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

Optimal. Leaf size=67 \[ -\frac{5 \sqrt [4]{2-3 x^2}}{8 x}-\frac{\sqrt [4]{2-3 x^2}}{6 x^3}+\frac{5 \sqrt{3} F\left (\left .\frac{1}{2} \sin ^{-1}\left (\sqrt{\frac{3}{2}} x\right )\right |2\right )}{8 \sqrt [4]{2}} \]

[Out]

-(2 - 3*x^2)^(1/4)/(6*x^3) - (5*(2 - 3*x^2)^(1/4))/(8*x) + (5*Sqrt[3]*EllipticF[
ArcSin[Sqrt[3/2]*x]/2, 2])/(8*2^(1/4))

_______________________________________________________________________________________

Rubi [A]  time = 0.0553558, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ -\frac{5 \sqrt [4]{2-3 x^2}}{8 x}-\frac{\sqrt [4]{2-3 x^2}}{6 x^3}+\frac{5 \sqrt{3} F\left (\left .\frac{1}{2} \sin ^{-1}\left (\sqrt{\frac{3}{2}} x\right )\right |2\right )}{8 \sqrt [4]{2}} \]

Antiderivative was successfully verified.

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

[Out]

-(2 - 3*x^2)^(1/4)/(6*x^3) - (5*(2 - 3*x^2)^(1/4))/(8*x) + (5*Sqrt[3]*EllipticF[
ArcSin[Sqrt[3/2]*x]/2, 2])/(8*2^(1/4))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 5.11973, size = 56, normalized size = 0.84 \[ \frac{5 \cdot 2^{\frac{3}{4}} \sqrt{3} F\left (\frac{\operatorname{asin}{\left (\frac{\sqrt{6} x}{2} \right )}}{2}\middle | 2\right )}{16} - \frac{5 \sqrt [4]{- 3 x^{2} + 2}}{8 x} - \frac{\sqrt [4]{- 3 x^{2} + 2}}{6 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5*2**(3/4)*sqrt(3)*elliptic_f(asin(sqrt(6)*x/2)/2, 2)/16 - 5*(-3*x**2 + 2)**(1/4
)/(8*x) - (-3*x**2 + 2)**(1/4)/(6*x**3)

_______________________________________________________________________________________

Mathematica [C]  time = 0.0299766, size = 55, normalized size = 0.82 \[ \frac{15 x \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{3}{2};\frac{3 x^2}{2}\right )}{16\ 2^{3/4}}+\sqrt [4]{2-3 x^2} \left (-\frac{1}{6 x^3}-\frac{5}{8 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-1/(6*x^3) - 5/(8*x))*(2 - 3*x^2)^(1/4) + (15*x*Hypergeometric2F1[1/2, 3/4, 3/2
, (3*x^2)/2])/(16*2^(3/4))

_______________________________________________________________________________________

Maple [F]  time = 0.025, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{4}} \left ( -3\,{x}^{2}+2 \right ) ^{-{\frac{3}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^4/(-3*x^2+2)^(3/4),x)

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 3.43907, size = 34, normalized size = 0.51 \[ - \frac{\sqrt [4]{2}{{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{2}, \frac{3}{4} \\ - \frac{1}{2} \end{matrix}\middle |{\frac{3 x^{2} e^{2 i \pi }}{2}} \right )}}{6 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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