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

Optimal. Leaf size=240 \[ \frac{8}{135} \left (3 x^2-2\right )^{3/4} x+\frac{32 \sqrt [4]{3 x^2-2} x}{135 \left (\sqrt{3 x^2-2}+\sqrt{2}\right )}+\frac{16 \sqrt [4]{2} \sqrt{\frac{x^2}{\left (\sqrt{3 x^2-2}+\sqrt{2}\right )^2}} \left (\sqrt{3 x^2-2}+\sqrt{2}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{3 x^2-2}}{\sqrt [4]{2}}\right )|\frac{1}{2}\right )}{135 \sqrt{3} x}-\frac{32 \sqrt [4]{2} \sqrt{\frac{x^2}{\left (\sqrt{3 x^2-2}+\sqrt{2}\right )^2}} \left (\sqrt{3 x^2-2}+\sqrt{2}\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{3 x^2-2}}{\sqrt [4]{2}}\right )|\frac{1}{2}\right )}{135 \sqrt{3} x}+\frac{2}{27} \left (3 x^2-2\right )^{3/4} x^3 \]

[Out]

(8*x*(-2 + 3*x^2)^(3/4))/135 + (2*x^3*(-2 + 3*x^2)^(3/4))/27 + (32*x*(-2 + 3*x^2
)^(1/4))/(135*(Sqrt[2] + Sqrt[-2 + 3*x^2])) - (32*2^(1/4)*Sqrt[x^2/(Sqrt[2] + Sq
rt[-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])/(135*Sqrt[3]*x) + (16*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])/(135*Sqrt[3]*x)

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Rubi [A]  time = 0.282993, antiderivative size = 240, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{8}{135} \left (3 x^2-2\right )^{3/4} x+\frac{32 \sqrt [4]{3 x^2-2} x}{135 \left (\sqrt{3 x^2-2}+\sqrt{2}\right )}+\frac{16 \sqrt [4]{2} \sqrt{\frac{x^2}{\left (\sqrt{3 x^2-2}+\sqrt{2}\right )^2}} \left (\sqrt{3 x^2-2}+\sqrt{2}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{3 x^2-2}}{\sqrt [4]{2}}\right )|\frac{1}{2}\right )}{135 \sqrt{3} x}-\frac{32 \sqrt [4]{2} \sqrt{\frac{x^2}{\left (\sqrt{3 x^2-2}+\sqrt{2}\right )^2}} \left (\sqrt{3 x^2-2}+\sqrt{2}\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{3 x^2-2}}{\sqrt [4]{2}}\right )|\frac{1}{2}\right )}{135 \sqrt{3} x}+\frac{2}{27} \left (3 x^2-2\right )^{3/4} x^3 \]

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*x*(-2 + 3*x^2
)^(1/4))/(135*(Sqrt[2] + Sqrt[-2 + 3*x^2])) - (32*2^(1/4)*Sqrt[x^2/(Sqrt[2] + Sq
rt[-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])/(135*Sqrt[3]*x) + (16*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])/(135*Sqrt[3]*x)

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Rubi in Sympy [A]  time = 5.85185, size = 75, normalized size = 0.31 \[ \frac{2 x^{3} \left (3 x^{2} - 2\right )^{\frac{3}{4}}}{27} + \frac{8 x \left (3 x^{2} - 2\right )^{\frac{3}{4}}}{135} + \frac{32 \sqrt{6} \sqrt [4]{- \frac{3 x^{2}}{2} + 1} E\left (\frac{\operatorname{asin}{\left (\frac{\sqrt{6} x}{2} \right )}}{2}\middle | 2\right )}{405 \sqrt [4]{3 x^{2} - 2}} \]

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*sqrt(6)*(-3*x**
2/2 + 1)**(1/4)*elliptic_e(asin(sqrt(6)*x/2)/2, 2)/(405*(3*x**2 - 2)**(1/4))

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Mathematica [C]  time = 0.0357802, size = 63, normalized size = 0.26 \[ \frac{2 x \left (4\ 2^{3/4} \sqrt [4]{2-3 x^2} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{3}{2};\frac{3 x^2}{2}\right )+15 x^4+2 x^2-8\right )}{135 \sqrt [4]{3 x^2-2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*x*(-8 + 2*x^2 + 15*x^4 + 4*2^(3/4)*(2 - 3*x^2)^(1/4)*Hypergeometric2F1[1/4, 1
/2, 3/2, (3*x^2)/2]))/(135*(-2 + 3*x^2)^(1/4))

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Maple [C]  time = 0.051, size = 60, normalized size = 0.3 \[{\frac{2\,x \left ( 5\,{x}^{2}+4 \right ) }{135} \left ( 3\,{x}^{2}-2 \right ) ^{{\frac{3}{4}}}}+{\frac{8\,{2}^{3/4}x}{135}\sqrt [4]{-{\it signum} \left ( -1+{\frac{3\,{x}^{2}}{2}} \right ) }{\mbox{$_2$F$_1$}({\frac{1}{4}},{\frac{1}{2}};\,{\frac{3}{2}};\,{\frac{3\,{x}^{2}}{2}})}{\frac{1}{\sqrt [4]{{\it signum} \left ( -1+{\frac{3\,{x}^{2}}{2}} \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4/(3*x^2-2)^(1/4),x)

[Out]

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

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 = 2.46017, size = 31, normalized size = 0.13 \[ \frac{2^{\frac{3}{4}} x^{5} e^{\frac{15 i \pi }{4}}{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{5}{2} \\ \frac{7}{2} \end{matrix}\middle |{\frac{3 x^{2}}{2}} \right )}}{10} \]

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*exp(15*I*pi/4)*hyper((1/4, 5/2), (7/2,), 3*x**2/2)/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)