3.1015 \(\int \frac{x^2}{\sqrt [3]{1-x^2} \left (3+x^2\right )} \, dx\)

Optimal. Leaf size=515 \[ -\frac{3 x}{-\sqrt [3]{1-x^2}-\sqrt{3}+1}-\frac{\sqrt{3} \tan ^{-1}\left (\frac{\sqrt{3} \left (1-\sqrt [3]{2} \sqrt [3]{1-x^2}\right )}{x}\right )}{2\ 2^{2/3}}-\frac{3 \tanh ^{-1}\left (\frac{x}{\sqrt [3]{2} \sqrt [3]{1-x^2}+1}\right )}{2\ 2^{2/3}}+\frac{\sqrt{2} 3^{3/4} \left (1-\sqrt [3]{1-x^2}\right ) \sqrt{\frac{\left (1-x^2\right )^{2/3}+\sqrt [3]{1-x^2}+1}{\left (-\sqrt [3]{1-x^2}-\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{-\sqrt [3]{1-x^2}+\sqrt{3}+1}{-\sqrt [3]{1-x^2}-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{\sqrt{-\frac{1-\sqrt [3]{1-x^2}}{\left (-\sqrt [3]{1-x^2}-\sqrt{3}+1\right )^2}} x}-\frac{3 \sqrt [4]{3} \sqrt{2+\sqrt{3}} \left (1-\sqrt [3]{1-x^2}\right ) \sqrt{\frac{\left (1-x^2\right )^{2/3}+\sqrt [3]{1-x^2}+1}{\left (-\sqrt [3]{1-x^2}-\sqrt{3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac{-\sqrt [3]{1-x^2}+\sqrt{3}+1}{-\sqrt [3]{1-x^2}-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{2 \sqrt{-\frac{1-\sqrt [3]{1-x^2}}{\left (-\sqrt [3]{1-x^2}-\sqrt{3}+1\right )^2}} x}-\frac{\sqrt{3} \tan ^{-1}\left (\frac{\sqrt{3}}{x}\right )}{2\ 2^{2/3}}+\frac{\tanh ^{-1}(x)}{2\ 2^{2/3}} \]

[Out]

(-3*x)/(1 - Sqrt[3] - (1 - x^2)^(1/3)) - (Sqrt[3]*ArcTan[Sqrt[3]/x])/(2*2^(2/3))
- (Sqrt[3]*ArcTan[(Sqrt[3]*(1 - 2^(1/3)*(1 - x^2)^(1/3)))/x])/(2*2^(2/3)) + Arc
Tanh[x]/(2*2^(2/3)) - (3*ArcTanh[x/(1 + 2^(1/3)*(1 - x^2)^(1/3))])/(2*2^(2/3)) -
(3*3^(1/4)*Sqrt[2 + Sqrt[3]]*(1 - (1 - x^2)^(1/3))*Sqrt[(1 + (1 - x^2)^(1/3) +
(1 - x^2)^(2/3))/(1 - Sqrt[3] - (1 - x^2)^(1/3))^2]*EllipticE[ArcSin[(1 + Sqrt[3
] - (1 - x^2)^(1/3))/(1 - Sqrt[3] - (1 - x^2)^(1/3))], -7 + 4*Sqrt[3]])/(2*x*Sqr
t[-((1 - (1 - x^2)^(1/3))/(1 - Sqrt[3] - (1 - x^2)^(1/3))^2)]) + (Sqrt[2]*3^(3/4
)*(1 - (1 - x^2)^(1/3))*Sqrt[(1 + (1 - x^2)^(1/3) + (1 - x^2)^(2/3))/(1 - Sqrt[3
] - (1 - x^2)^(1/3))^2]*EllipticF[ArcSin[(1 + Sqrt[3] - (1 - x^2)^(1/3))/(1 - Sq
rt[3] - (1 - x^2)^(1/3))], -7 + 4*Sqrt[3]])/(x*Sqrt[-((1 - (1 - x^2)^(1/3))/(1 -
Sqrt[3] - (1 - x^2)^(1/3))^2)])

_______________________________________________________________________________________

Rubi [A]  time = 0.447887, antiderivative size = 515, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ -\frac{3 x}{-\sqrt [3]{1-x^2}-\sqrt{3}+1}-\frac{\sqrt{3} \tan ^{-1}\left (\frac{\sqrt{3} \left (1-\sqrt [3]{2} \sqrt [3]{1-x^2}\right )}{x}\right )}{2\ 2^{2/3}}-\frac{3 \tanh ^{-1}\left (\frac{x}{\sqrt [3]{2} \sqrt [3]{1-x^2}+1}\right )}{2\ 2^{2/3}}+\frac{\sqrt{2} 3^{3/4} \left (1-\sqrt [3]{1-x^2}\right ) \sqrt{\frac{\left (1-x^2\right )^{2/3}+\sqrt [3]{1-x^2}+1}{\left (-\sqrt [3]{1-x^2}-\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{-\sqrt [3]{1-x^2}+\sqrt{3}+1}{-\sqrt [3]{1-x^2}-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{\sqrt{-\frac{1-\sqrt [3]{1-x^2}}{\left (-\sqrt [3]{1-x^2}-\sqrt{3}+1\right )^2}} x}-\frac{3 \sqrt [4]{3} \sqrt{2+\sqrt{3}} \left (1-\sqrt [3]{1-x^2}\right ) \sqrt{\frac{\left (1-x^2\right )^{2/3}+\sqrt [3]{1-x^2}+1}{\left (-\sqrt [3]{1-x^2}-\sqrt{3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac{-\sqrt [3]{1-x^2}+\sqrt{3}+1}{-\sqrt [3]{1-x^2}-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{2 \sqrt{-\frac{1-\sqrt [3]{1-x^2}}{\left (-\sqrt [3]{1-x^2}-\sqrt{3}+1\right )^2}} x}-\frac{\sqrt{3} \tan ^{-1}\left (\frac{\sqrt{3}}{x}\right )}{2\ 2^{2/3}}+\frac{\tanh ^{-1}(x)}{2\ 2^{2/3}} \]

Warning: Unable to verify antiderivative.

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

[Out]

(-3*x)/(1 - Sqrt[3] - (1 - x^2)^(1/3)) - (Sqrt[3]*ArcTan[Sqrt[3]/x])/(2*2^(2/3))
- (Sqrt[3]*ArcTan[(Sqrt[3]*(1 - 2^(1/3)*(1 - x^2)^(1/3)))/x])/(2*2^(2/3)) + Arc
Tanh[x]/(2*2^(2/3)) - (3*ArcTanh[x/(1 + 2^(1/3)*(1 - x^2)^(1/3))])/(2*2^(2/3)) -
(3*3^(1/4)*Sqrt[2 + Sqrt[3]]*(1 - (1 - x^2)^(1/3))*Sqrt[(1 + (1 - x^2)^(1/3) +
(1 - x^2)^(2/3))/(1 - Sqrt[3] - (1 - x^2)^(1/3))^2]*EllipticE[ArcSin[(1 + Sqrt[3
] - (1 - x^2)^(1/3))/(1 - Sqrt[3] - (1 - x^2)^(1/3))], -7 + 4*Sqrt[3]])/(2*x*Sqr
t[-((1 - (1 - x^2)^(1/3))/(1 - Sqrt[3] - (1 - x^2)^(1/3))^2)]) + (Sqrt[2]*3^(3/4
)*(1 - (1 - x^2)^(1/3))*Sqrt[(1 + (1 - x^2)^(1/3) + (1 - x^2)^(2/3))/(1 - Sqrt[3
] - (1 - x^2)^(1/3))^2]*EllipticF[ArcSin[(1 + Sqrt[3] - (1 - x^2)^(1/3))/(1 - Sq
rt[3] - (1 - x^2)^(1/3))], -7 + 4*Sqrt[3]])/(x*Sqrt[-((1 - (1 - x^2)^(1/3))/(1 -
Sqrt[3] - (1 - x^2)^(1/3))^2)])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 8.7077, size = 19, normalized size = 0.04 \[ \frac{x^{3} \operatorname{appellf_{1}}{\left (\frac{3}{2},\frac{1}{3},1,\frac{5}{2},x^{2},- \frac{x^{2}}{3} \right )}}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**3*appellf1(3/2, 1/3, 1, 5/2, x**2, -x**2/3)/9

_______________________________________________________________________________________

Mathematica [C]  time = 0.138656, size = 120, normalized size = 0.23 \[ -\frac{5 x^3 F_1\left (\frac{3}{2};\frac{1}{3},1;\frac{5}{2};x^2,-\frac{x^2}{3}\right )}{\sqrt [3]{1-x^2} \left (x^2+3\right ) \left (2 x^2 \left (F_1\left (\frac{5}{2};\frac{1}{3},2;\frac{7}{2};x^2,-\frac{x^2}{3}\right )-F_1\left (\frac{5}{2};\frac{4}{3},1;\frac{7}{2};x^2,-\frac{x^2}{3}\right )\right )-15 F_1\left (\frac{3}{2};\frac{1}{3},1;\frac{5}{2};x^2,-\frac{x^2}{3}\right )\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

(-5*x^3*AppellF1[3/2, 1/3, 1, 5/2, x^2, -x^2/3])/((1 - x^2)^(1/3)*(3 + x^2)*(-15
*AppellF1[3/2, 1/3, 1, 5/2, x^2, -x^2/3] + 2*x^2*(AppellF1[5/2, 1/3, 2, 7/2, x^2
, -x^2/3] - AppellF1[5/2, 4/3, 1, 7/2, x^2, -x^2/3])))

_______________________________________________________________________________________

Maple [F]  time = 0.05, size = 0, normalized size = 0. \[ \int{\frac{{x}^{2}}{{x}^{2}+3}{\frac{1}{\sqrt [3]{-{x}^{2}+1}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(-x^2+1)^(1/3)/(x^2+3),x)

[Out]

int(x^2/(-x^2+1)^(1/3)/(x^2+3),x)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x^2/((x^2 + 3)*(-x^2 + 1)^(1/3)), x)

_______________________________________________________________________________________

Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{\sqrt [3]{- \left (x - 1\right ) \left (x + 1\right )} \left (x^{2} + 3\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(x**2/((-(x - 1)*(x + 1))**(1/3)*(x**2 + 3)), x)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x^2/((x^2 + 3)*(-x^2 + 1)^(1/3)), x)