3.887 \(\int \frac{1}{(2-3 x^2)^{3/4}} \, dx\)

Optimal. Leaf size=27 \[ \frac{2^{3/4} \text{EllipticF}\left (\frac{1}{2} \sin ^{-1}\left (\sqrt{\frac{3}{2}} x\right ),2\right )}{\sqrt{3}} \]

[Out]

(2^(3/4)*EllipticF[ArcSin[Sqrt[3/2]*x]/2, 2])/Sqrt[3]

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Rubi [A]  time = 0.0033811, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {232} \[ \frac{2^{3/4} F\left (\left .\frac{1}{2} \sin ^{-1}\left (\sqrt{\frac{3}{2}} x\right )\right |2\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

[In]

Int[(2 - 3*x^2)^(-3/4),x]

[Out]

(2^(3/4)*EllipticF[ArcSin[Sqrt[3/2]*x]/2, 2])/Sqrt[3]

Rule 232

Int[((a_) + (b_.)*(x_)^2)^(-3/4), x_Symbol] :> Simp[(2*EllipticF[(1*ArcSin[Rt[-(b/a), 2]*x])/2, 2])/(a^(3/4)*R
t[-(b/a), 2]), x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && NegQ[b/a]

Rubi steps

\begin{align*} \int \frac{1}{\left (2-3 x^2\right )^{3/4}} \, dx &=\frac{2^{3/4} F\left (\left .\frac{1}{2} \sin ^{-1}\left (\sqrt{\frac{3}{2}} x\right )\right |2\right )}{\sqrt{3}}\\ \end{align*}

Mathematica [A]  time = 0.002695, size = 27, normalized size = 1. \[ \frac{2^{3/4} \text{EllipticF}\left (\frac{1}{2} \sin ^{-1}\left (\sqrt{\frac{3}{2}} x\right ),2\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

[In]

Integrate[(2 - 3*x^2)^(-3/4),x]

[Out]

(2^(3/4)*EllipticF[ArcSin[Sqrt[3/2]*x]/2, 2])/Sqrt[3]

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Maple [C]  time = 0.02, size = 18, normalized size = 0.7 \begin{align*}{\frac{\sqrt [4]{2}x}{2}{\mbox{$_2$F$_1$}({\frac{1}{2}},{\frac{3}{4}};\,{\frac{3}{2}};\,{\frac{3\,{x}^{2}}{2}})}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(-3*x^2+2)^(3/4),x)

[Out]

1/2*2^(1/4)*x*hypergeom([1/2,3/4],[3/2],3/2*x^2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (-3 \, x^{2} + 2\right )}^{\frac{3}{4}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-3*x^2+2)^(3/4),x, algorithm="maxima")

[Out]

integrate((-3*x^2 + 2)^(-3/4), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}}}{3 \, x^{2} - 2}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-3*x^2+2)^(3/4),x, algorithm="fricas")

[Out]

integral(-(-3*x^2 + 2)^(1/4)/(3*x^2 - 2), x)

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Sympy [C]  time = 0.610807, size = 27, normalized size = 1. \begin{align*} \frac{\sqrt [4]{2} x{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{3}{4} \\ \frac{3}{2} \end{matrix}\middle |{\frac{3 x^{2} e^{2 i \pi }}{2}} \right )}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-3*x**2+2)**(3/4),x)

[Out]

2**(1/4)*x*hyper((1/2, 3/4), (3/2,), 3*x**2*exp_polar(2*I*pi)/2)/2

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (-3 \, x^{2} + 2\right )}^{\frac{3}{4}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-3*x^2+2)^(3/4),x, algorithm="giac")

[Out]

integrate((-3*x^2 + 2)^(-3/4), x)