3.2576 \(\int \frac{1}{(1-x)^{2/3} \left (1+x+x^2\right )^{2/3}} \, dx\)

Optimal. Leaf size=45 \[ \frac{x \left (1-x^3\right )^{2/3} \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{4}{3};x^3\right )}{(1-x)^{2/3} \left (x^2+x+1\right )^{2/3}} \]

[Out]

(x*(1 - x^3)^(2/3)*Hypergeometric2F1[1/3, 2/3, 4/3, x^3])/((1 - x)^(2/3)*(1 + x
+ x^2)^(2/3))

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Rubi [A]  time = 0.0412461, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ \frac{x \left (1-x^3\right )^{2/3} \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{4}{3};x^3\right )}{(1-x)^{2/3} \left (x^2+x+1\right )^{2/3}} \]

Antiderivative was successfully verified.

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

[Out]

(x*(1 - x^3)^(2/3)*Hypergeometric2F1[1/3, 2/3, 4/3, x^3])/((1 - x)^(2/3)*(1 + x
+ x^2)^(2/3))

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Rubi in Sympy [A]  time = 5.26969, size = 36, normalized size = 0.8 \[ \frac{x \sqrt [3]{- x + 1} \sqrt [3]{x^{2} + x + 1}{{}_{2}F_{1}\left (\begin{matrix} \frac{2}{3}, \frac{1}{3} \\ \frac{4}{3} \end{matrix}\middle |{x^{3}} \right )}}{\sqrt [3]{- x^{3} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*(-x + 1)**(1/3)*(x**2 + x + 1)**(1/3)*hyper((2/3, 1/3), (4/3,), x**3)/(-x**3 +
1)**(1/3)

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Mathematica [C]  time = 0.176884, size = 163, normalized size = 3.62 \[ \frac{\sqrt [3]{1-x} \left (2 i x+\sqrt{3}+i\right ) \left (\frac{-\left (\sqrt{3}-3 i\right ) x+\sqrt{3}+3 i}{\left (\sqrt{3}+3 i\right ) x-\sqrt{3}+3 i}\right )^{2/3} \left (\left (\sqrt{3}+3 i\right ) x-\sqrt{3}+3 i\right )^2 \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{4}{3};-\frac{2 \sqrt{3} (1-x)}{\left (3 i+\sqrt{3}\right ) (x-1)+6 i}\right )}{4 \left (\sqrt{3}+3 i\right ) \left (x^2+x+1\right )^{5/3}} \]

Antiderivative was successfully verified.

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

[Out]

((1 - x)^(1/3)*(I + Sqrt[3] + (2*I)*x)*((3*I + Sqrt[3] - (-3*I + Sqrt[3])*x)/(3*
I - Sqrt[3] + (3*I + Sqrt[3])*x))^(2/3)*(3*I - Sqrt[3] + (3*I + Sqrt[3])*x)^2*Hy
pergeometric2F1[1/3, 2/3, 4/3, (-2*Sqrt[3]*(1 - x))/(6*I + (3*I + Sqrt[3])*(-1 +
x))])/(4*(3*I + Sqrt[3])*(1 + x + x^2)^(5/3))

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Maple [F]  time = 0.139, size = 0, normalized size = 0. \[ \int{1 \left ( 1-x \right ) ^{-{\frac{2}{3}}} \left ({x}^{2}+x+1 \right ) ^{-{\frac{2}{3}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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