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

Optimal. Leaf size=26 \[ \frac{1}{2} x^2 F_1\left (\frac{2}{3};\frac{1}{3},1;\frac{5}{3};x^3,-x^3\right ) \]

[Out]

(x^2*AppellF1[2/3, 1/3, 1, 5/3, x^3, -x^3])/2

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Rubi [A]  time = 0.0462526, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05 \[ \frac{1}{2} x^2 F_1\left (\frac{2}{3};\frac{1}{3},1;\frac{5}{3};x^3,-x^3\right ) \]

Antiderivative was successfully verified.

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

[Out]

(x^2*AppellF1[2/3, 1/3, 1, 5/3, x^3, -x^3])/2

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Rubi in Sympy [A]  time = 5.14259, size = 17, normalized size = 0.65 \[ \frac{x^{2} \operatorname{appellf_{1}}{\left (\frac{2}{3},\frac{1}{3},1,\frac{5}{3},x^{3},- x^{3} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [B]  time = 0.160272, size = 115, normalized size = 4.42 \[ -\frac{5 x^2 F_1\left (\frac{2}{3};\frac{1}{3},1;\frac{5}{3};x^3,-x^3\right )}{2 \sqrt [3]{1-x^3} \left (x^3+1\right ) \left (x^3 \left (3 F_1\left (\frac{5}{3};\frac{1}{3},2;\frac{8}{3};x^3,-x^3\right )-F_1\left (\frac{5}{3};\frac{4}{3},1;\frac{8}{3};x^3,-x^3\right )\right )-5 F_1\left (\frac{2}{3};\frac{1}{3},1;\frac{5}{3};x^3,-x^3\right )\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

(-5*x^2*AppellF1[2/3, 1/3, 1, 5/3, x^3, -x^3])/(2*(1 - x^3)^(1/3)*(1 + x^3)*(-5*
AppellF1[2/3, 1/3, 1, 5/3, x^3, -x^3] + x^3*(3*AppellF1[5/3, 1/3, 2, 8/3, x^3, -
x^3] - AppellF1[5/3, 4/3, 1, 8/3, x^3, -x^3])))

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Maple [F]  time = 0.056, size = 0, normalized size = 0. \[ \int{\frac{x}{{x}^{3}+1}{\frac{1}{\sqrt [3]{-{x}^{3}+1}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(-x^3+1)^(1/3)/(x^3+1),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 1.79286, size = 433, normalized size = 16.65 \[ \frac{1}{216} \, \sqrt{3} 2^{\frac{2}{3}}{\left (2 \, \sqrt{3} \left (-1\right )^{\frac{1}{3}} \log \left (-\frac{6 \cdot 2^{\frac{2}{3}}{\left (-x^{3} + 1\right )}^{\frac{2}{3}} x^{2} - 6 \, \left (-1\right )^{\frac{2}{3}}{\left (x^{4} - x\right )}{\left (-x^{3} + 1\right )}^{\frac{1}{3}} - 2^{\frac{1}{3}} \left (-1\right )^{\frac{1}{3}}{\left (x^{6} + 2 \, x^{3} + 1\right )}}{x^{6} + 2 \, x^{3} + 1}\right ) - \sqrt{3} \left (-1\right )^{\frac{1}{3}} \log \left (\frac{2^{\frac{2}{3}} \left (-1\right )^{\frac{2}{3}}{\left (x^{12} - 32 \, x^{9} + 78 \, x^{6} - 32 \, x^{3} + 1\right )} - 24 \, \left (-1\right )^{\frac{1}{3}}{\left (x^{8} - 4 \, x^{5} + x^{2}\right )}{\left (-x^{3} + 1\right )}^{\frac{2}{3}} + 6 \cdot 2^{\frac{1}{3}}{\left (x^{10} - 11 \, x^{7} + 11 \, x^{4} - x\right )}{\left (-x^{3} + 1\right )}^{\frac{1}{3}}}{x^{12} + 4 \, x^{9} + 6 \, x^{6} + 4 \, x^{3} + 1}\right ) + 6 \, \left (-1\right )^{\frac{1}{3}} \arctan \left (\frac{12 \, \sqrt{3} 2^{\frac{2}{3}}{\left (-x^{3} + 1\right )}^{\frac{2}{3}} x^{2} + 6 \, \sqrt{3} \left (-1\right )^{\frac{2}{3}}{\left (x^{4} - x\right )}{\left (-x^{3} + 1\right )}^{\frac{1}{3}} + \sqrt{3} 2^{\frac{1}{3}} \left (-1\right )^{\frac{1}{3}}{\left (x^{6} + 2 \, x^{3} + 1\right )}}{3 \,{\left (6 \, \left (-1\right )^{\frac{2}{3}}{\left (x^{4} - x\right )}{\left (-x^{3} + 1\right )}^{\frac{1}{3}} - 2^{\frac{1}{3}} \left (-1\right )^{\frac{1}{3}}{\left (x^{6} + 2 \, x^{3} + 1\right )}\right )}}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/216*sqrt(3)*2^(2/3)*(2*sqrt(3)*(-1)^(1/3)*log(-(6*2^(2/3)*(-x^3 + 1)^(2/3)*x^2
- 6*(-1)^(2/3)*(x^4 - x)*(-x^3 + 1)^(1/3) - 2^(1/3)*(-1)^(1/3)*(x^6 + 2*x^3 + 1
))/(x^6 + 2*x^3 + 1)) - sqrt(3)*(-1)^(1/3)*log((2^(2/3)*(-1)^(2/3)*(x^12 - 32*x^
9 + 78*x^6 - 32*x^3 + 1) - 24*(-1)^(1/3)*(x^8 - 4*x^5 + x^2)*(-x^3 + 1)^(2/3) +
6*2^(1/3)*(x^10 - 11*x^7 + 11*x^4 - x)*(-x^3 + 1)^(1/3))/(x^12 + 4*x^9 + 6*x^6 +
4*x^3 + 1)) + 6*(-1)^(1/3)*arctan(1/3*(12*sqrt(3)*2^(2/3)*(-x^3 + 1)^(2/3)*x^2
+ 6*sqrt(3)*(-1)^(2/3)*(x^4 - x)*(-x^3 + 1)^(1/3) + sqrt(3)*2^(1/3)*(-1)^(1/3)*(
x^6 + 2*x^3 + 1))/(6*(-1)^(2/3)*(x^4 - x)*(-x^3 + 1)^(1/3) - 2^(1/3)*(-1)^(1/3)*
(x^6 + 2*x^3 + 1))))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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