Integrand size = 25, antiderivative size = 194 \[ \int \frac {x^3 \left (d^2-e^2 x^2\right )^p}{(d+e x)^3} \, dx=\frac {2 d^5 \left (d^2-e^2 x^2\right )^{-2+p}}{e^4 (2-p)}+\frac {e x^5 \left (d^2-e^2 x^2\right )^{-2+p}}{1+2 p}-\frac {7 d^3 \left (d^2-e^2 x^2\right )^{-1+p}}{2 e^4 (1-p)}-\frac {3 d \left (d^2-e^2 x^2\right )^p}{2 e^4 p}-\frac {2 e (4+3 p) x^5 \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {5}{2},3-p,\frac {7}{2},\frac {e^2 x^2}{d^2}\right )}{5 d^4 (1+2 p)} \]
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Time = 0.15 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {866, 1666, 457, 78, 470, 372, 371} \[ \int \frac {x^3 \left (d^2-e^2 x^2\right )^p}{(d+e x)^3} \, dx=\frac {e x^5 \left (d^2-e^2 x^2\right )^{p-2}}{2 p+1}-\frac {3 d \left (d^2-e^2 x^2\right )^p}{2 e^4 p}+\frac {2 d^5 \left (d^2-e^2 x^2\right )^{p-2}}{e^4 (2-p)}-\frac {2 e (3 p+4) x^5 \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \left (d^2-e^2 x^2\right )^p \operatorname {Hypergeometric2F1}\left (\frac {5}{2},3-p,\frac {7}{2},\frac {e^2 x^2}{d^2}\right )}{5 d^4 (2 p+1)}-\frac {7 d^3 \left (d^2-e^2 x^2\right )^{p-1}}{2 e^4 (1-p)} \]
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Rule 78
Rule 371
Rule 372
Rule 457
Rule 470
Rule 866
Rule 1666
Rubi steps \begin{align*} \text {integral}& = \int x^3 (d-e x)^3 \left (d^2-e^2 x^2\right )^{-3+p} \, dx \\ & = \int x^3 \left (d^2-e^2 x^2\right )^{-3+p} \left (d^3+3 d e^2 x^2\right ) \, dx+\int x^4 \left (d^2-e^2 x^2\right )^{-3+p} \left (-3 d^2 e-e^3 x^2\right ) \, dx \\ & = \frac {e x^5 \left (d^2-e^2 x^2\right )^{-2+p}}{1+2 p}+\frac {1}{2} \text {Subst}\left (\int x \left (d^2-e^2 x\right )^{-3+p} \left (d^3+3 d e^2 x\right ) \, dx,x,x^2\right )-\frac {\left (2 d^2 e (4+3 p)\right ) \int x^4 \left (d^2-e^2 x^2\right )^{-3+p} \, dx}{1+2 p} \\ & = \frac {e x^5 \left (d^2-e^2 x^2\right )^{-2+p}}{1+2 p}+\frac {1}{2} \text {Subst}\left (\int \left (\frac {4 d^5 \left (d^2-e^2 x\right )^{-3+p}}{e^2}-\frac {7 d^3 \left (d^2-e^2 x\right )^{-2+p}}{e^2}+\frac {3 d \left (d^2-e^2 x\right )^{-1+p}}{e^2}\right ) \, dx,x,x^2\right )-\frac {\left (2 e (4+3 p) \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p}\right ) \int x^4 \left (1-\frac {e^2 x^2}{d^2}\right )^{-3+p} \, dx}{d^4 (1+2 p)} \\ & = \frac {2 d^5 \left (d^2-e^2 x^2\right )^{-2+p}}{e^4 (2-p)}+\frac {e x^5 \left (d^2-e^2 x^2\right )^{-2+p}}{1+2 p}-\frac {7 d^3 \left (d^2-e^2 x^2\right )^{-1+p}}{2 e^4 (1-p)}-\frac {3 d \left (d^2-e^2 x^2\right )^p}{2 e^4 p}-\frac {2 e (4+3 p) x^5 \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac {5}{2},3-p;\frac {7}{2};\frac {e^2 x^2}{d^2}\right )}{5 d^4 (1+2 p)} \\ \end{align*}
Time = 0.38 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.04 \[ \int \frac {x^3 \left (d^2-e^2 x^2\right )^p}{(d+e x)^3} \, dx=\frac {2^{-3+p} \left (1+\frac {e x}{d}\right )^{-p} \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \left (8 e (1+p) x \left (\frac {1}{2}+\frac {e x}{2 d}\right )^p \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-p,\frac {3}{2},\frac {e^2 x^2}{d^2}\right )+(d-e x) \left (1-\frac {e^2 x^2}{d^2}\right )^p \left (12 \operatorname {Hypergeometric2F1}\left (1-p,1+p,2+p,\frac {d-e x}{2 d}\right )-6 \operatorname {Hypergeometric2F1}\left (2-p,1+p,2+p,\frac {d-e x}{2 d}\right )+\operatorname {Hypergeometric2F1}\left (3-p,1+p,2+p,\frac {d-e x}{2 d}\right )\right )\right )}{e^4 (1+p)} \]
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\[\int \frac {x^{3} \left (-e^{2} x^{2}+d^{2}\right )^{p}}{\left (e x +d \right )^{3}}d x\]
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\[ \int \frac {x^3 \left (d^2-e^2 x^2\right )^p}{(d+e x)^3} \, dx=\int { \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{p} x^{3}}{{\left (e x + d\right )}^{3}} \,d x } \]
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\[ \int \frac {x^3 \left (d^2-e^2 x^2\right )^p}{(d+e x)^3} \, dx=\int \frac {x^{3} \left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{p}}{\left (d + e x\right )^{3}}\, dx \]
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\[ \int \frac {x^3 \left (d^2-e^2 x^2\right )^p}{(d+e x)^3} \, dx=\int { \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{p} x^{3}}{{\left (e x + d\right )}^{3}} \,d x } \]
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\[ \int \frac {x^3 \left (d^2-e^2 x^2\right )^p}{(d+e x)^3} \, dx=\int { \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{p} x^{3}}{{\left (e x + d\right )}^{3}} \,d x } \]
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Timed out. \[ \int \frac {x^3 \left (d^2-e^2 x^2\right )^p}{(d+e x)^3} \, dx=\int \frac {x^3\,{\left (d^2-e^2\,x^2\right )}^p}{{\left (d+e\,x\right )}^3} \,d x \]
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