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

Optimal. Leaf size=65 \[ -\frac{8}{135} \left (2-3 x^2\right )^{3/4} x-\frac{2}{27} \left (2-3 x^2\right )^{3/4} x^3+\frac{32 \sqrt [4]{2} E\left (\left .\frac{1}{2} \sin ^{-1}\left (\sqrt{\frac{3}{2}} x\right )\right |2\right )}{135 \sqrt{3}} \]

[Out]

(-8*x*(2 - 3*x^2)^(3/4))/135 - (2*x^3*(2 - 3*x^2)^(3/4))/27 + (32*2^(1/4)*Ellipt
icE[ArcSin[Sqrt[3/2]*x]/2, 2])/(135*Sqrt[3])

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Rubi [A]  time = 0.0565701, antiderivative size = 65, 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 \[ -\frac{8}{135} \left (2-3 x^2\right )^{3/4} x-\frac{2}{27} \left (2-3 x^2\right )^{3/4} x^3+\frac{32 \sqrt [4]{2} E\left (\left .\frac{1}{2} \sin ^{-1}\left (\sqrt{\frac{3}{2}} x\right )\right |2\right )}{135 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Int[x^4/(2 - 3*x^2)^(1/4),x]

[Out]

(-8*x*(2 - 3*x^2)^(3/4))/135 - (2*x^3*(2 - 3*x^2)^(3/4))/27 + (32*2^(1/4)*Ellipt
icE[ArcSin[Sqrt[3/2]*x]/2, 2])/(135*Sqrt[3])

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Rubi in Sympy [A]  time = 5.2437, size = 58, normalized size = 0.89 \[ - \frac{2 x^{3} \left (- 3 x^{2} + 2\right )^{\frac{3}{4}}}{27} - \frac{8 x \left (- 3 x^{2} + 2\right )^{\frac{3}{4}}}{135} + \frac{32 \sqrt [4]{2} \sqrt{3} E\left (\frac{\operatorname{asin}{\left (\frac{\sqrt{6} x}{2} \right )}}{2}\middle | 2\right )}{405} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*x**3*(-3*x**2 + 2)**(3/4)/27 - 8*x*(-3*x**2 + 2)**(3/4)/135 + 32*2**(1/4)*sqr
t(3)*elliptic_e(asin(sqrt(6)*x/2)/2, 2)/405

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Mathematica [C]  time = 0.0498281, size = 50, normalized size = 0.77 \[ \frac{2}{135} x \left (4\ 2^{3/4} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{3}{2};\frac{3 x^2}{2}\right )-\left (2-3 x^2\right )^{3/4} \left (5 x^2+4\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

(2*x*(-((2 - 3*x^2)^(3/4)*(4 + 5*x^2)) + 4*2^(3/4)*Hypergeometric2F1[1/4, 1/2, 3
/2, (3*x^2)/2]))/135

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Maple [C]  time = 0.039, size = 45, normalized size = 0.7 \[{\frac{2\,x \left ( 5\,{x}^{2}+4 \right ) \left ( 3\,{x}^{2}-2 \right ) }{135}{\frac{1}{\sqrt [4]{-3\,{x}^{2}+2}}}}+{\frac{8\,{2}^{3/4}x}{135}{\mbox{$_2$F$_1$}({\frac{1}{4}},{\frac{1}{2}};\,{\frac{3}{2}};\,{\frac{3\,{x}^{2}}{2}})}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4/(-3*x^2+2)^(1/4),x)

[Out]

2/135*x*(5*x^2+4)*(3*x^2-2)/(-3*x^2+2)^(1/4)+8/135*2^(3/4)*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. \[ \int \frac{x^{4}}{{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x^4/(-3*x^2 + 2)^(1/4), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(x^4/(-3*x^2 + 2)^(1/4), x)

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Sympy [A]  time = 2.39721, size = 29, normalized size = 0.45 \[ \frac{2^{\frac{3}{4}} x^{5}{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{5}{2} \\ \frac{7}{2} \end{matrix}\middle |{\frac{3 x^{2} e^{2 i \pi }}{2}} \right )}}{10} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x^4/(-3*x^2 + 2)^(1/4), x)