Integrand size = 25, antiderivative size = 185 \[ \int x^4 (d+e x)^2 \left (d^2-e^2 x^2\right )^p \, dx=-\frac {d^5 \left (d^2-e^2 x^2\right )^{1+p}}{e^5 (1+p)}-\frac {x^5 \left (d^2-e^2 x^2\right )^{1+p}}{7+2 p}+\frac {2 d^3 \left (d^2-e^2 x^2\right )^{2+p}}{e^5 (2+p)}-\frac {d \left (d^2-e^2 x^2\right )^{3+p}}{e^5 (3+p)}+\frac {2 d^2 (6+p) x^5 \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {5}{2},-p,\frac {7}{2},\frac {e^2 x^2}{d^2}\right )}{5 (7+2 p)} \]
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Time = 0.11 (sec) , antiderivative size = 185, 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 = {1666, 470, 372, 371, 12, 272, 45} \[ \int x^4 (d+e x)^2 \left (d^2-e^2 x^2\right )^p \, dx=\frac {2 d^2 (p+6) x^5 \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {5}{2},-p,\frac {7}{2},\frac {e^2 x^2}{d^2}\right )}{5 (2 p+7)}-\frac {x^5 \left (d^2-e^2 x^2\right )^{p+1}}{2 p+7}-\frac {d \left (d^2-e^2 x^2\right )^{p+3}}{e^5 (p+3)}-\frac {d^5 \left (d^2-e^2 x^2\right )^{p+1}}{e^5 (p+1)}+\frac {2 d^3 \left (d^2-e^2 x^2\right )^{p+2}}{e^5 (p+2)} \]
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Rule 12
Rule 45
Rule 272
Rule 371
Rule 372
Rule 470
Rule 1666
Rubi steps \begin{align*} \text {integral}& = \int 2 d e x^5 \left (d^2-e^2 x^2\right )^p \, dx+\int x^4 \left (d^2-e^2 x^2\right )^p \left (d^2+e^2 x^2\right ) \, dx \\ & = -\frac {x^5 \left (d^2-e^2 x^2\right )^{1+p}}{7+2 p}+(2 d e) \int x^5 \left (d^2-e^2 x^2\right )^p \, dx+\frac {\left (2 d^2 (6+p)\right ) \int x^4 \left (d^2-e^2 x^2\right )^p \, dx}{7+2 p} \\ & = -\frac {x^5 \left (d^2-e^2 x^2\right )^{1+p}}{7+2 p}+(d e) \text {Subst}\left (\int x^2 \left (d^2-e^2 x\right )^p \, dx,x,x^2\right )+\frac {\left (2 d^2 (6+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 )^p \, dx}{7+2 p} \\ & = -\frac {x^5 \left (d^2-e^2 x^2\right )^{1+p}}{7+2 p}+\frac {2 d^2 (6+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},-p;\frac {7}{2};\frac {e^2 x^2}{d^2}\right )}{5 (7+2 p)}+(d 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^5 \left (d^2-e^2 x^2\right )^{1+p}}{e^5 (1+p)}-\frac {x^5 \left (d^2-e^2 x^2\right )^{1+p}}{7+2 p}+\frac {2 d^3 \left (d^2-e^2 x^2\right )^{2+p}}{e^5 (2+p)}-\frac {d \left (d^2-e^2 x^2\right )^{3+p}}{e^5 (3+p)}+\frac {2 d^2 (6+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},-p;\frac {7}{2};\frac {e^2 x^2}{d^2}\right )}{5 (7+2 p)} \\ \end{align*}
Time = 0.29 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.01 \[ \int x^4 (d+e x)^2 \left (d^2-e^2 x^2\right )^p \, dx=\frac {1}{35} \left (d^2-e^2 x^2\right )^p \left (-\frac {35 d^5 \left (d^2-e^2 x^2\right )}{e^5 (1+p)}+\frac {70 d^3 \left (d^2-e^2 x^2\right )^2}{e^5 (2+p)}-\frac {35 d \left (d^2-e^2 x^2\right )^3}{e^5 (3+p)}+7 d^2 x^5 \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {5}{2},-p,\frac {7}{2},\frac {e^2 x^2}{d^2}\right )+5 e^2 x^7 \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {7}{2},-p,\frac {9}{2},\frac {e^2 x^2}{d^2}\right )\right ) \]
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\[\int x^{4} \left (e x +d \right )^{2} \left (-e^{2} x^{2}+d^{2}\right )^{p}d x\]
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\[ \int x^4 (d+e x)^2 \left (d^2-e^2 x^2\right )^p \, dx=\int { {\left (e x + d\right )}^{2} {\left (-e^{2} x^{2} + d^{2}\right )}^{p} x^{4} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 940 vs. \(2 (153) = 306\).
Time = 2.39 (sec) , antiderivative size = 1015, normalized size of antiderivative = 5.49 \[ \int x^4 (d+e x)^2 \left (d^2-e^2 x^2\right )^p \, dx=\frac {d^{2} 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} + 2 d 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 ) + \frac {d^{2 p} e^{2} x^{7} {{}_{2}F_{1}\left (\begin {matrix} \frac {7}{2}, - p \\ \frac {9}{2} \end {matrix}\middle | {\frac {e^{2} x^{2} e^{2 i \pi }}{d^{2}}} \right )}}{7} \]
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\[ \int x^4 (d+e x)^2 \left (d^2-e^2 x^2\right )^p \, dx=\int { {\left (e x + d\right )}^{2} {\left (-e^{2} x^{2} + d^{2}\right )}^{p} x^{4} \,d x } \]
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\[ \int x^4 (d+e x)^2 \left (d^2-e^2 x^2\right )^p \, dx=\int { {\left (e x + d\right )}^{2} {\left (-e^{2} x^{2} + d^{2}\right )}^{p} x^{4} \,d x } \]
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Timed out. \[ \int x^4 (d+e x)^2 \left (d^2-e^2 x^2\right )^p \, dx=\int x^4\,{\left (d^2-e^2\,x^2\right )}^p\,{\left (d+e\,x\right )}^2 \,d x \]
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