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

Optimal result

Integrand size = 15, antiderivative size = 22 \[ \int \left (1-\frac {e^2 x^2}{d^2}\right )^p \, dx=x \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-p,\frac {3}{2},\frac {e^2 x^2}{d^2}\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {251} \[ \int \left (1-\frac {e^2 x^2}{d^2}\right )^p \, dx=x \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-p,\frac {3}{2},\frac {e^2 x^2}{d^2}\right ) \]

[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*} \text {integral}& = x \, _2F_1\left (\frac {1}{2},-p;\frac {3}{2};\frac {e^2 x^2}{d^2}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \left (1-\frac {e^2 x^2}{d^2}\right )^p \, dx=x \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-p,\frac {3}{2},\frac {e^2 x^2}{d^2}\right ) \]

[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]

Maple [A] (verified)

Time = 2.35 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.95

method	result	size
meijerg	\(x {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (\frac {1}{2},-p ;\frac {3}{2};\frac {e^{2} x^{2}}{d^{2}}\right )\)	\(21\)

[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)

Fricas [F]

\[ \int \left (1-\frac {e^2 x^2}{d^2}\right )^p \, dx=\int { {\left (-\frac {e^{2} x^{2}}{d^{2}} + 1\right )}^{p} \,d x } \]

[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)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.58 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \left (1-\frac {e^2 x^2}{d^2}\right )^p \, dx=x {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, - p \\ \frac {3}{2} \end {matrix}\middle | {\frac {e^{2} x^{2} e^{2 i \pi }}{d^{2}}} \right )} \]

[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)

Maxima [F]

\[ \int \left (1-\frac {e^2 x^2}{d^2}\right )^p \, dx=\int { {\left (-\frac {e^{2} x^{2}}{d^{2}} + 1\right )}^{p} \,d x } \]

[In]

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

[Out]

integrate((-e^2*x^2/d^2 + 1)^p, x)

Giac [F]

\[ \int \left (1-\frac {e^2 x^2}{d^2}\right )^p \, dx=\int { {\left (-\frac {e^{2} x^{2}}{d^{2}} + 1\right )}^{p} \,d x } \]

[In]

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

[Out]

integrate((-e^2*x^2/d^2 + 1)^p, x)

Mupad [B] (verification not implemented)

Time = 10.86 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.86 \[ \int \left (1-\frac {e^2 x^2}{d^2}\right )^p \, dx=x\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},-p;\ \frac {3}{2};\ \frac {e^2\,x^2}{d^2}\right ) \]

[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)