Optimal. Leaf size=57 \[ -\frac {d^3 2^{p+2} \left (\frac {d-e x}{d}\right )^{p+1} \, _2F_1\left (-p-2,p+1;p+2;\frac {d-e x}{2 d}\right )}{e (p+1)} \]
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Rubi [A] time = 0.03, 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 = {676, 69} \[ -\frac {d^3 2^{p+2} \left (\frac {d-e x}{d}\right )^{p+1} \, _2F_1\left (-p-2,p+1;p+2;\frac {d-e x}{2 d}\right )}{e (p+1)} \]
Antiderivative was successfully verified.
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Rule 69
Rule 676
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 = 0.07, size = 86, normalized size = 1.51 \[ d^2 x \, _2F_1\left (\frac {1}{2},-p;\frac {3}{2};\frac {e^2 x^2}{d^2}\right )+\frac {1}{3} e^2 x^3 \, _2F_1\left (\frac {3}{2},-p;\frac {5}{2};\frac {e^2 x^2}{d^2}\right )-\frac {d^3 \left (1-\frac {e^2 x^2}{d^2}\right )^{p+1}}{e (p+1)} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.31, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (e^{2} x^{2} + 2 \, d e x + d^{2}\right )} \left (-\frac {e^{2} x^{2} - d^{2}}{d^{2}}\right )^{p}, x\right ) \]
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 \[ \int {\left (e x + d\right )}^{2} {\left (-\frac {e^{2} x^{2}}{d^{2}} + 1\right )}^{p}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.74, size = 75, normalized size = 1.32 \[ \frac {e^{2} x^{3} \hypergeom \left (\left [\frac {3}{2}, -p \right ], \left [\frac {5}{2}\right ], \frac {e^{2} x^{2}}{d^{2}}\right )}{3}+d e \,x^{2} \hypergeom \left (\left [1, -p \right ], \relax [2], \frac {e^{2} x^{2}}{d^{2}}\right )+d^{2} x \hypergeom \left (\left [\frac {1}{2}, -p \right ], \left [\frac {3}{2}\right ], \frac {e^{2} x^{2}}{d^{2}}\right ) \]
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 \[ \int {\left (e x + d\right )}^{2} {\left (-\frac {e^{2} x^{2}}{d^{2}} + 1\right )}^{p}\,{d x} \]
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 \[ \int {\left (1-\frac {e^2\,x^2}{d^2}\right )}^p\,{\left (d+e\,x\right )}^2 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 4.58, size = 116, normalized size = 2.04 \[ d^{2} x {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, - p \\ \frac {3}{2} \end {matrix}\middle | {\frac {e^{2} x^{2} e^{2 i \pi }}{d^{2}}} \right )} + 2 d e \left (\begin {cases} \frac {x^{2}}{2} & \text {for}\: e^{2} = 0 \\- \frac {d^{2} \left (\begin {cases} \frac {\left (1 - \frac {e^{2} x^{2}}{d^{2}}\right )^{p + 1}}{p + 1} & \text {for}\: p \neq -1 \\\log {\left (1 - \frac {e^{2} x^{2}}{d^{2}} \right )} & \text {otherwise} \end {cases}\right )}{2 e^{2}} & \text {otherwise} \end {cases}\right ) + \frac {e^{2} x^{3} {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{2}, - p \\ \frac {5}{2} \end {matrix}\middle | {\frac {e^{2} x^{2} e^{2 i \pi }}{d^{2}}} \right )}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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