Optimal. Leaf size=193 \[ -\frac {2 d^5 \left (d^2-e^2 x^2\right )^{1+p}}{e^4 (1+p)}-\frac {e x^5 \left (d^2-e^2 x^2\right )^{1+p}}{7+2 p}+\frac {7 d^3 \left (d^2-e^2 x^2\right )^{2+p}}{2 e^4 (2+p)}-\frac {3 d \left (d^2-e^2 x^2\right )^{3+p}}{2 e^4 (3+p)}+\frac {2 d^2 e (13+3 p) x^5 \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac {5}{2},-p;\frac {7}{2};\frac {e^2 x^2}{d^2}\right )}{5 (7+2 p)} \]
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Rubi [A]
time = 0.11, antiderivative size = 193, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {1666, 457, 78,
470, 372, 371} \begin {gather*} \frac {2 d^2 e (3 p+13) 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 (2 p+7)}-\frac {e x^5 \left (d^2-e^2 x^2\right )^{p+1}}{2 p+7}-\frac {3 d \left (d^2-e^2 x^2\right )^{p+3}}{2 e^4 (p+3)}-\frac {2 d^5 \left (d^2-e^2 x^2\right )^{p+1}}{e^4 (p+1)}+\frac {7 d^3 \left (d^2-e^2 x^2\right )^{p+2}}{2 e^4 (p+2)} \end {gather*}
Antiderivative was successfully verified.
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Rule 78
Rule 371
Rule 372
Rule 457
Rule 470
Rule 1666
Rubi steps
\begin {align*} \int x^3 (d+e x)^3 \left (d^2-e^2 x^2\right )^p \, dx &=\int x^3 \left (d^2-e^2 x^2\right )^p \left (d^3+3 d e^2 x^2\right ) \, dx+\int x^4 \left (d^2-e^2 x^2\right )^p \left (3 d^2 e+e^3 x^2\right ) \, dx\\ &=-\frac {e x^5 \left (d^2-e^2 x^2\right )^{1+p}}{7+2 p}+\frac {1}{2} \text {Subst}\left (\int x \left (d^2-e^2 x\right )^p \left (d^3+3 d e^2 x\right ) \, dx,x,x^2\right )+\frac {\left (2 d^2 e (13+3 p)\right ) \int x^4 \left (d^2-e^2 x^2\right )^p \, dx}{7+2 p}\\ &=-\frac {e x^5 \left (d^2-e^2 x^2\right )^{1+p}}{7+2 p}+\frac {1}{2} \text {Subst}\left (\int \left (\frac {4 d^5 \left (d^2-e^2 x\right )^p}{e^2}-\frac {7 d^3 \left (d^2-e^2 x\right )^{1+p}}{e^2}+\frac {3 d \left (d^2-e^2 x\right )^{2+p}}{e^2}\right ) \, dx,x,x^2\right )+\frac {\left (2 d^2 e (13+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 )^p \, dx}{7+2 p}\\ &=-\frac {2 d^5 \left (d^2-e^2 x^2\right )^{1+p}}{e^4 (1+p)}-\frac {e x^5 \left (d^2-e^2 x^2\right )^{1+p}}{7+2 p}+\frac {7 d^3 \left (d^2-e^2 x^2\right )^{2+p}}{2 e^4 (2+p)}-\frac {3 d \left (d^2-e^2 x^2\right )^{3+p}}{2 e^4 (3+p)}+\frac {2 d^2 e (13+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},-p;\frac {7}{2};\frac {e^2 x^2}{d^2}\right )}{5 (7+2 p)}\\ \end {align*}
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Mathematica [A]
time = 0.34, size = 187, normalized size = 0.97 \begin {gather*} \frac {\left (d^2-e^2 x^2\right )^p \left (-\frac {35 d \left (d^2-e^2 x^2\right ) \left (d^4 (9+p)+d^2 e^2 \left (9+10 p+p^2\right ) x^2+3 e^4 \left (2+3 p+p^2\right ) x^4\right )}{(1+p) (2+p) (3+p)}+42 d^2 e^5 x^5 \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 )+10 e^7 x^7 \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac {7}{2},-p;\frac {9}{2};\frac {e^2 x^2}{d^2}\right )\right )}{70 e^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int x^{3} \left (e x +d \right )^{3} \left (-e^{2} x^{2}+d^{2}\right )^{p}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 348 vs.
\(2 (165) = 330\).
time = 3.41, size = 1370, normalized size = 7.10 \begin {gather*} d^{3} \left (\begin {cases} \frac {x^{4} \left (d^{2}\right )^{p}}{4} & \text {for}\: e = 0 \\- \frac {d^{2} \log {\left (- \frac {d}{e} + x \right )}}{- 2 d^{2} e^{4} + 2 e^{6} x^{2}} - \frac {d^{2} \log {\left (\frac {d}{e} + x \right )}}{- 2 d^{2} e^{4} + 2 e^{6} x^{2}} - \frac {d^{2}}{- 2 d^{2} e^{4} + 2 e^{6} x^{2}} + \frac {e^{2} x^{2} \log {\left (- \frac {d}{e} + x \right )}}{- 2 d^{2} e^{4} + 2 e^{6} x^{2}} + \frac {e^{2} x^{2} \log {\left (\frac {d}{e} + x \right )}}{- 2 d^{2} e^{4} + 2 e^{6} x^{2}} & \text {for}\: p = -2 \\- \frac {d^{2} \log {\left (- \frac {d}{e} + x \right )}}{2 e^{4}} - \frac {d^{2} \log {\left (\frac {d}{e} + x \right )}}{2 e^{4}} - \frac {x^{2}}{2 e^{2}} & \text {for}\: p = -1 \\- \frac {d^{4} \left (d^{2} - e^{2} x^{2}\right )^{p}}{2 e^{4} p^{2} + 6 e^{4} p + 4 e^{4}} - \frac {d^{2} e^{2} p x^{2} \left (d^{2} - e^{2} x^{2}\right )^{p}}{2 e^{4} p^{2} + 6 e^{4} p + 4 e^{4}} + \frac {e^{4} p x^{4} \left (d^{2} - e^{2} x^{2}\right )^{p}}{2 e^{4} p^{2} + 6 e^{4} p + 4 e^{4}} + \frac {e^{4} x^{4} \left (d^{2} - e^{2} x^{2}\right )^{p}}{2 e^{4} p^{2} + 6 e^{4} p + 4 e^{4}} & \text {otherwise} \end {cases}\right ) + \frac {3 d^{2} d^{2 p} e 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} + 3 d e^{2} \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^{3} 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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int x^3\,{\left (d^2-e^2\,x^2\right )}^p\,{\left (d+e\,x\right )}^3 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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