3.913 \(\int \frac{x^4}{\left (-2-3 x^2\right )^{3/4}} \, dx\)

Optimal. Leaf size=121 \[ \frac{8}{63} \sqrt [4]{-3 x^2-2} x-\frac{8\ 2^{3/4} \sqrt{-\frac{x^2}{\left (\sqrt{-3 x^2-2}+\sqrt{2}\right )^2}} \left (\sqrt{-3 x^2-2}+\sqrt{2}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{-3 x^2-2}}{\sqrt [4]{2}}\right )|\frac{1}{2}\right )}{63 \sqrt{3} x}-\frac{2}{21} \sqrt [4]{-3 x^2-2} x^3 \]

[Out]

(8*x*(-2 - 3*x^2)^(1/4))/63 - (2*x^3*(-2 - 3*x^2)^(1/4))/21 - (8*2^(3/4)*Sqrt[-(
x^2/(Sqrt[2] + Sqrt[-2 - 3*x^2])^2)]*(Sqrt[2] + Sqrt[-2 - 3*x^2])*EllipticF[2*Ar
cTan[(-2 - 3*x^2)^(1/4)/2^(1/4)], 1/2])/(63*Sqrt[3]*x)

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Rubi [A]  time = 0.133272, antiderivative size = 121, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{8}{63} \sqrt [4]{-3 x^2-2} x-\frac{8\ 2^{3/4} \sqrt{-\frac{x^2}{\left (\sqrt{-3 x^2-2}+\sqrt{2}\right )^2}} \left (\sqrt{-3 x^2-2}+\sqrt{2}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{-3 x^2-2}}{\sqrt [4]{2}}\right )|\frac{1}{2}\right )}{63 \sqrt{3} x}-\frac{2}{21} \sqrt [4]{-3 x^2-2} x^3 \]

Antiderivative was successfully verified.

[In]  Int[x^4/(-2 - 3*x^2)^(3/4),x]

[Out]

(8*x*(-2 - 3*x^2)^(1/4))/63 - (2*x^3*(-2 - 3*x^2)^(1/4))/21 - (8*2^(3/4)*Sqrt[-(
x^2/(Sqrt[2] + Sqrt[-2 - 3*x^2])^2)]*(Sqrt[2] + Sqrt[-2 - 3*x^2])*EllipticF[2*Ar
cTan[(-2 - 3*x^2)^(1/4)/2^(1/4)], 1/2])/(63*Sqrt[3]*x)

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Rubi in Sympy [A]  time = 5.7245, size = 80, normalized size = 0.66 \[ - \frac{2 x^{3} \sqrt [4]{- 3 x^{2} - 2}}{21} + \frac{8 x \sqrt [4]{- 3 x^{2} - 2}}{63} + \frac{32 \sqrt{6} \left (\frac{3 x^{2}}{2} + 1\right )^{\frac{3}{4}} F\left (\frac{\operatorname{atan}{\left (\frac{\sqrt{6} x}{2} \right )}}{2}\middle | 2\right )}{189 \left (- 3 x^{2} - 2\right )^{\frac{3}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*x**3*(-3*x**2 - 2)**(1/4)/21 + 8*x*(-3*x**2 - 2)**(1/4)/63 + 32*sqrt(6)*(3*x*
*2/2 + 1)**(3/4)*elliptic_f(atan(sqrt(6)*x/2)/2, 2)/(189*(-3*x**2 - 2)**(3/4))

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Mathematica [C]  time = 0.0350301, size = 63, normalized size = 0.52 \[ \frac{2 x \left (4 \sqrt [4]{2} \left (3 x^2+2\right )^{3/4} \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{3}{2};-\frac{3 x^2}{2}\right )+9 x^4-6 x^2-8\right )}{63 \left (-3 x^2-2\right )^{3/4}} \]

Antiderivative was successfully verified.

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

[Out]

(2*x*(-8 - 6*x^2 + 9*x^4 + 4*2^(1/4)*(2 + 3*x^2)^(3/4)*Hypergeometric2F1[1/2, 3/
4, 3/2, (-3*x^2)/2]))/(63*(-2 - 3*x^2)^(3/4))

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Maple [F]  time = 0.021, size = 0, normalized size = 0. \[ \int{{x}^{4} \left ( -3\,{x}^{2}-2 \right ) ^{-{\frac{3}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4/(-3*x^2-2)^(3/4),x)

[Out]

int(x^4/(-3*x^2-2)^(3/4),x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x^4/(-3*x^2 - 2)^(3/4), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[ -\frac{2}{63} \,{\left (3 \, x^{3} - 4 \, x\right )}{\left (-3 \, x^{2} - 2\right )}^{\frac{1}{4}} +{\rm integral}\left (-\frac{16 \,{\left (-3 \, x^{2} - 2\right )}^{\frac{1}{4}}}{63 \,{\left (3 \, x^{2} + 2\right )}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/63*(3*x^3 - 4*x)*(-3*x^2 - 2)^(1/4) + integral(-16/63*(-3*x^2 - 2)^(1/4)/(3*x
^2 + 2), x)

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Sympy [A]  time = 2.26923, size = 36, normalized size = 0.3 \[ \frac{\sqrt [4]{2} x^{5} e^{- \frac{3 i \pi }{4}}{{}_{2}F_{1}\left (\begin{matrix} \frac{3}{4}, \frac{5}{2} \\ \frac{7}{2} \end{matrix}\middle |{\frac{3 x^{2} e^{i \pi }}{2}} \right )}}{10} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x^4/(-3*x^2 - 2)^(3/4), x)