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

Optimal. Leaf size=242 \[ \frac{8}{135} \left (-3 x^2-2\right )^{3/4} x+\frac{32 \sqrt [4]{-3 x^2-2} x}{135 \left (\sqrt{-3 x^2-2}+\sqrt{2}\right )}-\frac{16 \sqrt [4]{2} \sqrt{-\frac{x^2}{\left (\sqrt{-3 x^2-2}+\sqrt{2}\right )^2}} \left (\sqrt{-3 x^2-2}+\sqrt{2}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{-3 x^2-2}}{\sqrt [4]{2}}\right )|\frac{1}{2}\right )}{135 \sqrt{3} x}+\frac{32 \sqrt [4]{2} \sqrt{-\frac{x^2}{\left (\sqrt{-3 x^2-2}+\sqrt{2}\right )^2}} \left (\sqrt{-3 x^2-2}+\sqrt{2}\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{-3 x^2-2}}{\sqrt [4]{2}}\right )|\frac{1}{2}\right )}{135 \sqrt{3} x}-\frac{2}{27} \left (-3 x^2-2\right )^{3/4} x^3 \]

[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] +
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])/(135*Sqrt[3]*x) - (16*2^(1/4)*Sqrt[-(x^2/(Sqrt[2] + Sqr
t[-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 = 0.289821, antiderivative size = 242, 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] +
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])/(135*Sqrt[3]*x) - (16*2^(1/4)*Sqrt[-(x^2/(Sqrt[2] + Sqr
t[-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 [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{2 x^{3} \left (- 3 x^{2} - 2\right )^{\frac{3}{4}}}{27} + \frac{8 x \left (- 3 x^{2} - 2\right )^{\frac{3}{4}}}{135} + \frac{32 x}{135 \sqrt [4]{- 3 x^{2} - 2}} + \frac{32 \int \frac{1}{\left (- 3 x^{2} - 2\right )^{\frac{5}{4}}}\, dx}{135} \]

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*x/(135*(-3*x
**2 - 2)**(1/4)) + 32*Integral((-3*x**2 - 2)**(-5/4), x)/135

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Mathematica [C]  time = 0.0374198, size = 63, normalized size = 0.26 \[ \frac{2 x \left (4\ 2^{3/4} \sqrt [4]{3 x^2+2} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{3}{2};-\frac{3 x^2}{2}\right )+15 x^4-2 x^2-8\right )}{135 \sqrt [4]{-3 x^2-2}} \]

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 = 0.028, size = 48, normalized size = 0.2 \[{\frac{2\,x \left ( 5\,{x}^{2}-4 \right ) \left ( 3\,{x}^{2}+2 \right ) }{135}{\frac{1}{\sqrt [4]{-3\,{x}^{2}-2}}}}-{\frac{8\, \left ( -1 \right ) ^{3/4}{2}^{3/4}x}{135}{\mbox{$_2$F$_1$}({\frac{1}{4}},{\frac{1}{2}};\,{\frac{3}{2}};\,-{\frac{3\,{x}^{2}}{2}})}} \]

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*(-1)^(3/4)*2^(3/4)*x*hypergeo
m([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. \[ \frac{405 \, x{\rm integral}\left (-\frac{64 \,{\left (-3 \, x^{2} - 2\right )}^{\frac{3}{4}}}{405 \,{\left (3 \, x^{4} + 2 \, x^{2}\right )}}, x\right ) - 2 \,{\left (15 \, x^{4} - 12 \, x^{2} + 16\right )}{\left (-3 \, x^{2} - 2\right )}^{\frac{3}{4}}}{405 \, x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/405*(405*x*integral(-64/405*(-3*x^2 - 2)^(3/4)/(3*x^4 + 2*x^2), x) - 2*(15*x^4
- 12*x^2 + 16)*(-3*x^2 - 2)^(3/4))/x

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Sympy [A]  time = 2.34293, size = 34, normalized size = 0.14 \[ \frac{2^{\frac{3}{4}} x^{5} e^{- \frac{i \pi }{4}}{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{5}{2} \\ \frac{7}{2} \end{matrix}\middle |{\frac{3 x^{2} e^{i \pi }}{2}} \right )}}{10} \]

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*exp(-I*pi/4)*hyper((1/4, 5/2), (7/2,), 3*x**2*exp_polar(I*pi)/2)/1
0

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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)