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

Optimal. Leaf size=258 \[ \frac{32 \left (3 x^2-2\right )^{3/4} x}{1053}+\frac{128 \sqrt [4]{3 x^2-2} x}{1053 \left (\sqrt{3 x^2-2}+\sqrt{2}\right )}+\frac{64 \sqrt [4]{2} \sqrt{\frac{x^2}{\left (\sqrt{3 x^2-2}+\sqrt{2}\right )^2}} \left (\sqrt{3 x^2-2}+\sqrt{2}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{3 x^2-2}}{\sqrt [4]{2}}\right )|\frac{1}{2}\right )}{1053 \sqrt{3} x}-\frac{128 \sqrt [4]{2} \sqrt{\frac{x^2}{\left (\sqrt{3 x^2-2}+\sqrt{2}\right )^2}} \left (\sqrt{3 x^2-2}+\sqrt{2}\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{3 x^2-2}}{\sqrt [4]{2}}\right )|\frac{1}{2}\right )}{1053 \sqrt{3} x}+\frac{2}{39} \left (3 x^2-2\right )^{3/4} x^5+\frac{40 \left (3 x^2-2\right )^{3/4} x^3}{1053} \]

[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*x*(-2 + 3*x^2)^(1/4))/(1053*(Sqrt[2] + Sqrt[-2 + 3*x^2]
)) - (128*2^(1/4)*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])/(1053*Sqrt[3]*x) +
(64*2^(1/4)*Sqrt[x^2/(Sqrt[2] + Sqrt[-2 + 3*x^2])^2]*(Sqrt[2] + Sqrt[-2 + 3*x^2
])*EllipticF[2*ArcTan[(-2 + 3*x^2)^(1/4)/2^(1/4)], 1/2])/(1053*Sqrt[3]*x)

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Rubi [A]  time = 0.337327, antiderivative size = 258, 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 \[ \frac{32 \left (3 x^2-2\right )^{3/4} x}{1053}+\frac{128 \sqrt [4]{3 x^2-2} x}{1053 \left (\sqrt{3 x^2-2}+\sqrt{2}\right )}+\frac{64 \sqrt [4]{2} \sqrt{\frac{x^2}{\left (\sqrt{3 x^2-2}+\sqrt{2}\right )^2}} \left (\sqrt{3 x^2-2}+\sqrt{2}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{3 x^2-2}}{\sqrt [4]{2}}\right )|\frac{1}{2}\right )}{1053 \sqrt{3} x}-\frac{128 \sqrt [4]{2} \sqrt{\frac{x^2}{\left (\sqrt{3 x^2-2}+\sqrt{2}\right )^2}} \left (\sqrt{3 x^2-2}+\sqrt{2}\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{3 x^2-2}}{\sqrt [4]{2}}\right )|\frac{1}{2}\right )}{1053 \sqrt{3} x}+\frac{2}{39} \left (3 x^2-2\right )^{3/4} x^5+\frac{40 \left (3 x^2-2\right )^{3/4} x^3}{1053} \]

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*x*(-2 + 3*x^2)^(1/4))/(1053*(Sqrt[2] + Sqrt[-2 + 3*x^2]
)) - (128*2^(1/4)*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])/(1053*Sqrt[3]*x) +
(64*2^(1/4)*Sqrt[x^2/(Sqrt[2] + Sqrt[-2 + 3*x^2])^2]*(Sqrt[2] + Sqrt[-2 + 3*x^2
])*EllipticF[2*ArcTan[(-2 + 3*x^2)^(1/4)/2^(1/4)], 1/2])/(1053*Sqrt[3]*x)

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Rubi in Sympy [A]  time = 7.81062, size = 92, normalized size = 0.36 \[ \frac{2 x^{5} \left (3 x^{2} - 2\right )^{\frac{3}{4}}}{39} + \frac{40 x^{3} \left (3 x^{2} - 2\right )^{\frac{3}{4}}}{1053} + \frac{32 x \left (3 x^{2} - 2\right )^{\frac{3}{4}}}{1053} + \frac{128 \sqrt{6} \sqrt [4]{- \frac{3 x^{2}}{2} + 1} E\left (\frac{\operatorname{asin}{\left (\frac{\sqrt{6} x}{2} \right )}}{2}\middle | 2\right )}{3159 \sqrt [4]{3 x^{2} - 2}} \]

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*sqrt(6)*(-3*x**2/2 + 1)**(1/4)*elliptic_e(asin(sqrt(6)*x/
2)/2, 2)/(3159*(3*x**2 - 2)**(1/4))

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Mathematica [C]  time = 0.0434361, size = 68, normalized size = 0.26 \[ \frac{2 x \left (16\ 2^{3/4} \sqrt [4]{2-3 x^2} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{3}{2};\frac{3 x^2}{2}\right )+81 x^6+6 x^4+8 x^2-32\right )}{1053 \sqrt [4]{3 x^2-2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*x*(-32 + 8*x^2 + 6*x^4 + 81*x^6 + 16*2^(3/4)*(2 - 3*x^2)^(1/4)*Hypergeometric
2F1[1/4, 1/2, 3/2, (3*x^2)/2]))/(1053*(-2 + 3*x^2)^(1/4))

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Maple [C]  time = 0.06, size = 65, normalized size = 0.3 \[{\frac{2\,x \left ( 27\,{x}^{4}+20\,{x}^{2}+16 \right ) }{1053} \left ( 3\,{x}^{2}-2 \right ) ^{{\frac{3}{4}}}}+{\frac{32\,{2}^{3/4}x}{1053}\sqrt [4]{-{\it signum} \left ( -1+{\frac{3\,{x}^{2}}{2}} \right ) }{\mbox{$_2$F$_1$}({\frac{1}{4}},{\frac{1}{2}};\,{\frac{3}{2}};\,{\frac{3\,{x}^{2}}{2}})}{\frac{1}{\sqrt [4]{{\it signum} \left ( -1+{\frac{3\,{x}^{2}}{2}} \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^6/(3*x^2-2)^(1/4),x)

[Out]

2/1053*x*(27*x^4+20*x^2+16)*(3*x^2-2)^(3/4)+32/1053*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)

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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 = 2.86186, size = 31, normalized size = 0.12 \[ \frac{2^{\frac{3}{4}} x^{7} e^{\frac{15 i \pi }{4}}{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{7}{2} \\ \frac{9}{2} \end{matrix}\middle |{\frac{3 x^{2}}{2}} \right )}}{14} \]

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*exp(15*I*pi/4)*hyper((1/4, 7/2), (9/2,), 3*x**2/2)/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)