3.911 \(\int \frac{1}{x^6 \left (-2+3 x^2\right )^{3/4}} \, dx\)

Optimal. Leaf size=140 \[ \frac{27 \sqrt [4]{3 x^2-2}}{32 x}+\frac{27 \sqrt{3} \sqrt{\frac{x^2}{\left (\sqrt{3 x^2-2}+\sqrt{2}\right )^2}} \left (\sqrt{3 x^2-2}+\sqrt{2}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{3 x^2-2}}{\sqrt [4]{2}}\right )|\frac{1}{2}\right )}{64 \sqrt [4]{2} x}+\frac{\sqrt [4]{3 x^2-2}}{10 x^5}+\frac{9 \sqrt [4]{3 x^2-2}}{40 x^3} \]

[Out]

(-2 + 3*x^2)^(1/4)/(10*x^5) + (9*(-2 + 3*x^2)^(1/4))/(40*x^3) + (27*(-2 + 3*x^2)
^(1/4))/(32*x) + (27*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])/(64*2^(
1/4)*x)

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Rubi [A]  time = 0.15497, antiderivative size = 140, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{27 \sqrt [4]{3 x^2-2}}{32 x}+\frac{27 \sqrt{3} \sqrt{\frac{x^2}{\left (\sqrt{3 x^2-2}+\sqrt{2}\right )^2}} \left (\sqrt{3 x^2-2}+\sqrt{2}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{3 x^2-2}}{\sqrt [4]{2}}\right )|\frac{1}{2}\right )}{64 \sqrt [4]{2} x}+\frac{\sqrt [4]{3 x^2-2}}{10 x^5}+\frac{9 \sqrt [4]{3 x^2-2}}{40 x^3} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^6*(-2 + 3*x^2)^(3/4)),x]

[Out]

(-2 + 3*x^2)^(1/4)/(10*x^5) + (9*(-2 + 3*x^2)^(1/4))/(40*x^3) + (27*(-2 + 3*x^2)
^(1/4))/(32*x) + (27*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])/(64*2^(
1/4)*x)

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Rubi in Sympy [A]  time = 7.44869, size = 90, normalized size = 0.64 \[ \frac{27 \sqrt{6} \left (- \frac{3 x^{2}}{2} + 1\right )^{\frac{3}{4}} F\left (\frac{\operatorname{asin}{\left (\frac{\sqrt{6} x}{2} \right )}}{2}\middle | 2\right )}{32 \left (3 x^{2} - 2\right )^{\frac{3}{4}}} + \frac{27 \sqrt [4]{3 x^{2} - 2}}{32 x} + \frac{9 \sqrt [4]{3 x^{2} - 2}}{40 x^{3}} + \frac{\sqrt [4]{3 x^{2} - 2}}{10 x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**6/(3*x**2-2)**(3/4),x)

[Out]

27*sqrt(6)*(-3*x**2/2 + 1)**(3/4)*elliptic_f(asin(sqrt(6)*x/2)/2, 2)/(32*(3*x**2
- 2)**(3/4)) + 27*(3*x**2 - 2)**(1/4)/(32*x) + 9*(3*x**2 - 2)**(1/4)/(40*x**3)
+ (3*x**2 - 2)**(1/4)/(10*x**5)

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Mathematica [C]  time = 0.0295156, size = 73, normalized size = 0.52 \[ \frac{405 \sqrt [4]{2} \left (2-3 x^2\right )^{3/4} x^6 \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{3}{2};\frac{3 x^2}{2}\right )+1620 x^6-648 x^4-96 x^2-128}{640 x^5 \left (3 x^2-2\right )^{3/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^6*(-2 + 3*x^2)^(3/4)),x]

[Out]

(-128 - 96*x^2 - 648*x^4 + 1620*x^6 + 405*2^(1/4)*x^6*(2 - 3*x^2)^(3/4)*Hypergeo
metric2F1[1/2, 3/4, 3/2, (3*x^2)/2])/(640*x^5*(-2 + 3*x^2)^(3/4))

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Maple [C]  time = 0.056, size = 72, normalized size = 0.5 \[{\frac{405\,{x}^{6}-162\,{x}^{4}-24\,{x}^{2}-32}{160\,{x}^{5}} \left ( 3\,{x}^{2}-2 \right ) ^{-{\frac{3}{4}}}}+{\frac{81\,\sqrt [4]{2}x}{128} \left ( -{\it signum} \left ( -1+{\frac{3\,{x}^{2}}{2}} \right ) \right ) ^{{\frac{3}{4}}}{\mbox{$_2$F$_1$}({\frac{1}{2}},{\frac{3}{4}};\,{\frac{3}{2}};\,{\frac{3\,{x}^{2}}{2}})} \left ({\it signum} \left ( -1+{\frac{3\,{x}^{2}}{2}} \right ) \right ) ^{-{\frac{3}{4}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^6/(3*x^2-2)^(3/4),x)

[Out]

1/160*(405*x^6-162*x^4-24*x^2-32)/x^5/(3*x^2-2)^(3/4)+81/128*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. \[ \int \frac{1}{{\left (3 \, x^{2} - 2\right )}^{\frac{3}{4}} x^{6}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((3*x^2 - 2)^(3/4)*x^6),x, algorithm="maxima")

[Out]

integrate(1/((3*x^2 - 2)^(3/4)*x^6), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (3 \, x^{2} - 2\right )}^{\frac{3}{4}} x^{6}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((3*x^2 - 2)^(3/4)*x^6),x, algorithm="fricas")

[Out]

integral(1/((3*x^2 - 2)^(3/4)*x^6), x)

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Sympy [A]  time = 4.36326, size = 34, normalized size = 0.24 \[ \frac{\sqrt [4]{2} e^{- \frac{7 i \pi }{4}}{{}_{2}F_{1}\left (\begin{matrix} - \frac{5}{2}, \frac{3}{4} \\ - \frac{3}{2} \end{matrix}\middle |{\frac{3 x^{2}}{2}} \right )}}{10 x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**6/(3*x**2-2)**(3/4),x)

[Out]

2**(1/4)*exp(-7*I*pi/4)*hyper((-5/2, 3/4), (-3/2,), 3*x**2/2)/(10*x**5)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (3 \, x^{2} - 2\right )}^{\frac{3}{4}} x^{6}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((3*x^2 - 2)^(3/4)*x^6),x, algorithm="giac")

[Out]

integrate(1/((3*x^2 - 2)^(3/4)*x^6), x)