Optimal. Leaf size=57 \[ -\frac {2^{2+p} d^3 \left (\frac {d-e x}{d}\right )^{1+p} \, _2F_1\left (-2-p,1+p;2+p;\frac {d-e x}{2 d}\right )}{e (1+p)} \]
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Rubi [A]
time = 0.02, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {690, 71}
\begin {gather*} -\frac {d^3 2^{p+2} \left (\frac {d-e x}{d}\right )^{p+1} \, _2F_1\left (-p-2,p+1;p+2;\frac {d-e x}{2 d}\right )}{e (p+1)} \end {gather*}
Antiderivative was successfully verified.
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Rule 71
Rule 690
Rubi steps
\begin {align*} \int (d+e x)^2 \left (1-\frac {e^2 x^2}{d^2}\right )^p \, dx &=\left (d \left (\frac {d-e x}{d}\right )^{1+p} \left (\frac {1}{d}-\frac {e x}{d^2}\right )^{-1-p}\right ) \int \left (\frac {1}{d}-\frac {e x}{d^2}\right )^p \left (1+\frac {e x}{d}\right )^{2+p} \, dx\\ &=-\frac {2^{2+p} d^3 \left (\frac {d-e x}{d}\right )^{1+p} \, _2F_1\left (-2-p,1+p;2+p;\frac {d-e x}{2 d}\right )}{e (1+p)}\\ \end {align*}
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Mathematica [A]
time = 0.24, size = 86, normalized size = 1.51 \begin {gather*} -\frac {d^3 \left (1-\frac {e^2 x^2}{d^2}\right )^{1+p}}{e (1+p)}+d^2 x \, _2F_1\left (\frac {1}{2},-p;\frac {3}{2};\frac {e^2 x^2}{d^2}\right )+\frac {1}{3} e^2 x^3 \, _2F_1\left (\frac {3}{2},-p;\frac {5}{2};\frac {e^2 x^2}{d^2}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.56, size = 75, normalized size = 1.32
method | result | size |
meijerg | \(\frac {e^{2} x^{3} \hypergeom \left (\left [\frac {3}{2}, -p \right ], \left [\frac {5}{2}\right ], \frac {e^{2} x^{2}}{d^{2}}\right )}{3}+e d \,x^{2} \hypergeom \left (\left [1, -p \right ], \left [2\right ], \frac {e^{2} x^{2}}{d^{2}}\right )+d^{2} x \hypergeom \left (\left [\frac {1}{2}, -p \right ], \left [\frac {3}{2}\right ], \frac {e^{2} x^{2}}{d^{2}}\right )\) | \(75\) |
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 [C] Result contains complex when optimal does not.
time = 1.85, size = 116, normalized size = 2.04 \begin {gather*} d^{2} x {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, - p \\ \frac {3}{2} \end {matrix}\middle | {\frac {e^{2} x^{2} e^{2 i \pi }}{d^{2}}} \right )} + 2 d e \left (\begin {cases} \frac {x^{2}}{2} & \text {for}\: e^{2} = 0 \\- \frac {d^{2} \left (\begin {cases} \frac {\left (1 - \frac {e^{2} x^{2}}{d^{2}}\right )^{p + 1}}{p + 1} & \text {for}\: p \neq -1 \\\log {\left (1 - \frac {e^{2} x^{2}}{d^{2}} \right )} & \text {otherwise} \end {cases}\right )}{2 e^{2}} & \text {otherwise} \end {cases}\right ) + \frac {e^{2} x^{3} {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{2}, - p \\ \frac {5}{2} \end {matrix}\middle | {\frac {e^{2} x^{2} e^{2 i \pi }}{d^{2}}} \right )}}{3} \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 = -0.02 \begin {gather*} \int {\left (1-\frac {e^2\,x^2}{d^2}\right )}^p\,{\left (d+e\,x\right )}^2 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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