3.907 \(\int \frac{x^2}{\left (-2+3 x^2\right )^{3/4}} \, dx\)

Optimal. Leaf size=102 \[ \frac{2}{9} \sqrt [4]{3 x^2-2} x+\frac{2\ 2^{3/4} \sqrt{\frac{x^2}{\left (\sqrt{3 x^2-2}+\sqrt{2}\right )^2}} \left (\sqrt{3 x^2-2}+\sqrt{2}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{3 x^2-2}}{\sqrt [4]{2}}\right )|\frac{1}{2}\right )}{9 \sqrt{3} x} \]

[Out]

(2*x*(-2 + 3*x^2)^(1/4))/9 + (2*2^(3/4)*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])/(9*Sqrt[3]*x)

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Rubi [A]  time = 0.103013, antiderivative size = 102, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{2}{9} \sqrt [4]{3 x^2-2} x+\frac{2\ 2^{3/4} \sqrt{\frac{x^2}{\left (\sqrt{3 x^2-2}+\sqrt{2}\right )^2}} \left (\sqrt{3 x^2-2}+\sqrt{2}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{3 x^2-2}}{\sqrt [4]{2}}\right )|\frac{1}{2}\right )}{9 \sqrt{3} x} \]

Antiderivative was successfully verified.

[In]  Int[x^2/(-2 + 3*x^2)^(3/4),x]

[Out]

(2*x*(-2 + 3*x^2)^(1/4))/9 + (2*2^(3/4)*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])/(9*Sqrt[3]*x)

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Rubi in Sympy [A]  time = 4.14287, size = 58, normalized size = 0.57 \[ \frac{2 x \sqrt [4]{3 x^{2} - 2}}{9} + \frac{8 \sqrt{6} \left (- \frac{3 x^{2}}{2} + 1\right )^{\frac{3}{4}} F\left (\frac{\operatorname{asin}{\left (\frac{\sqrt{6} x}{2} \right )}}{2}\middle | 2\right )}{27 \left (3 x^{2} - 2\right )^{\frac{3}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*x*(3*x**2 - 2)**(1/4)/9 + 8*sqrt(6)*(-3*x**2/2 + 1)**(3/4)*elliptic_f(asin(sqr
t(6)*x/2)/2, 2)/(27*(3*x**2 - 2)**(3/4))

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Mathematica [C]  time = 0.0298899, size = 57, normalized size = 0.56 \[ \frac{2 x \left (\sqrt [4]{2} \left (2-3 x^2\right )^{3/4} \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{3}{2};\frac{3 x^2}{2}\right )+3 x^2-2\right )}{9 \left (3 x^2-2\right )^{3/4}} \]

Antiderivative was successfully verified.

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

[Out]

(2*x*(-2 + 3*x^2 + 2^(1/4)*(2 - 3*x^2)^(3/4)*Hypergeometric2F1[1/2, 3/4, 3/2, (3
*x^2)/2]))/(9*(-2 + 3*x^2)^(3/4))

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Maple [C]  time = 0.051, size = 53, normalized size = 0.5 \[{\frac{2\,x}{9}\sqrt [4]{3\,{x}^{2}-2}}+{\frac{2\,\sqrt [4]{2}x}{9} \left ( -{\it signum} \left ( -1+{\frac{3\,{x}^{2}}{2}} \right ) \right ) ^{{\frac{3}{4}}}{\mbox{$_2$F$_1$}({\frac{1}{2}},{\frac{3}{4}};\,{\frac{3}{2}};\,{\frac{3\,{x}^{2}}{2}})} \left ({\it signum} \left ( -1+{\frac{3\,{x}^{2}}{2}} \right ) \right ) ^{-{\frac{3}{4}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(3*x^2-2)^(3/4),x)

[Out]

2/9*x*(3*x^2-2)^(1/4)+2/9*2^(1/4)/signum(-1+3/2*x^2)^(3/4)*(-signum(-1+3/2*x^2))
^(3/4)*x*hypergeom([1/2,3/4],[3/2],3/2*x^2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x^2/(3*x^2 - 2)^(3/4), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(x^2/(3*x^2 - 2)^(3/4), x)

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Sympy [A]  time = 2.11622, size = 31, normalized size = 0.3 \[ \frac{\sqrt [4]{2} x^{3} e^{\frac{5 i \pi }{4}}{{}_{2}F_{1}\left (\begin{matrix} \frac{3}{4}, \frac{3}{2} \\ \frac{5}{2} \end{matrix}\middle |{\frac{3 x^{2}}{2}} \right )}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2**(1/4)*x**3*exp(5*I*pi/4)*hyper((3/4, 3/2), (5/2,), 3*x**2/2)/6

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x^2/(3*x^2 - 2)^(3/4), x)