3.10.63 \(\int (1-\frac {e^2 x^2}{d^2})^p \, dx\) [963]

Optimal. Leaf size=22 \[ x \, _2F_1\left (\frac {1}{2},-p;\frac {3}{2};\frac {e^2 x^2}{d^2}\right ) \]

[Out]

x*hypergeom([1/2, -p],[3/2],e^2*x^2/d^2)

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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 = {251} \begin {gather*} x \, _2F_1\left (\frac {1}{2},-p;\frac {3}{2};\frac {e^2 x^2}{d^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 - (e^2*x^2)/d^2)^p,x]

[Out]

x*Hypergeometric2F1[1/2, -p, 3/2, (e^2*x^2)/d^2]

Rule 251

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*x*Hypergeometric2F1[-p, 1/n, 1/n + 1, (-b)*(x^n/a)],
x] /; FreeQ[{a, b, n, p}, x] &&  !IGtQ[p, 0] &&  !IntegerQ[1/n] &&  !ILtQ[Simplify[1/n + p], 0] && (IntegerQ[p
] || GtQ[a, 0])

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 = 0.05, size = 22, normalized size = 1.00 \begin {gather*} x \, _2F_1\left (\frac {1}{2},-p;\frac {3}{2};\frac {e^2 x^2}{d^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 - (e^2*x^2)/d^2)^p,x]

[Out]

x*Hypergeometric2F1[1/2, -p, 3/2, (e^2*x^2)/d^2]

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Maple [A]
time = 0.44, size = 21, normalized size = 0.95

method	result	size
meijerg	\(x \hypergeom \left (\left [\frac {1}{2}, -p \right ], \left [\frac {3}{2}\right ], \frac {e^{2} x^{2}}{d^{2}}\right )\)	\(21\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1-e^2*x^2/d^2)^p,x,method=_RETURNVERBOSE)

[Out]

x*hypergeom([1/2,-p],[3/2],e^2*x^2/d^2)

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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.

[In]

integrate((1-e^2*x^2/d^2)^p,x, algorithm="maxima")

[Out]

integrate((-x^2*e^2/d^2 + 1)^p, x)

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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.

[In]

integrate((1-e^2*x^2/d^2)^p,x, algorithm="fricas")

[Out]

integral((-(x^2*e^2 - d^2)/d^2)^p, x)

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Sympy [C] Result contains complex when optimal does not.
time = 0.49, size = 24, normalized size = 1.09 \begin {gather*} 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 )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-e**2*x**2/d**2)**p,x)

[Out]

x*hyper((1/2, -p), (3/2,), e**2*x**2*exp_polar(2*I*pi)/d**2)

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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.

[In]

integrate((1-e^2*x^2/d^2)^p,x, algorithm="giac")

[Out]

integrate((-x^2*e^2/d^2 + 1)^p, x)

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Mupad [B]
time = 1.21, size = 19, normalized size = 0.86 \begin {gather*} x\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},-p;\ \frac {3}{2};\ \frac {e^2\,x^2}{d^2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1 - (e^2*x^2)/d^2)^p,x)

[Out]

x*hypergeom([1/2, -p], 3/2, (e^2*x^2)/d^2)

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