3.963 \(\int (1-\frac{e^2 x^2}{d^2})^p \, dx\)

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

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Rubi [A]  time = 0.0049424, antiderivative size = 22, normalized size of antiderivative = 1., 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.

[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 245

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*}

Mathematica [A]  time = 0.0020593, size = 22, normalized size = 1. \[ x \, _2F_1\left (\frac{1}{2},-p;\frac{3}{2};\frac{e^2 x^2}{d^2}\right ) \]

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 = 0.343, size = 21, normalized size = 1. \begin{align*} x{\mbox{$_2$F$_1$}({\frac{1}{2}},-p;\,{\frac{3}{2}};\,{\frac{{e}^{2}{x}^{2}}{{d}^{2}}})} \end{align*}

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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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (-\frac{e^{2} x^{2}}{d^{2}} + 1\right )}^{p}\,{d x} \end{align*}

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((-e^2*x^2/d^2 + 1)^p, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\left (-\frac{e^{2} x^{2} - d^{2}}{d^{2}}\right )^{p}, x\right ) \end{align*}

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((-(e^2*x^2 - d^2)/d^2)^p, x)

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Sympy [C]  time = 1.0558, size = 24, normalized size = 1.09 \begin{align*} 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{align*}

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., size = 0, normalized size = 0. \begin{align*} \int{\left (-\frac{e^{2} x^{2}}{d^{2}} + 1\right )}^{p}\,{d x} \end{align*}

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((-e^2*x^2/d^2 + 1)^p, x)