{c d^2+a e^2}\right )}{2 \left (c d^2+a e^2\right )^3 (1+p)} \]
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Time = 0.21 (sec) , antiderivative size = 322, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.529, Rules used = {771, 441, 440, 455, 70, 525, 524, 457, 79} \[ \int \frac {\left (a+c x^2\right )^p}{(d+e x)^3} \, dx=\frac {e^2 x^3 \left (a+c x^2\right )^p \left (\frac {c x^2}{a}+1\right )^{-p} \operatorname {AppellF1}\left (\frac {3}{2},-p,3,\frac {5}{2},-\frac {c x^2}{a},\frac {e^2 x^2}{d^2}\right )}{d^5}+\frac {x \left (a+c x^2\right )^p \left (\frac {c x^2}{a}+1\right )^{-p} \operatorname {AppellF1}\left (\frac {1}{2},-p,3,\frac {3}{2},-\frac {c x^2}{a},\frac {e^2 x^2}{d^2}\right )}{d^3}-\frac {3 c^2 d^2 e \left (a+c x^2\right )^{p+1} \operatorname {Hypergeometric2F1}\left (3,p+1,p+2,\frac {e^2 \left (c x^2+a\right )}{c d^2+a e^2}\right )}{2 (p+1) \left (a e^2+c d^2\right )^3}+\frac {c e \left (a+c x^2\right )^{p+1} \left (2 a e^2+c d^2 (p+1)\right ) \operatorname {Hypergeometric2F1}\left (2,p+1,p+2,\frac {e^2 \left (c x^2+a\right )}{c d^2+a e^2}\right )}{4 (p+1) \left (a e^2+c d^2\right )^3}-\frac {d^2 e \left (a+c x^2\right )^{p+1}}{4 \left (d^2-e^2 x^2\right )^2 \left (a e^2+c d^2\right )} \]
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Rule 70
Rule 79
Rule 440
Rule 441
Rule 455
Rule 457
Rule 524
Rule 525
Rule 771
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {d^3 \left (a+c x^2\right )^p}{\left (d^2-e^2 x^2\right )^3}-\frac {3 d^2 e x \left (a+c x^2\right )^p}{\left (d^2-e^2 x^2\right )^3}+\frac {3 d e^2 x^2 \left (a+c x^2\right )^p}{\left (d^2-e^2 x^2\right )^3}+\frac {e^3 x^3 \left (a+c x^2\right )^p}{\left (-d^2+e^2 x^2\right )^3}\right ) \, dx \\ & = d^3 \int \frac {\left (a+c x^2\right )^p}{\left (d^2-e^2 x^2\right )^3} \, dx-\left (3 d^2 e\right ) \int \frac {x \left (a+c x^2\right )^p}{\left (d^2-e^2 x^2\right )^3} \, dx+\left (3 d e^2\right ) \int \frac {x^2 \left (a+c x^2\right )^p}{\left (d^2-e^2 x^2\right )^3} \, dx+e^3 \int \frac {x^3 \left (a+c x^2\right )^p}{\left (-d^2+e^2 x^2\right )^3} \, dx \\ & = -\left (\frac {1}{2} \left (3 d^2 e\right ) \text {Subst}\left (\int \frac {(a+c x)^p}{\left (d^2-e^2 x\right )^3} \, dx,x,x^2\right )\right )+\frac {1}{2} e^3 \text {Subst}\left (\int \frac {x (a+c x)^p}{\left (-d^2+e^2 x\right )^3} \, dx,x,x^2\right )+\left (d^3 \left (a+c x^2\right )^p \left (1+\frac {c x^2}{a}\right )^{-p}\right ) \int \frac {\left (1+\frac {c x^2}{a}\right )^p}{\left (d^2-e^2 x^2\right )^3} \, dx+\left (3 d e^2 \left (a+c x^2\right )^p \left (1+\frac {c x^2}{a}\right )^{-p}\right ) \int \frac {x^2 \left (1+\frac {c x^2}{a}\right )^p}{\left (d^2-e^2 x^2\right )^3} \, dx \\ & = -\frac {d^2 e \left (a+c x^2\right )^{1+p}}{4 \left (c d^2+a e^2\right ) \left (d^2-e^2 x^2\right )^2}+\frac {x \left (a+c x^2\right )^p \left (1+\frac {c x^2}{a}\right )^{-p} F_1\left (\frac {1}{2};-p,3;\frac {3}{2};-\frac {c x^2}{a},\frac {e^2 x^2}{d^2}\right )}{d^3}+\frac {e^2 x^3 \left (a+c x^2\right )^p \left (1+\frac {c x^2}{a}\right )^{-p} F_1\left (\frac {3}{2};-p,3;\frac {5}{2};-\frac {c x^2}{a},\frac {e^2 x^2}{d^2}\right )}{d^5}-\frac {3 c^2 d^2 e \left (a+c x^2\right )^{1+p} \, _2F_1\left (3,1+p;2+p;\frac {e^2 \left (a+c x^2\right )}{c d^2+a e^2}\right )}{2 \left (c d^2+a e^2\right )^3 (1+p)}+\frac {\left (e \left (2 a e^2+c d^2 (1+p)\right )\right ) \text {Subst}\left (\int \frac {(a+c x)^p}{\left (-d^2+e^2 x\right )^2} \, dx,x,x^2\right )}{4 \left (c d^2+a e^2\right )} \\ & = -\frac {d^2 e \left (a+c x^2\right )^{1+p}}{4 \left (c d^2+a e^2\right ) \left (d^2-e^2 x^2\right )^2}+\frac {x \left (a+c x^2\right )^p \left (1+\frac {c x^2}{a}\right )^{-p} F_1\left (\frac {1}{2};-p,3;\frac {3}{2};-\frac {c x^2}{a},\frac {e^2 x^2}{d^2}\right )}{d^3}+\frac {e^2 x^3 \left (a+c x^2\right )^p \left (1+\frac {c x^2}{a}\right )^{-p} F_1\left (\frac {3}{2};-p,3;\frac {5}{2};-\frac {c x^2}{a},\frac {e^2 x^2}{d^2}\right )}{d^5}+\frac {c e \left (2 a e^2+c d^2 (1+p)\right ) \left (a+c x^2\right )^{1+p} \, _2F_1\left (2,1+p;2+p;\frac {e^2 \left (a+c x^2\right )}{c d^2+a e^2}\right )}{4 \left (c d^2+a e^2\right )^3 (1+p)}-\frac {3 c^2 d^2 e \left (a+c x^2\right )^{1+p} \, _2F_1\left (3,1+p;2+p;\frac {e^2 \left (a+c x^2\right )}{c d^2+a e^2}\right )}{2 \left (c d^2+a e^2\right )^3 (1+p)} \\ \end{align*}
Time = 0.39 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.44 \[ \int \frac {\left (a+c x^2\right )^p}{(d+e x)^3} \, dx=\frac {\left (\frac {e \left (-\sqrt {-\frac {a}{c}}+x\right )}{d+e x}\right )^{-p} \left (\frac {e \left (\sqrt {-\frac {a}{c}}+x\right )}{d+e x}\right )^{-p} \left (a+c x^2\right )^p \operatorname {AppellF1}\left (2-2 p,-p,-p,3-2 p,\frac {d-\sqrt {-\frac {a}{c}} e}{d+e x},\frac {d+\sqrt {-\frac {a}{c}} e}{d+e x}\right )}{2 e (-1+p) (d+e x)^2} \]
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\[\int \frac {\left (c \,x^{2}+a \right )^{p}}{\left (e x +d \right )^{3}}d x\]
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\[ \int \frac {\left (a+c x^2\right )^p}{(d+e x)^3} \, dx=\int { \frac {{\left (c x^{2} + a\right )}^{p}}{{\left (e x + d\right )}^{3}} \,d x } \]
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Timed out. \[ \int \frac {\left (a+c x^2\right )^p}{(d+e x)^3} \, dx=\text {Timed out} \]
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\[ \int \frac {\left (a+c x^2\right )^p}{(d+e x)^3} \, dx=\int { \frac {{\left (c x^{2} + a\right )}^{p}}{{\left (e x + d\right )}^{3}} \,d x } \]
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\[ \int \frac {\left (a+c x^2\right )^p}{(d+e x)^3} \, dx=\int { \frac {{\left (c x^{2} + a\right )}^{p}}{{\left (e x + d\right )}^{3}} \,d x } \]
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Timed out. \[ \int \frac {\left (a+c x^2\right )^p}{(d+e x)^3} \, dx=\int \frac {{\left (c\,x^2+a\right )}^p}{{\left (d+e\,x\right )}^3} \,d x \]
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