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

Optimal. Leaf size=18 \[ \frac{\text{EllipticF}\left (\sin ^{-1}\left (\sqrt [4]{\frac{2}{3}} x\right ),-1\right )}{\sqrt [4]{6}} \]

[Out]

EllipticF[ArcSin[(2/3)^(1/4)*x], -1]/6^(1/4)

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Rubi [A]  time = 0.0060963, antiderivative size = 18, 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 = {221} \[ \frac{F\left (\left .\sin ^{-1}\left (\sqrt [4]{\frac{2}{3}} x\right )\right |-1\right )}{\sqrt [4]{6}} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[3 - 2*x^4],x]

[Out]

EllipticF[ArcSin[(2/3)^(1/4)*x], -1]/6^(1/4)

Rule 221

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

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{3-2 x^4}} \, dx &=\frac{F\left (\left .\sin ^{-1}\left (\sqrt [4]{\frac{2}{3}} x\right )\right |-1\right )}{\sqrt [4]{6}}\\ \end{align*}

Mathematica [A]  time = 0.0210378, size = 18, normalized size = 1. \[ \frac{\text{EllipticF}\left (\sin ^{-1}\left (\sqrt [4]{\frac{2}{3}} x\right ),-1\right )}{\sqrt [4]{6}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[3 - 2*x^4],x]

[Out]

EllipticF[ArcSin[(2/3)^(1/4)*x], -1]/6^(1/4)

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Maple [B]  time = 0.148, size = 54, normalized size = 3. \begin{align*}{\frac{\sqrt{3}{6}^{{\frac{3}{4}}}}{54}\sqrt{9-3\,{x}^{2}\sqrt{6}}\sqrt{9+3\,{x}^{2}\sqrt{6}}{\it EllipticF} \left ({\frac{x\sqrt{3}\sqrt [4]{6}}{3}},i \right ){\frac{1}{\sqrt{-2\,{x}^{4}+3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/54*3^(1/2)*6^(3/4)*(9-3*x^2*6^(1/2))^(1/2)*(9+3*x^2*6^(1/2))^(1/2)/(-2*x^4+3)^(1/2)*EllipticF(1/3*x*3^(1/2)*
6^(1/4),I)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/sqrt(-2*x^4 + 3), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.639526, size = 37, normalized size = 2.06 \begin{align*} \frac{\sqrt{3} x \Gamma \left (\frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{1}{2} \\ \frac{5}{4} \end{matrix}\middle |{\frac{2 x^{4} e^{2 i \pi }}{3}} \right )}}{12 \Gamma \left (\frac{5}{4}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(3)*x*gamma(1/4)*hyper((1/4, 1/2), (5/4,), 2*x**4*exp_polar(2*I*pi)/3)/(12*gamma(5/4))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/sqrt(-2*x^4 + 3), x)