3.595 \(\int \frac{x^3}{\left (1-x^3\right )^{2/3} \left (1+x^3\right )} \, dx\)

Optimal. Leaf size=26 \[ \frac{1}{4} x^4 F_1\left (\frac{4}{3};\frac{2}{3},1;\frac{7}{3};x^3,-x^3\right ) \]

[Out]

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

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Rubi [A]  time = 0.0623624, 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 \[ \frac{1}{4} x^4 F_1\left (\frac{4}{3};\frac{2}{3},1;\frac{7}{3};x^3,-x^3\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 6.85313, size = 17, normalized size = 0.65 \[ \frac{x^{4} \operatorname{appellf_{1}}{\left (\frac{4}{3},\frac{2}{3},1,\frac{7}{3},x^{3},- x^{3} \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**4*appellf1(4/3, 2/3, 1, 7/3, x**3, -x**3)/4

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Mathematica [B]  time = 0.174404, size = 115, normalized size = 4.42 \[ -\frac{7 x^4 F_1\left (\frac{4}{3};\frac{2}{3},1;\frac{7}{3};x^3,-x^3\right )}{4 \left (1-x^3\right )^{2/3} \left (x^3+1\right ) \left (x^3 \left (3 F_1\left (\frac{7}{3};\frac{2}{3},2;\frac{10}{3};x^3,-x^3\right )-2 F_1\left (\frac{7}{3};\frac{5}{3},1;\frac{10}{3};x^3,-x^3\right )\right )-7 F_1\left (\frac{4}{3};\frac{2}{3},1;\frac{7}{3};x^3,-x^3\right )\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

(-7*x^4*AppellF1[4/3, 2/3, 1, 7/3, x^3, -x^3])/(4*(1 - x^3)^(2/3)*(1 + x^3)*(-7*
AppellF1[4/3, 2/3, 1, 7/3, x^3, -x^3] + x^3*(3*AppellF1[7/3, 2/3, 2, 10/3, x^3,
-x^3] - 2*AppellF1[7/3, 5/3, 1, 10/3, x^3, -x^3])))

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Maple [F]  time = 0.059, size = 0, normalized size = 0. \[ \int{\frac{{x}^{3}}{{x}^{3}+1} \left ( -{x}^{3}+1 \right ) ^{-{\frac{2}{3}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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