3.873 \(\int \frac{1}{\sqrt [4]{2-3 x^2}} \, dx\)

Optimal. Leaf size=28 \[ \frac{2 \sqrt [4]{2} E\left (\left .\frac{1}{2} \sin ^{-1}\left (\sqrt{\frac{3}{2}} x\right )\right |2\right )}{\sqrt{3}} \]

[Out]

(2*2^(1/4)*EllipticE[ArcSin[Sqrt[3/2]*x]/2, 2])/Sqrt[3]

________________________________________________________________________________________

Rubi [A]  time = 0.0030895, antiderivative size = 28, 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 = {228} \[ \frac{2 \sqrt [4]{2} E\left (\left .\frac{1}{2} \sin ^{-1}\left (\sqrt{\frac{3}{2}} x\right )\right |2\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

[In]

Int[(2 - 3*x^2)^(-1/4),x]

[Out]

(2*2^(1/4)*EllipticE[ArcSin[Sqrt[3/2]*x]/2, 2])/Sqrt[3]

Rule 228

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

Rubi steps

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

Mathematica [C]  time = 0.0030215, size = 24, normalized size = 0.86 \[ \frac{x \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{3}{2};\frac{3 x^2}{2}\right )}{\sqrt [4]{2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(2 - 3*x^2)^(-1/4),x]

[Out]

(x*Hypergeometric2F1[1/4, 1/2, 3/2, (3*x^2)/2])/2^(1/4)

________________________________________________________________________________________

Maple [C]  time = 0.018, size = 18, normalized size = 0.6 \begin{align*}{\frac{{2}^{{\frac{3}{4}}}x}{2}{\mbox{$_2$F$_1$}({\frac{1}{4}},{\frac{1}{2}};\,{\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)^(1/4),x)

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((-3*x^2 + 2)^(-1/4), x)

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

Sympy [C]  time = 0.589562, size = 27, normalized size = 0.96 \begin{align*} \frac{2^{\frac{3}{4}} x{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{1}{2} \\ \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)**(1/4),x)

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((-3*x^2 + 2)^(-1/4), x)