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

Optimal. Leaf size=242 \[ -\frac{9 \sqrt [4]{3 x^2-2} x}{8 \left (\sqrt{3 x^2-2}+\sqrt{2}\right )}+\frac{3 \left (3 x^2-2\right )^{3/4}}{8 x}-\frac{3 \sqrt{3} \sqrt{\frac{x^2}{\left (\sqrt{3 x^2-2}+\sqrt{2}\right )^2}} \left (\sqrt{3 x^2-2}+\sqrt{2}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{3 x^2-2}}{\sqrt [4]{2}}\right )|\frac{1}{2}\right )}{8\ 2^{3/4} x}+\frac{3 \sqrt{3} \sqrt{\frac{x^2}{\left (\sqrt{3 x^2-2}+\sqrt{2}\right )^2}} \left (\sqrt{3 x^2-2}+\sqrt{2}\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{3 x^2-2}}{\sqrt [4]{2}}\right )|\frac{1}{2}\right )}{4\ 2^{3/4} x}+\frac{\left (3 x^2-2\right )^{3/4}}{6 x^3} \]

[Out]

(-2 + 3*x^2)^(3/4)/(6*x^3) + (3*(-2 + 3*x^2)^(3/4))/(8*x) - (9*x*(-2 + 3*x^2)^(1
/4))/(8*(Sqrt[2] + Sqrt[-2 + 3*x^2])) + (3*Sqrt[3]*Sqrt[x^2/(Sqrt[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])/(4*2^(3/4)*x) - (3*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])/(8*2^(3/4)*x)

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Rubi [A]  time = 0.282946, antiderivative size = 242, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ -\frac{9 \sqrt [4]{3 x^2-2} x}{8 \left (\sqrt{3 x^2-2}+\sqrt{2}\right )}+\frac{3 \left (3 x^2-2\right )^{3/4}}{8 x}-\frac{3 \sqrt{3} \sqrt{\frac{x^2}{\left (\sqrt{3 x^2-2}+\sqrt{2}\right )^2}} \left (\sqrt{3 x^2-2}+\sqrt{2}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{3 x^2-2}}{\sqrt [4]{2}}\right )|\frac{1}{2}\right )}{8\ 2^{3/4} x}+\frac{3 \sqrt{3} \sqrt{\frac{x^2}{\left (\sqrt{3 x^2-2}+\sqrt{2}\right )^2}} \left (\sqrt{3 x^2-2}+\sqrt{2}\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{3 x^2-2}}{\sqrt [4]{2}}\right )|\frac{1}{2}\right )}{4\ 2^{3/4} x}+\frac{\left (3 x^2-2\right )^{3/4}}{6 x^3} \]

Antiderivative was successfully verified.

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

[Out]

(-2 + 3*x^2)^(3/4)/(6*x^3) + (3*(-2 + 3*x^2)^(3/4))/(8*x) - (9*x*(-2 + 3*x^2)^(1
/4))/(8*(Sqrt[2] + Sqrt[-2 + 3*x^2])) + (3*Sqrt[3]*Sqrt[x^2/(Sqrt[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])/(4*2^(3/4)*x) - (3*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])/(8*2^(3/4)*x)

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Rubi in Sympy [A]  time = 5.71533, size = 73, normalized size = 0.3 \[ - \frac{3 \sqrt{6} \sqrt [4]{- \frac{3 x^{2}}{2} + 1} E\left (\frac{\operatorname{asin}{\left (\frac{\sqrt{6} x}{2} \right )}}{2}\middle | 2\right )}{8 \sqrt [4]{3 x^{2} - 2}} + \frac{3 \left (3 x^{2} - 2\right )^{\frac{3}{4}}}{8 x} + \frac{\left (3 x^{2} - 2\right )^{\frac{3}{4}}}{6 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*sqrt(6)*(-3*x**2/2 + 1)**(1/4)*elliptic_e(asin(sqrt(6)*x/2)/2, 2)/(8*(3*x**2
- 2)**(1/4)) + 3*(3*x**2 - 2)**(3/4)/(8*x) + (3*x**2 - 2)**(3/4)/(6*x**3)

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Mathematica [C]  time = 0.034633, size = 71, normalized size = 0.29 \[ \frac{4 \left (27 x^4-6 x^2-8\right )-27\ 2^{3/4} x^4 \sqrt [4]{2-3 x^2} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{3}{2};\frac{3 x^2}{2}\right )}{96 x^3 \sqrt [4]{3 x^2-2}} \]

Antiderivative was successfully verified.

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

[Out]

(4*(-8 - 6*x^2 + 27*x^4) - 27*2^(3/4)*x^4*(2 - 3*x^2)^(1/4)*Hypergeometric2F1[1/
4, 1/2, 3/2, (3*x^2)/2])/(96*x^3*(-2 + 3*x^2)^(1/4))

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Maple [C]  time = 0.062, size = 67, normalized size = 0.3 \[{\frac{27\,{x}^{4}-6\,{x}^{2}-8}{24\,{x}^{3}}{\frac{1}{\sqrt [4]{3\,{x}^{2}-2}}}}-{\frac{9\,{2}^{3/4}x}{32}\sqrt [4]{-{\it signum} \left ( -1+{\frac{3\,{x}^{2}}{2}} \right ) }{\mbox{$_2$F$_1$}({\frac{1}{4}},{\frac{1}{2}};\,{\frac{3}{2}};\,{\frac{3\,{x}^{2}}{2}})}{\frac{1}{\sqrt [4]{{\it signum} \left ( -1+{\frac{3\,{x}^{2}}{2}} \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^4/(3*x^2-2)^(1/4),x)

[Out]

1/24*(27*x^4-6*x^2-8)/x^3/(3*x^2-2)^(1/4)-9/32*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. \[ \int \frac{1}{{\left (3 \, x^{2} - 2\right )}^{\frac{1}{4}} x^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 2.93385, size = 34, normalized size = 0.14 \[ \frac{2^{\frac{3}{4}} e^{- \frac{5 i \pi }{4}}{{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{2}, \frac{1}{4} \\ - \frac{1}{2} \end{matrix}\middle |{\frac{3 x^{2}}{2}} \right )}}{6 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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