3.909 \(\int \frac{1}{x^2 \left (-2+3 x^2\right )^{3/4}} \, dx\)

Optimal. Leaf size=104 \[ \frac{\sqrt [4]{3 x^2-2}}{2 x}+\frac{\sqrt{3} \sqrt{\frac{x^2}{\left (\sqrt{3 x^2-2}+\sqrt{2}\right )^2}} \left (\sqrt{3 x^2-2}+\sqrt{2}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{3 x^2-2}}{\sqrt [4]{2}}\right )|\frac{1}{2}\right )}{4 \sqrt [4]{2} x} \]

[Out]

(-2 + 3*x^2)^(1/4)/(2*x) + (Sqrt[3]*Sqrt[x^2/(Sqrt[2] + Sqrt[-2 + 3*x^2])^2]*(Sq
rt[2] + Sqrt[-2 + 3*x^2])*EllipticF[2*ArcTan[(-2 + 3*x^2)^(1/4)/2^(1/4)], 1/2])/
(4*2^(1/4)*x)

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Rubi [A]  time = 0.0977231, antiderivative size = 104, 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 \[ \frac{\sqrt [4]{3 x^2-2}}{2 x}+\frac{\sqrt{3} \sqrt{\frac{x^2}{\left (\sqrt{3 x^2-2}+\sqrt{2}\right )^2}} \left (\sqrt{3 x^2-2}+\sqrt{2}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{3 x^2-2}}{\sqrt [4]{2}}\right )|\frac{1}{2}\right )}{4 \sqrt [4]{2} x} \]

Antiderivative was successfully verified.

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

[Out]

(-2 + 3*x^2)^(1/4)/(2*x) + (Sqrt[3]*Sqrt[x^2/(Sqrt[2] + Sqrt[-2 + 3*x^2])^2]*(Sq
rt[2] + Sqrt[-2 + 3*x^2])*EllipticF[2*ArcTan[(-2 + 3*x^2)^(1/4)/2^(1/4)], 1/2])/
(4*2^(1/4)*x)

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Rubi in Sympy [A]  time = 3.88935, size = 54, normalized size = 0.52 \[ \frac{\sqrt{6} \left (- \frac{3 x^{2}}{2} + 1\right )^{\frac{3}{4}} F\left (\frac{\operatorname{asin}{\left (\frac{\sqrt{6} x}{2} \right )}}{2}\middle | 2\right )}{2 \left (3 x^{2} - 2\right )^{\frac{3}{4}}} + \frac{\sqrt [4]{3 x^{2} - 2}}{2 x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.0244137, size = 63, normalized size = 0.61 \[ \frac{3 \sqrt [4]{2} \left (2-3 x^2\right )^{3/4} x^2 \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{3}{2};\frac{3 x^2}{2}\right )+12 x^2-8}{8 x \left (3 x^2-2\right )^{3/4}} \]

Antiderivative was successfully verified.

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

[Out]

(-8 + 12*x^2 + 3*2^(1/4)*x^2*(2 - 3*x^2)^(3/4)*Hypergeometric2F1[1/2, 3/4, 3/2,
(3*x^2)/2])/(8*x*(-2 + 3*x^2)^(3/4))

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Maple [C]  time = 0.056, size = 55, normalized size = 0.5 \[{\frac{1}{2\,x}\sqrt [4]{3\,{x}^{2}-2}}+{\frac{3\,\sqrt [4]{2}x}{8} \left ( -{\it signum} \left ( -1+{\frac{3\,{x}^{2}}{2}} \right ) \right ) ^{{\frac{3}{4}}}{\mbox{$_2$F$_1$}({\frac{1}{2}},{\frac{3}{4}};\,{\frac{3}{2}};\,{\frac{3\,{x}^{2}}{2}})} \left ({\it signum} \left ( -1+{\frac{3\,{x}^{2}}{2}} \right ) \right ) ^{-{\frac{3}{4}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^2/(3*x^2-2)^(3/4),x)

[Out]

1/2*(3*x^2-2)^(1/4)/x+3/8*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. \[ \int \frac{1}{{\left (3 \, x^{2} - 2\right )}^{\frac{3}{4}} x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 2.68562, size = 29, normalized size = 0.28 \[ \frac{\sqrt [4]{2} e^{\frac{i \pi }{4}}{{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{3}{4} \\ \frac{1}{2} \end{matrix}\middle |{\frac{3 x^{2}}{2}} \right )}}{2 x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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