3.876 \(\int \frac{1}{x^6 \sqrt [4]{2-3 x^2}} \, dx\)

Optimal. Leaf size=85 \[ -\frac{63 \left (2-3 x^2\right )^{3/4}}{160 x}-\frac{\left (2-3 x^2\right )^{3/4}}{10 x^5}-\frac{7 \left (2-3 x^2\right )^{3/4}}{40 x^3}-\frac{63 \sqrt{3} E\left (\left .\frac{1}{2} \sin ^{-1}\left (\sqrt{\frac{3}{2}} x\right )\right |2\right )}{80\ 2^{3/4}} \]

[Out]

-(2 - 3*x^2)^(3/4)/(10*x^5) - (7*(2 - 3*x^2)^(3/4))/(40*x^3) - (63*(2 - 3*x^2)^(
3/4))/(160*x) - (63*Sqrt[3]*EllipticE[ArcSin[Sqrt[3/2]*x]/2, 2])/(80*2^(3/4))

_______________________________________________________________________________________

Rubi [A]  time = 0.0747803, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ -\frac{63 \left (2-3 x^2\right )^{3/4}}{160 x}-\frac{\left (2-3 x^2\right )^{3/4}}{10 x^5}-\frac{7 \left (2-3 x^2\right )^{3/4}}{40 x^3}-\frac{63 \sqrt{3} E\left (\left .\frac{1}{2} \sin ^{-1}\left (\sqrt{\frac{3}{2}} x\right )\right |2\right )}{80\ 2^{3/4}} \]

Antiderivative was successfully verified.

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

[Out]

-(2 - 3*x^2)^(3/4)/(10*x^5) - (7*(2 - 3*x^2)^(3/4))/(40*x^3) - (63*(2 - 3*x^2)^(
3/4))/(160*x) - (63*Sqrt[3]*EllipticE[ArcSin[Sqrt[3/2]*x]/2, 2])/(80*2^(3/4))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 6.77728, size = 75, normalized size = 0.88 \[ - \frac{63 \sqrt [4]{2} \sqrt{3} E\left (\frac{\operatorname{asin}{\left (\frac{\sqrt{6} x}{2} \right )}}{2}\middle | 2\right )}{160} - \frac{63 \left (- 3 x^{2} + 2\right )^{\frac{3}{4}}}{160 x} - \frac{7 \left (- 3 x^{2} + 2\right )^{\frac{3}{4}}}{40 x^{3}} - \frac{\left (- 3 x^{2} + 2\right )^{\frac{3}{4}}}{10 x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-63*2**(1/4)*sqrt(3)*elliptic_e(asin(sqrt(6)*x/2)/2, 2)/160 - 63*(-3*x**2 + 2)**
(3/4)/(160*x) - 7*(-3*x**2 + 2)**(3/4)/(40*x**3) - (-3*x**2 + 2)**(3/4)/(10*x**5
)

_______________________________________________________________________________________

Mathematica [C]  time = 0.0473031, size = 62, normalized size = 0.73 \[ \left (-\frac{1}{10 x^5}-\frac{7}{40 x^3}-\frac{63}{160 x}\right ) \left (2-3 x^2\right )^{3/4}-\frac{189 x \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{3}{2};\frac{3 x^2}{2}\right )}{320 \sqrt [4]{2}} \]

Antiderivative was successfully verified.

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

[Out]

(-1/(10*x^5) - 7/(40*x^3) - 63/(160*x))*(2 - 3*x^2)^(3/4) - (189*x*Hypergeometri
c2F1[1/4, 1/2, 3/2, (3*x^2)/2])/(320*2^(1/4))

_______________________________________________________________________________________

Maple [C]  time = 0.045, size = 50, normalized size = 0.6 \[{\frac{189\,{x}^{6}-42\,{x}^{4}-8\,{x}^{2}-32}{160\,{x}^{5}}{\frac{1}{\sqrt [4]{-3\,{x}^{2}+2}}}}-{\frac{189\,{2}^{3/4}x}{640}{\mbox{$_2$F$_1$}({\frac{1}{4}},{\frac{1}{2}};\,{\frac{3}{2}};\,{\frac{3\,{x}^{2}}{2}})}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^6/(-3*x^2+2)^(1/4),x)

[Out]

1/160*(189*x^6-42*x^4-8*x^2-32)/x^5/(-3*x^2+2)^(1/4)-189/640*2^(3/4)*x*hypergeom
([1/4,1/2],[3/2],3/2*x^2)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 3.6624, size = 34, normalized size = 0.4 \[ - \frac{2^{\frac{3}{4}}{{}_{2}F_{1}\left (\begin{matrix} - \frac{5}{2}, \frac{1}{4} \\ - \frac{3}{2} \end{matrix}\middle |{\frac{3 x^{2} e^{2 i \pi }}{2}} \right )}}{10 x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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