3.1027 \(\int \frac{x^2}{\sqrt [3]{1-x^2} \left (3+x^2\right )^2} \, dx\)

Optimal. Leaf size=543 \[ -\frac{\left (1-x^2\right )^{2/3} x}{8 \left (x^2+3\right )}+\frac{x}{8 \left (-\sqrt [3]{1-x^2}-\sqrt{3}+1\right )}+\frac{\tan ^{-1}\left (\frac{\sqrt{3} \left (1-\sqrt [3]{2} \sqrt [3]{1-x^2}\right )}{x}\right )}{8\ 2^{2/3} \sqrt{3}}+\frac{\tanh ^{-1}\left (\frac{x}{\sqrt [3]{2} \sqrt [3]{1-x^2}+1}\right )}{8\ 2^{2/3}}-\frac{\left (1-\sqrt [3]{1-x^2}\right ) \sqrt{\frac{\left (1-x^2\right )^{2/3}+\sqrt [3]{1-x^2}+1}{\left (-\sqrt [3]{1-x^2}-\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{-\sqrt [3]{1-x^2}+\sqrt{3}+1}{-\sqrt [3]{1-x^2}-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{4 \sqrt{2} \sqrt [4]{3} \sqrt{-\frac{1-\sqrt [3]{1-x^2}}{\left (-\sqrt [3]{1-x^2}-\sqrt{3}+1\right )^2}} x}+\frac{\sqrt [4]{3} \sqrt{2+\sqrt{3}} \left (1-\sqrt [3]{1-x^2}\right ) \sqrt{\frac{\left (1-x^2\right )^{2/3}+\sqrt [3]{1-x^2}+1}{\left (-\sqrt [3]{1-x^2}-\sqrt{3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac{-\sqrt [3]{1-x^2}+\sqrt{3}+1}{-\sqrt [3]{1-x^2}-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{16 \sqrt{-\frac{1-\sqrt [3]{1-x^2}}{\left (-\sqrt [3]{1-x^2}-\sqrt{3}+1\right )^2}} x}+\frac{\tan ^{-1}\left (\frac{\sqrt{3}}{x}\right )}{8\ 2^{2/3} \sqrt{3}}-\frac{\tanh ^{-1}(x)}{24\ 2^{2/3}} \]

[Out]

-(x*(1 - x^2)^(2/3))/(8*(3 + x^2)) + x/(8*(1 - Sqrt[3] - (1 - x^2)^(1/3))) + Arc
Tan[Sqrt[3]/x]/(8*2^(2/3)*Sqrt[3]) + ArcTan[(Sqrt[3]*(1 - 2^(1/3)*(1 - x^2)^(1/3
)))/x]/(8*2^(2/3)*Sqrt[3]) - ArcTanh[x]/(24*2^(2/3)) + ArcTanh[x/(1 + 2^(1/3)*(1
- x^2)^(1/3))]/(8*2^(2/3)) + (3^(1/4)*Sqrt[2 + Sqrt[3]]*(1 - (1 - x^2)^(1/3))*S
qrt[(1 + (1 - x^2)^(1/3) + (1 - x^2)^(2/3))/(1 - Sqrt[3] - (1 - x^2)^(1/3))^2]*E
llipticE[ArcSin[(1 + Sqrt[3] - (1 - x^2)^(1/3))/(1 - Sqrt[3] - (1 - x^2)^(1/3))]
, -7 + 4*Sqrt[3]])/(16*x*Sqrt[-((1 - (1 - x^2)^(1/3))/(1 - Sqrt[3] - (1 - x^2)^(
1/3))^2)]) - ((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 - Sqrt[3] - (1 - x^2)^(1/3))], -7 + 4*Sqrt[3]])/(4*Sqrt[2]*3^(1/4)*x*Sqr
t[-((1 - (1 - x^2)^(1/3))/(1 - Sqrt[3] - (1 - x^2)^(1/3))^2)])

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Rubi [A]  time = 0.630415, antiderivative size = 543, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.318 \[ -\frac{\left (1-x^2\right )^{2/3} x}{8 \left (x^2+3\right )}+\frac{x}{8 \left (-\sqrt [3]{1-x^2}-\sqrt{3}+1\right )}+\frac{\tan ^{-1}\left (\frac{\sqrt{3} \left (1-\sqrt [3]{2} \sqrt [3]{1-x^2}\right )}{x}\right )}{8\ 2^{2/3} \sqrt{3}}+\frac{\tanh ^{-1}\left (\frac{x}{\sqrt [3]{2} \sqrt [3]{1-x^2}+1}\right )}{8\ 2^{2/3}}-\frac{\left (1-\sqrt [3]{1-x^2}\right ) \sqrt{\frac{\left (1-x^2\right )^{2/3}+\sqrt [3]{1-x^2}+1}{\left (-\sqrt [3]{1-x^2}-\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{-\sqrt [3]{1-x^2}+\sqrt{3}+1}{-\sqrt [3]{1-x^2}-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{4 \sqrt{2} \sqrt [4]{3} \sqrt{-\frac{1-\sqrt [3]{1-x^2}}{\left (-\sqrt [3]{1-x^2}-\sqrt{3}+1\right )^2}} x}+\frac{\sqrt [4]{3} \sqrt{2+\sqrt{3}} \left (1-\sqrt [3]{1-x^2}\right ) \sqrt{\frac{\left (1-x^2\right )^{2/3}+\sqrt [3]{1-x^2}+1}{\left (-\sqrt [3]{1-x^2}-\sqrt{3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac{-\sqrt [3]{1-x^2}+\sqrt{3}+1}{-\sqrt [3]{1-x^2}-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{16 \sqrt{-\frac{1-\sqrt [3]{1-x^2}}{\left (-\sqrt [3]{1-x^2}-\sqrt{3}+1\right )^2}} x}+\frac{\tan ^{-1}\left (\frac{\sqrt{3}}{x}\right )}{8\ 2^{2/3} \sqrt{3}}-\frac{\tanh ^{-1}(x)}{24\ 2^{2/3}} \]

Warning: Unable to verify antiderivative.

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

[Out]

-(x*(1 - x^2)^(2/3))/(8*(3 + x^2)) + x/(8*(1 - Sqrt[3] - (1 - x^2)^(1/3))) + Arc
Tan[Sqrt[3]/x]/(8*2^(2/3)*Sqrt[3]) + ArcTan[(Sqrt[3]*(1 - 2^(1/3)*(1 - x^2)^(1/3
)))/x]/(8*2^(2/3)*Sqrt[3]) - ArcTanh[x]/(24*2^(2/3)) + ArcTanh[x/(1 + 2^(1/3)*(1
- x^2)^(1/3))]/(8*2^(2/3)) + (3^(1/4)*Sqrt[2 + Sqrt[3]]*(1 - (1 - x^2)^(1/3))*S
qrt[(1 + (1 - x^2)^(1/3) + (1 - x^2)^(2/3))/(1 - Sqrt[3] - (1 - x^2)^(1/3))^2]*E
llipticE[ArcSin[(1 + Sqrt[3] - (1 - x^2)^(1/3))/(1 - Sqrt[3] - (1 - x^2)^(1/3))]
, -7 + 4*Sqrt[3]])/(16*x*Sqrt[-((1 - (1 - x^2)^(1/3))/(1 - Sqrt[3] - (1 - x^2)^(
1/3))^2)]) - ((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 - Sqrt[3] - (1 - x^2)^(1/3))], -7 + 4*Sqrt[3]])/(4*Sqrt[2]*3^(1/4)*x*Sqr
t[-((1 - (1 - x^2)^(1/3))/(1 - Sqrt[3] - (1 - x^2)^(1/3))^2)])

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Rubi in Sympy [A]  time = 8.43138, size = 19, normalized size = 0.03 \[ \frac{x^{3} \operatorname{appellf_{1}}{\left (\frac{3}{2},\frac{1}{3},2,\frac{5}{2},x^{2},- \frac{x^{2}}{3} \right )}}{27} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**3*appellf1(3/2, 1/3, 2, 5/2, x**2, -x**2/3)/27

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Mathematica [C]  time = 0.203204, size = 231, normalized size = 0.43 \[ \frac{x \left (\frac{5 x^2 F_1\left (\frac{3}{2};\frac{1}{3},1;\frac{5}{2};x^2,-\frac{x^2}{3}\right )}{2 x^2 \left (F_1\left (\frac{5}{2};\frac{1}{3},2;\frac{7}{2};x^2,-\frac{x^2}{3}\right )-F_1\left (\frac{5}{2};\frac{4}{3},1;\frac{7}{2};x^2,-\frac{x^2}{3}\right )\right )-15 F_1\left (\frac{3}{2};\frac{1}{3},1;\frac{5}{2};x^2,-\frac{x^2}{3}\right )}+\frac{27 F_1\left (\frac{1}{2};\frac{1}{3},1;\frac{3}{2};x^2,-\frac{x^2}{3}\right )}{2 x^2 \left (F_1\left (\frac{3}{2};\frac{4}{3},1;\frac{5}{2};x^2,-\frac{x^2}{3}\right )-F_1\left (\frac{3}{2};\frac{1}{3},2;\frac{5}{2};x^2,-\frac{x^2}{3}\right )\right )+9 F_1\left (\frac{1}{2};\frac{1}{3},1;\frac{3}{2};x^2,-\frac{x^2}{3}\right )}+3 x^2-3\right )}{24 \sqrt [3]{1-x^2} \left (x^2+3\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

(x*(-3 + 3*x^2 + (27*AppellF1[1/2, 1/3, 1, 3/2, x^2, -x^2/3])/(9*AppellF1[1/2, 1
/3, 1, 3/2, x^2, -x^2/3] + 2*x^2*(-AppellF1[3/2, 1/3, 2, 5/2, x^2, -x^2/3] + App
ellF1[3/2, 4/3, 1, 5/2, x^2, -x^2/3])) + (5*x^2*AppellF1[3/2, 1/3, 1, 5/2, x^2,
-x^2/3])/(-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]))))/(24*(1 - x^
2)^(1/3)*(3 + x^2))

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Maple [F]  time = 0.084, size = 0, normalized size = 0. \[ \int{\frac{{x}^{2}}{ \left ({x}^{2}+3 \right ) ^{2}}{\frac{1}{\sqrt [3]{-{x}^{2}+1}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

[Out]

Timed out

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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