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

Optimal. Leaf size=105 \[ \frac{\sqrt [4]{-3 x^2-2}}{2 x}+\frac{\sqrt{3} \sqrt{-\frac{x^2}{\left (\sqrt{-3 x^2-2}+\sqrt{2}\right )^2}} \left (\sqrt{-3 x^2-2}+\sqrt{2}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{-3 x^2-2}}{\sqrt [4]{2}}\right )|\frac{1}{2}\right )}{4 \sqrt [4]{2} x} \]

[Out]

(-2 - 3*x^2)^(1/4)/(2*x) + (Sqrt[3]*Sqrt[-(x^2/(Sqrt[2] + Sqrt[-2 - 3*x^2])^2)]*
(Sqrt[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 = 0.103299, antiderivative size = 105, 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)]*
(Sqrt[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 = 3.82858, size = 58, normalized size = 0.55 \[ - \frac{\sqrt{6} \left (\frac{3 x^{2}}{2} + 1\right )^{\frac{3}{4}} F\left (\frac{\operatorname{atan}{\left (\frac{\sqrt{6} x}{2} \right )}}{2}\middle | 2\right )}{2 \left (- 3 x^{2} - 2\right )^{\frac{3}{4}}} + \frac{\sqrt [4]{- 3 x^{2} - 2}}{2 x} \]

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(atan(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 = 0.0259372, size = 63, normalized size = 0.6 \[ \frac{-3 \sqrt [4]{2} \left (3 x^2+2\right )^{3/4} x^2 \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{3}{2};-\frac{3 x^2}{2}\right )-12 x^2-8}{8 x \left (-3 x^2-2\right )^{3/4}} \]

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 [F]  time = 0.025, 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. \[ \frac{2 \, x{\rm integral}\left (\frac{3 \,{\left (-3 \, x^{2} - 2\right )}^{\frac{1}{4}}}{4 \,{\left (3 \, x^{2} + 2\right )}}, x\right ) +{\left (-3 \, x^{2} - 2\right )}^{\frac{1}{4}}}{2 \, x} \]

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]

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

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Sympy [A]  time = 2.6721, size = 34, normalized size = 0.32 \[ \frac{\sqrt [4]{2} e^{\frac{i \pi }{4}}{{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{3}{4} \\ \frac{1}{2} \end{matrix}\middle |{\frac{3 x^{2} e^{i \pi }}{2}} \right )}}{2 x} \]

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