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

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

[Out]

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

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Rubi [A]  time = 0.0101381, 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} \[ \frac{F\left (\left .\sin ^{-1}\left (\sqrt [4]{\frac{3}{2}} x\right )\right |-1\right )}{\sqrt [4]{6}} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[2 - 3*x^4],x]

[Out]

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

Mathematica [A]  time = 0.0220728, size = 18, normalized size = 1. \[ \frac{\text{EllipticF}\left (\sin ^{-1}\left (\sqrt [4]{\frac{3}{2}} x\right ),-1\right )}{\sqrt [4]{6}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[2 - 3*x^4],x]

[Out]

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

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Maple [B]  time = 0.191, size = 54, normalized size = 3. \begin{align*}{\frac{\sqrt{2}{6}^{{\frac{3}{4}}}}{24}\sqrt{4-2\,{x}^{2}\sqrt{6}}\sqrt{4+2\,{x}^{2}\sqrt{6}}{\it EllipticF} \left ({\frac{x\sqrt{2}\sqrt [4]{6}}{2}},i \right ){\frac{1}{\sqrt{-3\,{x}^{4}+2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*2^(1/2)*6^(3/4)*(4-2*x^2*6^(1/2))^(1/2)*(4+2*x^2*6^(1/2))^(1/2)/(-3*x^4+2)^(1/2)*EllipticF(1/2*x*2^(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{-3 \, x^{4} + 2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/sqrt(-3*x^4 + 2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.624791, size = 37, normalized size = 2.06 \begin{align*} \frac{\sqrt{2} 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{3 x^{4} e^{2 i \pi }}{2}} \right )}}{8 \Gamma \left (\frac{5}{4}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(2)*x*gamma(1/4)*hyper((1/4, 1/2), (5/4,), 3*x**4*exp_polar(2*I*pi)/2)/(8*gamma(5/4))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/sqrt(-3*x^4 + 2), x)