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

Optimal. Leaf size=83 \[ -\frac{32 \left (2-3 x^2\right )^{3/4} x}{1053}-\frac{2}{39} \left (2-3 x^2\right )^{3/4} x^5-\frac{40 \left (2-3 x^2\right )^{3/4} x^3}{1053}+\frac{128 \sqrt [4]{2} E\left (\left .\frac{1}{2} \sin ^{-1}\left (\sqrt{\frac{3}{2}} x\right )\right |2\right )}{1053 \sqrt{3}} \]

[Out]

(-32*x*(2 - 3*x^2)^(3/4))/1053 - (40*x^3*(2 - 3*x^2)^(3/4))/1053 - (2*x^5*(2 - 3
*x^2)^(3/4))/39 + (128*2^(1/4)*EllipticE[ArcSin[Sqrt[3/2]*x]/2, 2])/(1053*Sqrt[3
])

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Rubi [A]  time = 0.0776138, antiderivative size = 83, 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 \[ -\frac{32 \left (2-3 x^2\right )^{3/4} x}{1053}-\frac{2}{39} \left (2-3 x^2\right )^{3/4} x^5-\frac{40 \left (2-3 x^2\right )^{3/4} x^3}{1053}+\frac{128 \sqrt [4]{2} E\left (\left .\frac{1}{2} \sin ^{-1}\left (\sqrt{\frac{3}{2}} x\right )\right |2\right )}{1053 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Int[x^6/(2 - 3*x^2)^(1/4),x]

[Out]

(-32*x*(2 - 3*x^2)^(3/4))/1053 - (40*x^3*(2 - 3*x^2)^(3/4))/1053 - (2*x^5*(2 - 3
*x^2)^(3/4))/39 + (128*2^(1/4)*EllipticE[ArcSin[Sqrt[3/2]*x]/2, 2])/(1053*Sqrt[3
])

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Rubi in Sympy [A]  time = 7.01607, size = 75, normalized size = 0.9 \[ - \frac{2 x^{5} \left (- 3 x^{2} + 2\right )^{\frac{3}{4}}}{39} - \frac{40 x^{3} \left (- 3 x^{2} + 2\right )^{\frac{3}{4}}}{1053} - \frac{32 x \left (- 3 x^{2} + 2\right )^{\frac{3}{4}}}{1053} + \frac{128 \sqrt [4]{2} \sqrt{3} E\left (\frac{\operatorname{asin}{\left (\frac{\sqrt{6} x}{2} \right )}}{2}\middle | 2\right )}{3159} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*x**5*(-3*x**2 + 2)**(3/4)/39 - 40*x**3*(-3*x**2 + 2)**(3/4)/1053 - 32*x*(-3*x
**2 + 2)**(3/4)/1053 + 128*2**(1/4)*sqrt(3)*elliptic_e(asin(sqrt(6)*x/2)/2, 2)/3
159

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Mathematica [C]  time = 0.0609184, size = 55, normalized size = 0.66 \[ \frac{2 x \left (16\ 2^{3/4} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{3}{2};\frac{3 x^2}{2}\right )-\left (2-3 x^2\right )^{3/4} \left (27 x^4+20 x^2+16\right )\right )}{1053} \]

Antiderivative was successfully verified.

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

[Out]

(2*x*(-((2 - 3*x^2)^(3/4)*(16 + 20*x^2 + 27*x^4)) + 16*2^(3/4)*Hypergeometric2F1
[1/4, 1/2, 3/2, (3*x^2)/2]))/1053

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Maple [C]  time = 0.056, size = 50, normalized size = 0.6 \[{\frac{2\,x \left ( 27\,{x}^{4}+20\,{x}^{2}+16 \right ) \left ( 3\,{x}^{2}-2 \right ) }{1053}{\frac{1}{\sqrt [4]{-3\,{x}^{2}+2}}}}+{\frac{32\,{2}^{3/4}x}{1053}{\mbox{$_2$F$_1$}({\frac{1}{4}},{\frac{1}{2}};\,{\frac{3}{2}};\,{\frac{3\,{x}^{2}}{2}})}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^6/(-3*x^2+2)^(1/4),x)

[Out]

2/1053*x*(27*x^4+20*x^2+16)*(3*x^2-2)/(-3*x^2+2)^(1/4)+32/1053*2^(3/4)*x*hyperge
om([1/4,1/2],[3/2],3/2*x^2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x^6/(-3*x^2 + 2)^(1/4), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(x^6/(-3*x^2 + 2)^(1/4), x)

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Sympy [A]  time = 2.87001, size = 29, normalized size = 0.35 \[ \frac{2^{\frac{3}{4}} x^{7}{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{7}{2} \\ \frac{9}{2} \end{matrix}\middle |{\frac{3 x^{2} e^{2 i \pi }}{2}} \right )}}{14} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2**(3/4)*x**7*hyper((1/4, 7/2), (9/2,), 3*x**2*exp_polar(2*I*pi)/2)/14

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x^6/(-3*x^2 + 2)^(1/4), x)