Optimal. Leaf size=22 \[ x \, _2F_1\left (\frac {1}{2},-p;\frac {3}{2};\frac {e^2 x^2}{d^2}\right ) \]
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Rubi [A] time = 0.00, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {245} \[ x \, _2F_1\left (\frac {1}{2},-p;\frac {3}{2};\frac {e^2 x^2}{d^2}\right ) \]
Antiderivative was successfully verified.
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Rule 245
Rubi steps
\begin {align*} \int \left (1-\frac {e^2 x^2}{d^2}\right )^p \, dx &=x \, _2F_1\left (\frac {1}{2},-p;\frac {3}{2};\frac {e^2 x^2}{d^2}\right )\\ \end {align*}
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Mathematica [A] time = 0.00, size = 22, normalized size = 1.00 \[ x \, _2F_1\left (\frac {1}{2},-p;\frac {3}{2};\frac {e^2 x^2}{d^2}\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 1.09, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (-\frac {e^{2} x^{2} - d^{2}}{d^{2}}\right )^{p}, x\right ) \]
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 \[ \int {\left (-\frac {e^{2} x^{2}}{d^{2}} + 1\right )}^{p}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.57, size = 21, normalized size = 0.95 \[ x \hypergeom \left (\left [\frac {1}{2}, -p \right ], \left [\frac {3}{2}\right ], \frac {e^{2} x^{2}}{d^{2}}\right ) \]
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 \[ \int {\left (-\frac {e^{2} x^{2}}{d^{2}} + 1\right )}^{p}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.21, size = 19, normalized size = 0.86 \[ x\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},-p;\ \frac {3}{2};\ \frac {e^2\,x^2}{d^2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 1.07, size = 24, normalized size = 1.09 \[ x {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, - p \\ \frac {3}{2} \end {matrix}\middle | {\frac {e^{2} x^{2} e^{2 i \pi }}{d^{2}}} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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