Integrand size = 23, antiderivative size = 147 \[ \int x^4 (d+e x) \left (d^2-e^2 x^2\right )^p \, dx=-\frac {d^4 \left (d^2-e^2 x^2\right )^{1+p}}{2 e^5 (1+p)}+\frac {d^2 \left (d^2-e^2 x^2\right )^{2+p}}{e^5 (2+p)}-\frac {\left (d^2-e^2 x^2\right )^{3+p}}{2 e^5 (3+p)}+\frac {1}{5} d x^5 \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {5}{2},-p,\frac {7}{2},\frac {e^2 x^2}{d^2}\right ) \]
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Time = 0.06 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {778, 372, 371, 272, 45} \[ \int x^4 (d+e x) \left (d^2-e^2 x^2\right )^p \, dx=\frac {1}{5} d x^5 \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {5}{2},-p,\frac {7}{2},\frac {e^2 x^2}{d^2}\right )+\frac {d^2 \left (d^2-e^2 x^2\right )^{p+2}}{e^5 (p+2)}-\frac {\left (d^2-e^2 x^2\right )^{p+3}}{2 e^5 (p+3)}-\frac {d^4 \left (d^2-e^2 x^2\right )^{p+1}}{2 e^5 (p+1)} \]
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Rule 45
Rule 272
Rule 371
Rule 372
Rule 778
Rubi steps \begin{align*} \text {integral}& = d \int x^4 \left (d^2-e^2 x^2\right )^p \, dx+e \int x^5 \left (d^2-e^2 x^2\right )^p \, dx \\ & = \frac {1}{2} e \text {Subst}\left (\int x^2 \left (d^2-e^2 x\right )^p \, dx,x,x^2\right )+\left (d \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 )^p \, dx \\ & = \frac {1}{5} d 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},-p;\frac {7}{2};\frac {e^2 x^2}{d^2}\right )+\frac {1}{2} e \text {Subst}\left (\int \left (\frac {d^4 \left (d^2-e^2 x\right )^p}{e^4}-\frac {2 d^2 \left (d^2-e^2 x\right )^{1+p}}{e^4}+\frac {\left (d^2-e^2 x\right )^{2+p}}{e^4}\right ) \, dx,x,x^2\right ) \\ & = -\frac {d^4 \left (d^2-e^2 x^2\right )^{1+p}}{2 e^5 (1+p)}+\frac {d^2 \left (d^2-e^2 x^2\right )^{2+p}}{e^5 (2+p)}-\frac {\left (d^2-e^2 x^2\right )^{3+p}}{2 e^5 (3+p)}+\frac {1}{5} d 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},-p;\frac {7}{2};\frac {e^2 x^2}{d^2}\right ) \\ \end{align*}
Time = 0.21 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.88 \[ \int x^4 (d+e x) \left (d^2-e^2 x^2\right )^p \, dx=\frac {1}{10} \left (d^2-e^2 x^2\right )^p \left (-\frac {5 \left (d^2-e^2 x^2\right ) \left (2 d^4+2 d^2 e^2 (1+p) x^2+e^4 \left (2+3 p+p^2\right ) x^4\right )}{e^5 (1+p) (2+p) (3+p)}+2 d x^5 \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {5}{2},-p,\frac {7}{2},\frac {e^2 x^2}{d^2}\right )\right ) \]
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\[\int x^{4} \left (e x +d \right ) \left (-e^{2} x^{2}+d^{2}\right )^{p}d x\]
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\[ \int x^4 (d+e x) \left (d^2-e^2 x^2\right )^p \, dx=\int { {\left (e x + d\right )} {\left (-e^{2} x^{2} + d^{2}\right )}^{p} x^{4} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 937 vs. \(2 (119) = 238\).
Time = 1.87 (sec) , antiderivative size = 972, normalized size of antiderivative = 6.61 \[ \int x^4 (d+e x) \left (d^2-e^2 x^2\right )^p \, dx=\frac {d d^{2 p} x^{5} {{}_{2}F_{1}\left (\begin {matrix} \frac {5}{2}, - p \\ \frac {7}{2} \end {matrix}\middle | {\frac {e^{2} x^{2} e^{2 i \pi }}{d^{2}}} \right )}}{5} + e \left (\begin {cases} \frac {x^{6} \left (d^{2}\right )^{p}}{6} & \text {for}\: e = 0 \\- \frac {2 d^{4} \log {\left (- \frac {d}{e} + x \right )}}{4 d^{4} e^{6} - 8 d^{2} e^{8} x^{2} + 4 e^{10} x^{4}} - \frac {2 d^{4} \log {\left (\frac {d}{e} + x \right )}}{4 d^{4} e^{6} - 8 d^{2} e^{8} x^{2} + 4 e^{10} x^{4}} - \frac {3 d^{4}}{4 d^{4} e^{6} - 8 d^{2} e^{8} x^{2} + 4 e^{10} x^{4}} + \frac {4 d^{2} e^{2} x^{2} \log {\left (- \frac {d}{e} + x \right )}}{4 d^{4} e^{6} - 8 d^{2} e^{8} x^{2} + 4 e^{10} x^{4}} + \frac {4 d^{2} e^{2} x^{2} \log {\left (\frac {d}{e} + x \right )}}{4 d^{4} e^{6} - 8 d^{2} e^{8} x^{2} + 4 e^{10} x^{4}} + \frac {4 d^{2} e^{2} x^{2}}{4 d^{4} e^{6} - 8 d^{2} e^{8} x^{2} + 4 e^{10} x^{4}} - \frac {2 e^{4} x^{4} \log {\left (- \frac {d}{e} + x \right )}}{4 d^{4} e^{6} - 8 d^{2} e^{8} x^{2} + 4 e^{10} x^{4}} - \frac {2 e^{4} x^{4} \log {\left (\frac {d}{e} + x \right )}}{4 d^{4} e^{6} - 8 d^{2} e^{8} x^{2} + 4 e^{10} x^{4}} & \text {for}\: p = -3 \\- \frac {2 d^{4} \log {\left (- \frac {d}{e} + x \right )}}{- 2 d^{2} e^{6} + 2 e^{8} x^{2}} - \frac {2 d^{4} \log {\left (\frac {d}{e} + x \right )}}{- 2 d^{2} e^{6} + 2 e^{8} x^{2}} - \frac {2 d^{4}}{- 2 d^{2} e^{6} + 2 e^{8} x^{2}} + \frac {2 d^{2} e^{2} x^{2} \log {\left (- \frac {d}{e} + x \right )}}{- 2 d^{2} e^{6} + 2 e^{8} x^{2}} + \frac {2 d^{2} e^{2} x^{2} \log {\left (\frac {d}{e} + x \right )}}{- 2 d^{2} e^{6} + 2 e^{8} x^{2}} + \frac {e^{4} x^{4}}{- 2 d^{2} e^{6} + 2 e^{8} x^{2}} & \text {for}\: p = -2 \\- \frac {d^{4} \log {\left (- \frac {d}{e} + x \right )}}{2 e^{6}} - \frac {d^{4} \log {\left (\frac {d}{e} + x \right )}}{2 e^{6}} - \frac {d^{2} x^{2}}{2 e^{4}} - \frac {x^{4}}{4 e^{2}} & \text {for}\: p = -1 \\- \frac {2 d^{6} \left (d^{2} - e^{2} x^{2}\right )^{p}}{2 e^{6} p^{3} + 12 e^{6} p^{2} + 22 e^{6} p + 12 e^{6}} - \frac {2 d^{4} e^{2} p x^{2} \left (d^{2} - e^{2} x^{2}\right )^{p}}{2 e^{6} p^{3} + 12 e^{6} p^{2} + 22 e^{6} p + 12 e^{6}} - \frac {d^{2} e^{4} p^{2} x^{4} \left (d^{2} - e^{2} x^{2}\right )^{p}}{2 e^{6} p^{3} + 12 e^{6} p^{2} + 22 e^{6} p + 12 e^{6}} - \frac {d^{2} e^{4} p x^{4} \left (d^{2} - e^{2} x^{2}\right )^{p}}{2 e^{6} p^{3} + 12 e^{6} p^{2} + 22 e^{6} p + 12 e^{6}} + \frac {e^{6} p^{2} x^{6} \left (d^{2} - e^{2} x^{2}\right )^{p}}{2 e^{6} p^{3} + 12 e^{6} p^{2} + 22 e^{6} p + 12 e^{6}} + \frac {3 e^{6} p x^{6} \left (d^{2} - e^{2} x^{2}\right )^{p}}{2 e^{6} p^{3} + 12 e^{6} p^{2} + 22 e^{6} p + 12 e^{6}} + \frac {2 e^{6} x^{6} \left (d^{2} - e^{2} x^{2}\right )^{p}}{2 e^{6} p^{3} + 12 e^{6} p^{2} + 22 e^{6} p + 12 e^{6}} & \text {otherwise} \end {cases}\right ) \]
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\[ \int x^4 (d+e x) \left (d^2-e^2 x^2\right )^p \, dx=\int { {\left (e x + d\right )} {\left (-e^{2} x^{2} + d^{2}\right )}^{p} x^{4} \,d x } \]
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\[ \int x^4 (d+e x) \left (d^2-e^2 x^2\right )^p \, dx=\int { {\left (e x + d\right )} {\left (-e^{2} x^{2} + d^{2}\right )}^{p} x^{4} \,d x } \]
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Timed out. \[ \int x^4 (d+e x) \left (d^2-e^2 x^2\right )^p \, dx=\int x^4\,{\left (d^2-e^2\,x^2\right )}^p\,\left (d+e\,x\right ) \,d x \]
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