3.580 \(\int \frac{x^4}{\sqrt [3]{1-x^3} \left (1+x^3\right )} \, dx\)

Optimal. Leaf size=26 \[ \frac{1}{5} x^5 F_1\left (\frac{5}{3};\frac{1}{3},1;\frac{8}{3};x^3,-x^3\right ) \]

[Out]

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

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Rubi [A]  time = 0.0591777, antiderivative size = 26, 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 \[ \frac{1}{5} x^5 F_1\left (\frac{5}{3};\frac{1}{3},1;\frac{8}{3};x^3,-x^3\right ) \]

Antiderivative was successfully verified.

[In]  Int[x^4/((1 - x^3)^(1/3)*(1 + x^3)),x]

[Out]

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

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Rubi in Sympy [A]  time = 6.17415, size = 17, normalized size = 0.65 \[ \frac{x^{5} \operatorname{appellf_{1}}{\left (\frac{5}{3},\frac{1}{3},1,\frac{8}{3},x^{3},- x^{3} \right )}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**4/(-x**3+1)**(1/3)/(x**3+1),x)

[Out]

x**5*appellf1(5/3, 1/3, 1, 8/3, x**3, -x**3)/5

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Mathematica [B]  time = 0.16789, size = 115, normalized size = 4.42 \[ -\frac{8 x^5 F_1\left (\frac{5}{3};\frac{1}{3},1;\frac{8}{3};x^3,-x^3\right )}{5 \sqrt [3]{1-x^3} \left (x^3+1\right ) \left (x^3 \left (3 F_1\left (\frac{8}{3};\frac{1}{3},2;\frac{11}{3};x^3,-x^3\right )-F_1\left (\frac{8}{3};\frac{4}{3},1;\frac{11}{3};x^3,-x^3\right )\right )-8 F_1\left (\frac{5}{3};\frac{1}{3},1;\frac{8}{3};x^3,-x^3\right )\right )} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[x^4/((1 - x^3)^(1/3)*(1 + x^3)),x]

[Out]

(-8*x^5*AppellF1[5/3, 1/3, 1, 8/3, x^3, -x^3])/(5*(1 - x^3)^(1/3)*(1 + x^3)*(-8*
AppellF1[5/3, 1/3, 1, 8/3, x^3, -x^3] + x^3*(3*AppellF1[8/3, 1/3, 2, 11/3, x^3,
-x^3] - AppellF1[8/3, 4/3, 1, 11/3, x^3, -x^3])))

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Maple [F]  time = 0.059, size = 0, normalized size = 0. \[ \int{\frac{{x}^{4}}{{x}^{3}+1}{\frac{1}{\sqrt [3]{-{x}^{3}+1}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4/(-x^3+1)^(1/3)/(x^3+1),x)

[Out]

int(x^4/(-x^3+1)^(1/3)/(x^3+1),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{4}}{{\left (x^{3} + 1\right )}{\left (-x^{3} + 1\right )}^{\frac{1}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/((x^3 + 1)*(-x^3 + 1)^(1/3)),x, algorithm="maxima")

[Out]

integrate(x^4/((x^3 + 1)*(-x^3 + 1)^(1/3)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/((x^3 + 1)*(-x^3 + 1)^(1/3)),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{4}}{\sqrt [3]{- \left (x - 1\right ) \left (x^{2} + x + 1\right )} \left (x + 1\right ) \left (x^{2} - x + 1\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**4/(-x**3+1)**(1/3)/(x**3+1),x)

[Out]

Integral(x**4/((-(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. \[ \int \frac{x^{4}}{{\left (x^{3} + 1\right )}{\left (-x^{3} + 1\right )}^{\frac{1}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/((x^3 + 1)*(-x^3 + 1)^(1/3)),x, algorithm="giac")

[Out]

integrate(x^4/((x^3 + 1)*(-x^3 + 1)^(1/3)), x)