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

Optimal. Leaf size=260 \[ -\frac{189 \sqrt [4]{3 x^2-2} x}{160 \left (\sqrt{3 x^2-2}+\sqrt{2}\right )}+\frac{63 \left (3 x^2-2\right )^{3/4}}{160 x}-\frac{63 \sqrt{3} \sqrt{\frac{x^2}{\left (\sqrt{3 x^2-2}+\sqrt{2}\right )^2}} \left (\sqrt{3 x^2-2}+\sqrt{2}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{3 x^2-2}}{\sqrt [4]{2}}\right )|\frac{1}{2}\right )}{160\ 2^{3/4} x}+\frac{63 \sqrt{3} \sqrt{\frac{x^2}{\left (\sqrt{3 x^2-2}+\sqrt{2}\right )^2}} \left (\sqrt{3 x^2-2}+\sqrt{2}\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{3 x^2-2}}{\sqrt [4]{2}}\right )|\frac{1}{2}\right )}{80\ 2^{3/4} x}+\frac{\left (3 x^2-2\right )^{3/4}}{10 x^5}+\frac{7 \left (3 x^2-2\right )^{3/4}}{40 x^3} \]

[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) - (189*x*(-2 + 3*x^2)^(1/4))/(160*(Sqrt[2] + Sqrt[-2 + 3*x^2]))
+ (63*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])/(80*2^(3/4)*x) - (63*S
qrt[3]*Sqrt[x^2/(Sqrt[2] + Sqrt[-2 + 3*x^2])^2]*(Sqrt[2] + Sqrt[-2 + 3*x^2])*Ell
ipticF[2*ArcTan[(-2 + 3*x^2)^(1/4)/2^(1/4)], 1/2])/(160*2^(3/4)*x)

_______________________________________________________________________________________

Rubi [A]  time = 0.327219, antiderivative size = 260, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ -\frac{189 \sqrt [4]{3 x^2-2} x}{160 \left (\sqrt{3 x^2-2}+\sqrt{2}\right )}+\frac{63 \left (3 x^2-2\right )^{3/4}}{160 x}-\frac{63 \sqrt{3} \sqrt{\frac{x^2}{\left (\sqrt{3 x^2-2}+\sqrt{2}\right )^2}} \left (\sqrt{3 x^2-2}+\sqrt{2}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{3 x^2-2}}{\sqrt [4]{2}}\right )|\frac{1}{2}\right )}{160\ 2^{3/4} x}+\frac{63 \sqrt{3} \sqrt{\frac{x^2}{\left (\sqrt{3 x^2-2}+\sqrt{2}\right )^2}} \left (\sqrt{3 x^2-2}+\sqrt{2}\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{3 x^2-2}}{\sqrt [4]{2}}\right )|\frac{1}{2}\right )}{80\ 2^{3/4} x}+\frac{\left (3 x^2-2\right )^{3/4}}{10 x^5}+\frac{7 \left (3 x^2-2\right )^{3/4}}{40 x^3} \]

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) - (189*x*(-2 + 3*x^2)^(1/4))/(160*(Sqrt[2] + Sqrt[-2 + 3*x^2]))
+ (63*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])/(80*2^(3/4)*x) - (63*S
qrt[3]*Sqrt[x^2/(Sqrt[2] + Sqrt[-2 + 3*x^2])^2]*(Sqrt[2] + Sqrt[-2 + 3*x^2])*Ell
ipticF[2*ArcTan[(-2 + 3*x^2)^(1/4)/2^(1/4)], 1/2])/(160*2^(3/4)*x)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 7.42916, size = 90, normalized size = 0.35 \[ - \frac{63 \sqrt{6} \sqrt [4]{- \frac{3 x^{2}}{2} + 1} E\left (\frac{\operatorname{asin}{\left (\frac{\sqrt{6} x}{2} \right )}}{2}\middle | 2\right )}{160 \sqrt [4]{3 x^{2} - 2}} + \frac{63 \left (3 x^{2} - 2\right )^{\frac{3}{4}}}{160 x} + \frac{7 \left (3 x^{2} - 2\right )^{\frac{3}{4}}}{40 x^{3}} + \frac{\left (3 x^{2} - 2\right )^{\frac{3}{4}}}{10 x^{5}} \]

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*sqrt(6)*(-3*x**2/2 + 1)**(1/4)*elliptic_e(asin(sqrt(6)*x/2)/2, 2)/(160*(3*x*
*2 - 2)**(1/4)) + 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 = 0.0372006, size = 76, normalized size = 0.29 \[ \frac{4 \left (189 x^6-42 x^4-8 x^2-32\right )-189\ 2^{3/4} x^6 \sqrt [4]{2-3 x^2} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{3}{2};\frac{3 x^2}{2}\right )}{640 x^5 \sqrt [4]{3 x^2-2}} \]

Antiderivative was successfully verified.

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

[Out]

(4*(-32 - 8*x^2 - 42*x^4 + 189*x^6) - 189*2^(3/4)*x^6*(2 - 3*x^2)^(1/4)*Hypergeo
metric2F1[1/4, 1/2, 3/2, (3*x^2)/2])/(640*x^5*(-2 + 3*x^2)^(1/4))

_______________________________________________________________________________________

Maple [C]  time = 0.053, size = 72, normalized size = 0.3 \[{\frac{189\,{x}^{6}-42\,{x}^{4}-8\,{x}^{2}-32}{160\,{x}^{5}}{\frac{1}{\sqrt [4]{3\,{x}^{2}-2}}}}-{\frac{189\,{2}^{3/4}x}{640}\sqrt [4]{-{\it signum} \left ( -1+{\frac{3\,{x}^{2}}{2}} \right ) }{\mbox{$_2$F$_1$}({\frac{1}{4}},{\frac{1}{2}};\,{\frac{3}{2}};\,{\frac{3\,{x}^{2}}{2}})}{\frac{1}{\sqrt [4]{{\it signum} \left ( -1+{\frac{3\,{x}^{2}}{2}} \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^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)/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)

_______________________________________________________________________________________

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 = 3.77551, size = 34, normalized size = 0.13 \[ \frac{2^{\frac{3}{4}} e^{- \frac{5 i \pi }{4}}{{}_{2}F_{1}\left (\begin{matrix} - \frac{5}{2}, \frac{1}{4} \\ - \frac{3}{2} \end{matrix}\middle |{\frac{3 x^{2}}{2}} \right )}}{10 x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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