3.1.21 \(\int \frac {1}{\sqrt {2-3 x^4}} \, dx\) [21]

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

[Out]

1/6*EllipticF(1/2*3^(1/4)*2^(3/4)*x,I)*6^(3/4)

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Rubi [A]
time = 0.01, antiderivative size = 18, normalized size of antiderivative = 1.00, 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 = {227} \begin {gather*} \frac {F\left (\left .\text {ArcSin}\left (\sqrt [4]{\frac {3}{2}} x\right )\right |-1\right )}{\sqrt [4]{6}} \end {gather*}

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 227

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*}

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Mathematica [A]
time = 10.04, size = 18, normalized size = 1.00 \begin {gather*} \frac {F\left (\left .\sin ^{-1}\left (\sqrt [4]{\frac {3}{2}} x\right )\right |-1\right )}{\sqrt [4]{6}} \end {gather*}

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] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (17 ) = 34\).
time = 0.12, size = 54, normalized size = 3.00

method	result	size
meijerg	\(\frac {\sqrt {2}\, x \hypergeom \left (\left [\frac {1}{4}, \frac {1}{2}\right ], \left [\frac {5}{4}\right ], \frac {3 x^{4}}{2}\right )}{2}\)	\(18\)
default	\(\frac {\sqrt {2}\, 6^{\frac {3}{4}} \sqrt {4-2 x^{2} \sqrt {6}}\, \sqrt {4+2 x^{2} \sqrt {6}}\, \EllipticF \left (\frac {x \sqrt {2}\, 6^{\frac {1}{4}}}{2}, i\right )}{24 \sqrt {-3 x^{4}+2}}\)	\(54\)
elliptic	\(\frac {\sqrt {2}\, 6^{\frac {3}{4}} \sqrt {4-2 x^{2} \sqrt {6}}\, \sqrt {4+2 x^{2} \sqrt {6}}\, \EllipticF \left (\frac {x \sqrt {2}\, 6^{\frac {1}{4}}}{2}, i\right )}{24 \sqrt {-3 x^{4}+2}}\)	\(54\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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 [A]
time = 0.07, size = 16, normalized size = 0.89 \begin {gather*} \frac {1}{6} \cdot 6^{\frac {3}{4}} {\rm ellipticF}\left (\frac {1}{2} \cdot 6^{\frac {1}{4}} \sqrt {2} x, -1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*6^(3/4)*ellipticF(1/2*6^(1/4)*sqrt(2)*x, -1)

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Sympy [A]
time = 0.36, size = 37, normalized size = 2.06 \begin {gather*} \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 {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [B]
time = 4.20, size = 16, normalized size = 0.89 \begin {gather*} \frac {\sqrt {2}\,x\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{4},\frac {1}{2};\ \frac {5}{4};\ \frac {3\,x^4}{2}\right )}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(2 - 3*x^4)^(1/2),x)

[Out]

(2^(1/2)*x*hypergeom([1/4, 1/2], 5/4, (3*x^4)/2))/2

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