Integrand size = 20, antiderivative size = 163 \[ \int \frac {x^3 \left (a+b x^2\right )^p}{d+e x} \, dx=-\frac {d \left (a+b x^2\right )^{1+p}}{2 b e^2 (1+p)}-\frac {e x^5 \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \operatorname {AppellF1}\left (\frac {5}{2},-p,1,\frac {7}{2},-\frac {b x^2}{a},\frac {e^2 x^2}{d^2}\right )}{5 d^2}+\frac {d^3 \left (a+b x^2\right )^{1+p} \operatorname {Hypergeometric2F1}\left (1,1+p,2+p,\frac {e^2 \left (a+b x^2\right )}{b d^2+a e^2}\right )}{2 e^2 \left (b d^2+a e^2\right ) (1+p)} \]
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Time = 0.10 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {973, 457, 81, 70, 525, 524} \[ \int \frac {x^3 \left (a+b x^2\right )^p}{d+e x} \, dx=-\frac {e x^5 \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \operatorname {AppellF1}\left (\frac {5}{2},-p,1,\frac {7}{2},-\frac {b x^2}{a},\frac {e^2 x^2}{d^2}\right )}{5 d^2}+\frac {d^3 \left (a+b x^2\right )^{p+1} \operatorname {Hypergeometric2F1}\left (1,p+1,p+2,\frac {e^2 \left (b x^2+a\right )}{b d^2+a e^2}\right )}{2 e^2 (p+1) \left (a e^2+b d^2\right )}-\frac {d \left (a+b x^2\right )^{p+1}}{2 b e^2 (p+1)} \]
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Rule 70
Rule 81
Rule 457
Rule 524
Rule 525
Rule 973
Rubi steps \begin{align*} \text {integral}& = d \int \frac {x^3 \left (a+b x^2\right )^p}{d^2-e^2 x^2} \, dx-e \int \frac {x^4 \left (a+b x^2\right )^p}{d^2-e^2 x^2} \, dx \\ & = \frac {1}{2} d \text {Subst}\left (\int \frac {x (a+b x)^p}{d^2-e^2 x} \, dx,x,x^2\right )-\left (e \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p}\right ) \int \frac {x^4 \left (1+\frac {b x^2}{a}\right )^p}{d^2-e^2 x^2} \, dx \\ & = -\frac {d \left (a+b x^2\right )^{1+p}}{2 b e^2 (1+p)}-\frac {e x^5 \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} F_1\left (\frac {5}{2};-p,1;\frac {7}{2};-\frac {b x^2}{a},\frac {e^2 x^2}{d^2}\right )}{5 d^2}+\frac {d^3 \text {Subst}\left (\int \frac {(a+b x)^p}{d^2-e^2 x} \, dx,x,x^2\right )}{2 e^2} \\ & = -\frac {d \left (a+b x^2\right )^{1+p}}{2 b e^2 (1+p)}-\frac {e x^5 \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} F_1\left (\frac {5}{2};-p,1;\frac {7}{2};-\frac {b x^2}{a},\frac {e^2 x^2}{d^2}\right )}{5 d^2}+\frac {d^3 \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^2 \left (b d^2+a e^2\right ) (1+p)} \\ \end{align*}
Time = 0.33 (sec) , antiderivative size = 260, normalized size of antiderivative = 1.60 \[ \int \frac {x^3 \left (a+b x^2\right )^p}{d+e x} \, dx=\frac {\left (a+b x^2\right )^p \left (-\frac {3 d^3 \left (\frac {e \left (-\sqrt {-\frac {a}{b}}+x\right )}{d+e x}\right )^{-p} \left (\frac {e \left (\sqrt {-\frac {a}{b}}+x\right )}{d+e x}\right )^{-p} \operatorname {AppellF1}\left (-2 p,-p,-p,1-2 p,\frac {d-\sqrt {-\frac {a}{b}} e}{d+e x},\frac {d+\sqrt {-\frac {a}{b}} e}{d+e x}\right )}{p}+\frac {e \left (1+\frac {b x^2}{a}\right )^{-p} \left (6 b d^2 (1+p) x \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-p,\frac {3}{2},-\frac {b x^2}{a}\right )+e \left (-3 d \left (b x^2 \left (1+\frac {b x^2}{a}\right )^p+a \left (-1+\left (1+\frac {b x^2}{a}\right )^p\right )\right )+2 b e (1+p) x^3 \operatorname {Hypergeometric2F1}\left (\frac {3}{2},-p,\frac {5}{2},-\frac {b x^2}{a}\right )\right )\right )}{b (1+p)}\right )}{6 e^4} \]
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\[\int \frac {x^{3} \left (b \,x^{2}+a \right )^{p}}{e x +d}d x\]
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\[ \int \frac {x^3 \left (a+b x^2\right )^p}{d+e x} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{p} x^{3}}{e x + d} \,d x } \]
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Timed out. \[ \int \frac {x^3 \left (a+b x^2\right )^p}{d+e x} \, dx=\text {Timed out} \]
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\[ \int \frac {x^3 \left (a+b x^2\right )^p}{d+e x} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{p} x^{3}}{e x + d} \,d x } \]
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\[ \int \frac {x^3 \left (a+b x^2\right )^p}{d+e x} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{p} x^{3}}{e x + d} \,d x } \]
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Timed out. \[ \int \frac {x^3 \left (a+b x^2\right )^p}{d+e x} \, dx=\int \frac {x^3\,{\left (b\,x^2+a\right )}^p}{d+e\,x} \,d x \]
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