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

Optimal. Leaf size=199 \[ \frac{2 \sqrt [4]{3 x^2-2} x}{\sqrt{3 x^2-2}+\sqrt{2}}+\frac{\sqrt [4]{2} \sqrt{\frac{x^2}{\left (\sqrt{3 x^2-2}+\sqrt{2}\right )^2}} \left (\sqrt{3 x^2-2}+\sqrt{2}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{3 x^2-2}}{\sqrt [4]{2}}\right )|\frac{1}{2}\right )}{\sqrt{3} x}-\frac{2 \sqrt [4]{2} \sqrt{\frac{x^2}{\left (\sqrt{3 x^2-2}+\sqrt{2}\right )^2}} \left (\sqrt{3 x^2-2}+\sqrt{2}\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{3 x^2-2}}{\sqrt [4]{2}}\right )|\frac{1}{2}\right )}{\sqrt{3} x} \]

[Out]

(2*x*(-2 + 3*x^2)^(1/4))/(Sqrt[2] + Sqrt[-2 + 3*x^2]) - (2*2^(1/4)*Sqrt[x^2/(Sqr
t[2] + Sqrt[-2 + 3*x^2])^2]*(Sqrt[2] + Sqrt[-2 + 3*x^2])*EllipticE[2*ArcTan[(-2
+ 3*x^2)^(1/4)/2^(1/4)], 1/2])/(Sqrt[3]*x) + (2^(1/4)*Sqrt[x^2/(Sqrt[2] + Sqrt[-
2 + 3*x^2])^2]*(Sqrt[2] + Sqrt[-2 + 3*x^2])*EllipticF[2*ArcTan[(-2 + 3*x^2)^(1/4
)/2^(1/4)], 1/2])/(Sqrt[3]*x)

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Rubi [A]  time = 0.206198, antiderivative size = 199, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364 \[ \frac{2 \sqrt [4]{3 x^2-2} x}{\sqrt{3 x^2-2}+\sqrt{2}}+\frac{\sqrt [4]{2} \sqrt{\frac{x^2}{\left (\sqrt{3 x^2-2}+\sqrt{2}\right )^2}} \left (\sqrt{3 x^2-2}+\sqrt{2}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{3 x^2-2}}{\sqrt [4]{2}}\right )|\frac{1}{2}\right )}{\sqrt{3} x}-\frac{2 \sqrt [4]{2} \sqrt{\frac{x^2}{\left (\sqrt{3 x^2-2}+\sqrt{2}\right )^2}} \left (\sqrt{3 x^2-2}+\sqrt{2}\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{3 x^2-2}}{\sqrt [4]{2}}\right )|\frac{1}{2}\right )}{\sqrt{3} x} \]

Antiderivative was successfully verified.

[In]  Int[(-2 + 3*x^2)^(-1/4),x]

[Out]

(2*x*(-2 + 3*x^2)^(1/4))/(Sqrt[2] + Sqrt[-2 + 3*x^2]) - (2*2^(1/4)*Sqrt[x^2/(Sqr
t[2] + Sqrt[-2 + 3*x^2])^2]*(Sqrt[2] + Sqrt[-2 + 3*x^2])*EllipticE[2*ArcTan[(-2
+ 3*x^2)^(1/4)/2^(1/4)], 1/2])/(Sqrt[3]*x) + (2^(1/4)*Sqrt[x^2/(Sqrt[2] + Sqrt[-
2 + 3*x^2])^2]*(Sqrt[2] + Sqrt[-2 + 3*x^2])*EllipticF[2*ArcTan[(-2 + 3*x^2)^(1/4
)/2^(1/4)], 1/2])/(Sqrt[3]*x)

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Rubi in Sympy [A]  time = 1.55473, size = 42, normalized size = 0.21 \[ \frac{2 \sqrt{6} \sqrt [4]{- \frac{3 x^{2}}{2} + 1} E\left (\frac{\operatorname{asin}{\left (\frac{\sqrt{6} x}{2} \right )}}{2}\middle | 2\right )}{3 \sqrt [4]{3 x^{2} - 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(3*x**2-2)**(1/4),x)

[Out]

2*sqrt(6)*(-3*x**2/2 + 1)**(1/4)*elliptic_e(asin(sqrt(6)*x/2)/2, 2)/(3*(3*x**2 -
2)**(1/4))

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Mathematica [C]  time = 0.0190982, size = 41, normalized size = 0.21 \[ \frac{x \sqrt [4]{2-3 x^2} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{3}{2};\frac{3 x^2}{2}\right )}{\sqrt [4]{6 x^2-4}} \]

Antiderivative was successfully verified.

[In]  Integrate[(-2 + 3*x^2)^(-1/4),x]

[Out]

(x*(2 - 3*x^2)^(1/4)*Hypergeometric2F1[1/4, 1/2, 3/2, (3*x^2)/2])/(-4 + 6*x^2)^(
1/4)

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Maple [C]  time = 0.04, size = 40, normalized size = 0.2 \[{\frac{{2}^{{\frac{3}{4}}}x}{2}\sqrt [4]{-{\it signum} \left ( -1+{\frac{3\,{x}^{2}}{2}} \right ) }{\mbox{$_2$F$_1$}({\frac{1}{4}},{\frac{1}{2}};\,{\frac{3}{2}};\,{\frac{3\,{x}^{2}}{2}})}{\frac{1}{\sqrt [4]{{\it signum} \left ( -1+{\frac{3\,{x}^{2}}{2}} \right ) }}}} \]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((3*x^2 - 2)^(-1/4), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((3*x^2 - 2)^(-1/4), x)

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Sympy [A]  time = 2.14467, size = 29, normalized size = 0.15 \[ \frac{2^{\frac{3}{4}} x e^{\frac{7 i \pi }{4}}{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{1}{2} \\ \frac{3}{2} \end{matrix}\middle |{\frac{3 x^{2}}{2}} \right )}}{2} \]

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*exp(7*I*pi/4)*hyper((1/4, 1/2), (3/2,), 3*x**2/2)/2

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (3 \, x^{2} - 2\right )}^{\frac{1}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((3*x^2 - 2)^(-1/4), x)