Optimal. Leaf size=120 \[ \frac {1}{5} e x^5 \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac {5}{2},-p;\frac {7}{2};\frac {e^2 x^2}{d^2}\right )+\frac {d \left (d^2-e^2 x^2\right )^{p+2}}{2 e^4 (p+2)}-\frac {d^3 \left (d^2-e^2 x^2\right )^{p+1}}{2 e^4 (p+1)} \]
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Rubi [A] time = 0.07, antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {764, 266, 43, 365, 364} \[ \frac {1}{5} e x^5 \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac {5}{2},-p;\frac {7}{2};\frac {e^2 x^2}{d^2}\right )-\frac {d^3 \left (d^2-e^2 x^2\right )^{p+1}}{2 e^4 (p+1)}+\frac {d \left (d^2-e^2 x^2\right )^{p+2}}{2 e^4 (p+2)} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rule 364
Rule 365
Rule 764
Rubi steps
\begin {align*} \int x^3 (d+e x) \left (d^2-e^2 x^2\right )^p \, dx &=d \int x^3 \left (d^2-e^2 x^2\right )^p \, dx+e \int x^4 \left (d^2-e^2 x^2\right )^p \, dx\\ &=\frac {1}{2} d \operatorname {Subst}\left (\int x \left (d^2-e^2 x\right )^p \, dx,x,x^2\right )+\left (e \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} e 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} d \operatorname {Subst}\left (\int \left (\frac {d^2 \left (d^2-e^2 x\right )^p}{e^2}-\frac {\left (d^2-e^2 x\right )^{1+p}}{e^2}\right ) \, dx,x,x^2\right )\\ &=-\frac {d^3 \left (d^2-e^2 x^2\right )^{1+p}}{2 e^4 (1+p)}+\frac {d \left (d^2-e^2 x^2\right )^{2+p}}{2 e^4 (2+p)}+\frac {1}{5} e 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*}
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Mathematica [A] time = 0.08, size = 106, normalized size = 0.88 \[ \frac {\left (d^2-e^2 x^2\right )^p \left (2 e^5 x^5 \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac {5}{2},-p;\frac {7}{2};\frac {e^2 x^2}{d^2}\right )-\frac {5 d \left (d^2-e^2 x^2\right ) \left (d^2+e^2 (p+1) x^2\right )}{(p+1) (p+2)}\right )}{10 e^4} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.92, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (e x^{4} + d x^{3}\right )} {\left (-e^{2} x^{2} + d^{2}\right )}^{p}, x\right ) \]
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 \[ \int {\left (e x + d\right )} {\left (-e^{2} x^{2} + d^{2}\right )}^{p} x^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.05, size = 0, normalized size = 0.00 \[ \int \left (e x +d \right ) x^{3} \left (-e^{2} x^{2}+d^{2}\right )^{p}\, dx \]
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 \[ e \int x^{4} e^{\left (p \log \left (e x + d\right ) + p \log \left (-e x + d\right )\right )}\,{d x} + \frac {{\left (e^{4} {\left (p + 1\right )} x^{4} - d^{2} e^{2} p x^{2} - d^{4}\right )} {\left (-e^{2} x^{2} + d^{2}\right )}^{p} d}{2 \, {\left (p^{2} + 3 \, p + 2\right )} e^{4}} \]
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 \[ \int x^3\,{\left (d^2-e^2\,x^2\right )}^p\,\left (d+e\,x\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 5.46, size = 382, normalized size = 3.18 \[ d \left (\begin {cases} \frac {x^{4} \left (d^{2}\right )^{p}}{4} & \text {for}\: e = 0 \\- \frac {d^{2} \log {\left (- \frac {d}{e} + x \right )}}{- 2 d^{2} e^{4} + 2 e^{6} x^{2}} - \frac {d^{2} \log {\left (\frac {d}{e} + x \right )}}{- 2 d^{2} e^{4} + 2 e^{6} x^{2}} - \frac {d^{2}}{- 2 d^{2} e^{4} + 2 e^{6} x^{2}} + \frac {e^{2} x^{2} \log {\left (- \frac {d}{e} + x \right )}}{- 2 d^{2} e^{4} + 2 e^{6} x^{2}} + \frac {e^{2} x^{2} \log {\left (\frac {d}{e} + x \right )}}{- 2 d^{2} e^{4} + 2 e^{6} x^{2}} & \text {for}\: p = -2 \\- \frac {d^{2} \log {\left (- \frac {d}{e} + x \right )}}{2 e^{4}} - \frac {d^{2} \log {\left (\frac {d}{e} + x \right )}}{2 e^{4}} - \frac {x^{2}}{2 e^{2}} & \text {for}\: p = -1 \\- \frac {d^{4} \left (d^{2} - e^{2} x^{2}\right )^{p}}{2 e^{4} p^{2} + 6 e^{4} p + 4 e^{4}} - \frac {d^{2} e^{2} p x^{2} \left (d^{2} - e^{2} x^{2}\right )^{p}}{2 e^{4} p^{2} + 6 e^{4} p + 4 e^{4}} + \frac {e^{4} p x^{4} \left (d^{2} - e^{2} x^{2}\right )^{p}}{2 e^{4} p^{2} + 6 e^{4} p + 4 e^{4}} + \frac {e^{4} x^{4} \left (d^{2} - e^{2} x^{2}\right )^{p}}{2 e^{4} p^{2} + 6 e^{4} p + 4 e^{4}} & \text {otherwise} \end {cases}\right ) + \frac {d^{2 p} e x^{5} {{}_{2}F_{1}\left (\begin {matrix} \frac {5}{2}, - p \\ \frac {7}{2} \end {matrix}\middle | {\frac {e^{2} x^{2} e^{2 i \pi }}{d^{2}}} \right )}}{5} \]
Verification of antiderivative is not currently implemented for this CAS.
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