3.594 \(\int \frac{x^6}{\left (1-x^3\right )^{2/3} \left (1+x^3\right )} \, dx\)

Optimal. Leaf size=26 \[ \frac{1}{7} x^7 F_1\left (\frac{7}{3};\frac{2}{3},1;\frac{10}{3};x^3,-x^3\right ) \]

[Out]

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

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Rubi [A]  time = 0.0609548, 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 \[ \frac{1}{7} x^7 F_1\left (\frac{7}{3};\frac{2}{3},1;\frac{10}{3};x^3,-x^3\right ) \]

Antiderivative was successfully verified.

[In]  Int[x^6/((1 - x^3)^(2/3)*(1 + x^3)),x]

[Out]

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

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Rubi in Sympy [A]  time = 6.15043, size = 17, normalized size = 0.65 \[ \frac{x^{7} \operatorname{appellf_{1}}{\left (\frac{7}{3},\frac{2}{3},1,\frac{10}{3},x^{3},- x^{3} \right )}}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**6/(-x**3+1)**(2/3)/(x**3+1),x)

[Out]

x**7*appellf1(7/3, 2/3, 1, 10/3, x**3, -x**3)/7

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Mathematica [B]  time = 0.195612, size = 115, normalized size = 4.42 \[ \frac{1}{2} x \sqrt [3]{1-x^3} \left (-\frac{4 F_1\left (\frac{1}{3};-\frac{1}{3},1;\frac{4}{3};x^3,-x^3\right )}{\left (x^3+1\right ) \left (x^3 \left (3 F_1\left (\frac{4}{3};-\frac{1}{3},2;\frac{7}{3};x^3,-x^3\right )+F_1\left (\frac{4}{3};\frac{2}{3},1;\frac{7}{3};x^3,-x^3\right )\right )-4 F_1\left (\frac{1}{3};-\frac{1}{3},1;\frac{4}{3};x^3,-x^3\right )\right )}-1\right ) \]

Warning: Unable to verify antiderivative.

[In]  Integrate[x^6/((1 - x^3)^(2/3)*(1 + x^3)),x]

[Out]

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

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Maple [F]  time = 0.09, size = 0, normalized size = 0. \[ \int{\frac{{x}^{6}}{{x}^{3}+1} \left ( -{x}^{3}+1 \right ) ^{-{\frac{2}{3}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^6/(-x^3+1)^(2/3)/(x^3+1),x)

[Out]

int(x^6/(-x^3+1)^(2/3)/(x^3+1),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{6}}{{\left (x^{3} + 1\right )}{\left (-x^{3} + 1\right )}^{\frac{2}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^6/((x^3 + 1)*(-x^3 + 1)^(2/3)),x, algorithm="maxima")

[Out]

integrate(x^6/((x^3 + 1)*(-x^3 + 1)^(2/3)), x)

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Fricas [A]  time = 1.87377, size = 402, normalized size = 15.46 \[ -\frac{1}{432} \cdot 4^{\frac{2}{3}} \sqrt{3}{\left (18 \cdot 4^{\frac{1}{3}} \sqrt{3}{\left (-x^{3} + 1\right )}^{\frac{1}{3}} x + \sqrt{3} \log \left (\frac{x^{12} - 32 \, x^{9} + 78 \, x^{6} - 32 \, x^{3} + 6 \cdot 4^{\frac{2}{3}}{\left (x^{8} - 4 \, x^{5} + x^{2}\right )}{\left (-x^{3} + 1\right )}^{\frac{2}{3}} + 3 \cdot 4^{\frac{1}{3}}{\left (x^{10} - 11 \, x^{7} + 11 \, x^{4} - x\right )}{\left (-x^{3} + 1\right )}^{\frac{1}{3}} + 1}{x^{12} + 4 \, x^{9} + 6 \, x^{6} + 4 \, x^{3} + 1}\right ) - 2 \, \sqrt{3} \log \left (-\frac{x^{6} + 3 \cdot 4^{\frac{2}{3}}{\left (-x^{3} + 1\right )}^{\frac{2}{3}} x^{2} + 2 \, x^{3} - 3 \cdot 4^{\frac{1}{3}}{\left (x^{4} - x\right )}{\left (-x^{3} + 1\right )}^{\frac{1}{3}} + 1}{x^{6} + 2 \, x^{3} + 1}\right ) + 6 \, \arctan \left (\frac{6 \cdot 4^{\frac{2}{3}} \sqrt{3}{\left (-x^{3} + 1\right )}^{\frac{2}{3}} x^{2} + 3 \cdot 4^{\frac{1}{3}} \sqrt{3}{\left (x^{4} - x\right )}{\left (-x^{3} + 1\right )}^{\frac{1}{3}} - \sqrt{3}{\left (x^{6} + 2 \, x^{3} + 1\right )}}{3 \,{\left (x^{6} + 2 \, x^{3} + 3 \cdot 4^{\frac{1}{3}}{\left (x^{4} - x\right )}{\left (-x^{3} + 1\right )}^{\frac{1}{3}} + 1\right )}}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^6/((x^3 + 1)*(-x^3 + 1)^(2/3)),x, algorithm="fricas")

[Out]

-1/432*4^(2/3)*sqrt(3)*(18*4^(1/3)*sqrt(3)*(-x^3 + 1)^(1/3)*x + sqrt(3)*log((x^1
2 - 32*x^9 + 78*x^6 - 32*x^3 + 6*4^(2/3)*(x^8 - 4*x^5 + x^2)*(-x^3 + 1)^(2/3) +
3*4^(1/3)*(x^10 - 11*x^7 + 11*x^4 - x)*(-x^3 + 1)^(1/3) + 1)/(x^12 + 4*x^9 + 6*x
^6 + 4*x^3 + 1)) - 2*sqrt(3)*log(-(x^6 + 3*4^(2/3)*(-x^3 + 1)^(2/3)*x^2 + 2*x^3
- 3*4^(1/3)*(x^4 - x)*(-x^3 + 1)^(1/3) + 1)/(x^6 + 2*x^3 + 1)) + 6*arctan(1/3*(6
*4^(2/3)*sqrt(3)*(-x^3 + 1)^(2/3)*x^2 + 3*4^(1/3)*sqrt(3)*(x^4 - x)*(-x^3 + 1)^(
1/3) - sqrt(3)*(x^6 + 2*x^3 + 1))/(x^6 + 2*x^3 + 3*4^(1/3)*(x^4 - x)*(-x^3 + 1)^
(1/3) + 1)))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{6}}{\left (- \left (x - 1\right ) \left (x^{2} + x + 1\right )\right )^{\frac{2}{3}} \left (x + 1\right ) \left (x^{2} - x + 1\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**6/(-x**3+1)**(2/3)/(x**3+1),x)

[Out]

Integral(x**6/((-(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. \[ \int \frac{x^{6}}{{\left (x^{3} + 1\right )}{\left (-x^{3} + 1\right )}^{\frac{2}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^6/((x^3 + 1)*(-x^3 + 1)^(2/3)),x, algorithm="giac")

[Out]

integrate(x^6/((x^3 + 1)*(-x^3 + 1)^(2/3)), x)