Integrand size = 20, antiderivative size = 396 \[ \int \frac {x^2 \left (a+b x^2\right )^p}{(d+e x)^3} \, dx=-\frac {d^2 \left (a+b x^2\right )^{1+p}}{2 e \left (b d^2+a e^2\right ) (d+e x)^2}+\frac {d \left (2 a e^2+b d^2 (1+p)\right ) \left (a+b x^2\right )^{1+p}}{e \left (b d^2+a e^2\right )^2 (d+e x)}+\frac {\left (a^2 e^4+a b d^2 e^2 (2+5 p)+b^2 d^4 \left (1+3 p+2 p^2\right )\right ) x \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \operatorname {AppellF1}\left (\frac {1}{2},-p,1,\frac {3}{2},-\frac {b x^2}{a},\frac {e^2 x^2}{d^2}\right )}{d e^2 \left (b d^2+a e^2\right )^2}-\frac {b d (1+2 p) \left (2 a e^2+b d^2 (1+p)\right ) x \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-p,\frac {3}{2},-\frac {b x^2}{a}\right )}{e^2 \left (b d^2+a e^2\right )^2}-\frac {\left (a^2 e^4+a b d^2 e^2 (2+5 p)+b^2 d^4 \left (1+3 p+2 p^2\right )\right ) \left (a+b x^2\right )^{1+p} \operatorname {Hypergeometric2F1}\left (1,1+p,2+p,\frac {e^2 \left (a+b x^2\right )}{b d^2+a e^2}\right )}{2 e \left (b d^2+a e^2\right )^3 (1+p)} \]
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Time = 0.36 (sec) , antiderivative size = 396, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {1665, 849, 858, 252, 251, 771, 441, 440, 455, 70} \[ \int \frac {x^2 \left (a+b x^2\right )^p}{(d+e x)^3} \, dx=\frac {x \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \left (a^2 e^4+a b d^2 e^2 (5 p+2)+b^2 d^4 \left (2 p^2+3 p+1\right )\right ) \operatorname {AppellF1}\left (\frac {1}{2},-p,1,\frac {3}{2},-\frac {b x^2}{a},\frac {e^2 x^2}{d^2}\right )}{d e^2 \left (a e^2+b d^2\right )^2}-\frac {\left (a+b x^2\right )^{p+1} \left (a^2 e^4+a b d^2 e^2 (5 p+2)+b^2 d^4 \left (2 p^2+3 p+1\right )\right ) \operatorname {Hypergeometric2F1}\left (1,p+1,p+2,\frac {e^2 \left (b x^2+a\right )}{b d^2+a e^2}\right )}{2 e (p+1) \left (a e^2+b d^2\right )^3}-\frac {b d (2 p+1) x \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \left (2 a e^2+b d^2 (p+1)\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-p,\frac {3}{2},-\frac {b x^2}{a}\right )}{e^2 \left (a e^2+b d^2\right )^2}+\frac {d \left (a+b x^2\right )^{p+1} \left (2 a e^2+b d^2 (p+1)\right )}{e (d+e x) \left (a e^2+b d^2\right )^2}-\frac {d^2 \left (a+b x^2\right )^{p+1}}{2 e (d+e x)^2 \left (a e^2+b d^2\right )} \]
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Rule 70
Rule 251
Rule 252
Rule 440
Rule 441
Rule 455
Rule 771
Rule 849
Rule 858
Rule 1665
Rubi steps \begin{align*} \text {integral}& = -\frac {d^2 \left (a+b x^2\right )^{1+p}}{2 e \left (b d^2+a e^2\right ) (d+e x)^2}-\frac {\int \frac {\left (2 a d-\frac {2 \left (a e^2+b d^2 (1+p)\right ) x}{e}\right ) \left (a+b x^2\right )^p}{(d+e x)^2} \, dx}{2 \left (b d^2+a e^2\right )} \\ & = -\frac {d^2 \left (a+b x^2\right )^{1+p}}{2 e \left (b d^2+a e^2\right ) (d+e x)^2}+\frac {d \left (2 a e^2+b d^2 (1+p)\right ) \left (a+b x^2\right )^{1+p}}{e \left (b d^2+a e^2\right )^2 (d+e x)}+\frac {\int \frac {\left (2 a \left (a e^2+b d^2 p\right )-\frac {2 b d (1+2 p) \left (2 a e^2+b d^2 (1+p)\right ) x}{e}\right ) \left (a+b x^2\right )^p}{d+e x} \, dx}{2 \left (b d^2+a e^2\right )^2} \\ & = -\frac {d^2 \left (a+b x^2\right )^{1+p}}{2 e \left (b d^2+a e^2\right ) (d+e x)^2}+\frac {d \left (2 a e^2+b d^2 (1+p)\right ) \left (a+b x^2\right )^{1+p}}{e \left (b d^2+a e^2\right )^2 (d+e x)}-\frac {\left (b d (1+2 p) \left (2 a e^2+b d^2 (1+p)\right )\right ) \int \left (a+b x^2\right )^p \, dx}{e^2 \left (b d^2+a e^2\right )^2}+\frac {\left (a^2 e^4+a b d^2 e^2 (2+5 p)+b^2 d^4 \left (1+3 p+2 p^2\right )\right ) \int \frac {\left (a+b x^2\right )^p}{d+e x} \, dx}{e^2 \left (b d^2+a e^2\right )^2} \\ & = -\frac {d^2 \left (a+b x^2\right )^{1+p}}{2 e \left (b d^2+a e^2\right ) (d+e x)^2}+\frac {d \left (2 a e^2+b d^2 (1+p)\right ) \left (a+b x^2\right )^{1+p}}{e \left (b d^2+a e^2\right )^2 (d+e x)}+\frac {\left (a^2 e^4+a b d^2 e^2 (2+5 p)+b^2 d^4 \left (1+3 p+2 p^2\right )\right ) \int \left (\frac {d \left (a+b x^2\right )^p}{d^2-e^2 x^2}+\frac {e x \left (a+b x^2\right )^p}{-d^2+e^2 x^2}\right ) \, dx}{e^2 \left (b d^2+a e^2\right )^2}-\frac {\left (b d (1+2 p) \left (2 a e^2+b d^2 (1+p)\right ) \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p}\right ) \int \left (1+\frac {b x^2}{a}\right )^p \, dx}{e^2 \left (b d^2+a e^2\right )^2} \\ & = -\frac {d^2 \left (a+b x^2\right )^{1+p}}{2 e \left (b d^2+a e^2\right ) (d+e x)^2}+\frac {d \left (2 a e^2+b d^2 (1+p)\right ) \left (a+b x^2\right )^{1+p}}{e \left (b d^2+a e^2\right )^2 (d+e x)}-\frac {b d (1+2 p) \left (2 a e^2+b d^2 (1+p)\right ) x \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \, _2F_1\left (\frac {1}{2},-p;\frac {3}{2};-\frac {b x^2}{a}\right )}{e^2 \left (b d^2+a e^2\right )^2}+\frac {\left (d \left (a^2 e^4+a b d^2 e^2 (2+5 p)+b^2 d^4 \left (1+3 p+2 p^2\right )\right )\right ) \int \frac {\left (a+b x^2\right )^p}{d^2-e^2 x^2} \, dx}{e^2 \left (b d^2+a e^2\right )^2}+\frac {\left (a^2 e^4+a b d^2 e^2 (2+5 p)+b^2 d^4 \left (1+3 p+2 p^2\right )\right ) \int \frac {x \left (a+b x^2\right )^p}{-d^2+e^2 x^2} \, dx}{e \left (b d^2+a e^2\right )^2} \\ & = -\frac {d^2 \left (a+b x^2\right )^{1+p}}{2 e \left (b d^2+a e^2\right ) (d+e x)^2}+\frac {d \left (2 a e^2+b d^2 (1+p)\right ) \left (a+b x^2\right )^{1+p}}{e \left (b d^2+a e^2\right )^2 (d+e x)}-\frac {b d (1+2 p) \left (2 a e^2+b d^2 (1+p)\right ) x \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \, _2F_1\left (\frac {1}{2},-p;\frac {3}{2};-\frac {b x^2}{a}\right )}{e^2 \left (b d^2+a e^2\right )^2}+\frac {\left (a^2 e^4+a b d^2 e^2 (2+5 p)+b^2 d^4 \left (1+3 p+2 p^2\right )\right ) \text {Subst}\left (\int \frac {(a+b x)^p}{-d^2+e^2 x} \, dx,x,x^2\right )}{2 e \left (b d^2+a e^2\right )^2}+\frac {\left (d \left (a^2 e^4+a b d^2 e^2 (2+5 p)+b^2 d^4 \left (1+3 p+2 p^2\right )\right ) \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p}\right ) \int \frac {\left (1+\frac {b x^2}{a}\right )^p}{d^2-e^2 x^2} \, dx}{e^2 \left (b d^2+a e^2\right )^2} \\ & = -\frac {d^2 \left (a+b x^2\right )^{1+p}}{2 e \left (b d^2+a e^2\right ) (d+e x)^2}+\frac {d \left (2 a e^2+b d^2 (1+p)\right ) \left (a+b x^2\right )^{1+p}}{e \left (b d^2+a e^2\right )^2 (d+e x)}+\frac {\left (a^2 e^4+a b d^2 e^2 (2+5 p)+b^2 d^4 \left (1+3 p+2 p^2\right )\right ) x \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} F_1\left (\frac {1}{2};-p,1;\frac {3}{2};-\frac {b x^2}{a},\frac {e^2 x^2}{d^2}\right )}{d e^2 \left (b d^2+a e^2\right )^2}-\frac {b d (1+2 p) \left (2 a e^2+b d^2 (1+p)\right ) x \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \, _2F_1\left (\frac {1}{2},-p;\frac {3}{2};-\frac {b x^2}{a}\right )}{e^2 \left (b d^2+a e^2\right )^2}-\frac {\left (a^2 e^4+a b d^2 e^2 (2+5 p)+b^2 d^4 \left (1+3 p+2 p^2\right )\right ) \left (a+b x^2\right )^{1+p} \, _2F_1\left (1,1+p;2+p;\frac {e^2 \left (a+b x^2\right )}{b d^2+a e^2}\right )}{2 e \left (b d^2+a e^2\right )^3 (1+p)} \\ \end{align*}
Time = 0.41 (sec) , antiderivative size = 290, normalized size of antiderivative = 0.73 \[ \int \frac {x^2 \left (a+b x^2\right )^p}{(d+e x)^3} \, dx=\frac {\left (\frac {e \left (-\sqrt {-\frac {a}{b}}+x\right )}{d+e x}\right )^{-p} \left (\frac {e \left (\sqrt {-\frac {a}{b}}+x\right )}{d+e x}\right )^{-p} \left (a+b x^2\right )^p \left (-\frac {4 d \operatorname {AppellF1}\left (1-2 p,-p,-p,2-2 p,\frac {d-\sqrt {-\frac {a}{b}} e}{d+e x},\frac {d+\sqrt {-\frac {a}{b}} e}{d+e x}\right )}{(-1+2 p) (d+e x)}+\frac {d^2 \operatorname {AppellF1}\left (2-2 p,-p,-p,3-2 p,\frac {d-\sqrt {-\frac {a}{b}} e}{d+e x},\frac {d+\sqrt {-\frac {a}{b}} e}{d+e x}\right )}{(-1+p) (d+e x)^2}+\frac {\operatorname {AppellF1}\left (-2 p,-p,-p,1-2 p,\frac {d-\sqrt {-\frac {a}{b}} e}{d+e x},\frac {d+\sqrt {-\frac {a}{b}} e}{d+e x}\right )}{p}\right )}{2 e^3} \]
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\[\int \frac {x^{2} \left (b \,x^{2}+a \right )^{p}}{\left (e x +d \right )^{3}}d x\]
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\[ \int \frac {x^2 \left (a+b x^2\right )^p}{(d+e x)^3} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{p} x^{2}}{{\left (e x + d\right )}^{3}} \,d x } \]
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Timed out. \[ \int \frac {x^2 \left (a+b x^2\right )^p}{(d+e x)^3} \, dx=\text {Timed out} \]
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\[ \int \frac {x^2 \left (a+b x^2\right )^p}{(d+e x)^3} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{p} x^{2}}{{\left (e x + d\right )}^{3}} \,d x } \]
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\[ \int \frac {x^2 \left (a+b x^2\right )^p}{(d+e x)^3} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{p} x^{2}}{{\left (e x + d\right )}^{3}} \,d x } \]
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Timed out. \[ \int \frac {x^2 \left (a+b x^2\right )^p}{(d+e x)^3} \, dx=\int \frac {x^2\,{\left (b\,x^2+a\right )}^p}{{\left (d+e\,x\right )}^3} \,d x \]
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