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

Optimal. Leaf size=49 \[ \frac{\sqrt{3} F\left (\left .\frac{1}{2} \sin ^{-1}\left (\sqrt{\frac{3}{2}} x\right )\right |2\right )}{2 \sqrt [4]{2}}-\frac{\sqrt [4]{2-3 x^2}}{2 x} \]

[Out]

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

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Rubi [A]  time = 0.0345063, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{\sqrt{3} F\left (\left .\frac{1}{2} \sin ^{-1}\left (\sqrt{\frac{3}{2}} x\right )\right |2\right )}{2 \sqrt [4]{2}}-\frac{\sqrt [4]{2-3 x^2}}{2 x} \]

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

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Rubi in Sympy [A]  time = 3.44901, size = 37, normalized size = 0.76 \[ \frac{2^{\frac{3}{4}} \sqrt{3} F\left (\frac{\operatorname{asin}{\left (\frac{\sqrt{6} x}{2} \right )}}{2}\middle | 2\right )}{4} - \frac{\sqrt [4]{- 3 x^{2} + 2}}{2 x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.0292384, size = 46, normalized size = 0.94 \[ \frac{3 x \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{3}{2};\frac{3 x^2}{2}\right )}{4\ 2^{3/4}}-\frac{\sqrt [4]{2-3 x^2}}{2 x} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.024, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{2}} \left ( -3\,{x}^{2}+2 \right ) ^{-{\frac{3}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^2/(-3*x^2+2)^(3/4),x)

[Out]

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

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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 = 2.72694, size = 31, normalized size = 0.63 \[ - \frac{\sqrt [4]{2}{{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{3}{4} \\ \frac{1}{2} \end{matrix}\middle |{\frac{3 x^{2} e^{2 i \pi }}{2}} \right )}}{2 x} \]

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