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

Optimal. Leaf size=24 \[ -\frac{F_1\left (-\frac{1}{3};\frac{1}{3},1;\frac{2}{3};x^3,-x^3\right )}{x} \]

[Out]

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

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Rubi [A]  time = 0.0622655, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045 \[ -\frac{F_1\left (-\frac{1}{3};\frac{1}{3},1;\frac{2}{3};x^3,-x^3\right )}{x} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 6.22787, size = 17, normalized size = 0.71 \[ - \frac{\operatorname{appellf_{1}}{\left (- \frac{1}{3},\frac{1}{3},1,\frac{2}{3},x^{3},- x^{3} \right )}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [B]  time = 0.320955, size = 229, normalized size = 9.54 \[ \frac{\frac{25 x^3 F_1\left (\frac{2}{3};\frac{1}{3},1;\frac{5}{3};x^3,-x^3\right )}{\left (x^3+1\right ) \left (x^3 \left (3 F_1\left (\frac{5}{3};\frac{1}{3},2;\frac{8}{3};x^3,-x^3\right )-F_1\left (\frac{5}{3};\frac{4}{3},1;\frac{8}{3};x^3,-x^3\right )\right )-5 F_1\left (\frac{2}{3};\frac{1}{3},1;\frac{5}{3};x^3,-x^3\right )\right )}+\frac{8 x^6 F_1\left (\frac{5}{3};\frac{1}{3},1;\frac{8}{3};x^3,-x^3\right )}{\left (x^3+1\right ) \left (x^3 \left (3 F_1\left (\frac{8}{3};\frac{1}{3},2;\frac{11}{3};x^3,-x^3\right )-F_1\left (\frac{8}{3};\frac{4}{3},1;\frac{11}{3};x^3,-x^3\right )\right )-8 F_1\left (\frac{5}{3};\frac{1}{3},1;\frac{8}{3};x^3,-x^3\right )\right )}+5 x^3-5}{5 x \sqrt [3]{1-x^3}} \]

Warning: Unable to verify antiderivative.

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

[Out]

(-5 + 5*x^3 + (25*x^3*AppellF1[2/3, 1/3, 1, 5/3, x^3, -x^3])/((1 + x^3)*(-5*Appe
llF1[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]))) + (8*x^6*AppellF1[5/3, 1/3, 1, 8/3,
x^3, -x^3])/((1 + x^3)*(-8*AppellF1[5/3, 1/3, 1, 8/3, x^3, -x^3] + x^3*(3*Appell
F1[8/3, 1/3, 2, 11/3, x^3, -x^3] - AppellF1[8/3, 4/3, 1, 11/3, x^3, -x^3]))))/(5
*x*(1 - x^3)^(1/3))

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Maple [F]  time = 0.082, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{2} \left ({x}^{3}+1 \right ) }{\frac{1}{\sqrt [3]{-{x}^{3}+1}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (x^{5} + x^{2}\right )}{\left (-x^{3} + 1\right )}^{\frac{1}{3}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(1/((x^5 + x^2)*(-x^3 + 1)^(1/3)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/(x**2*(-(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. \[ \int \frac{1}{{\left (x^{3} + 1\right )}{\left (-x^{3} + 1\right )}^{\frac{1}{3}} x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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