3.2573 \(\int \frac{1}{(1+x)^{2/3} \left (1-x+x^2\right )^{2/3}} \, dx\)

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

[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 = 0.0371923, 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 \[ \frac{x \left (x^3+1\right )^{2/3} \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{4}{3};-x^3\right )}{(x+1)^{2/3} \left (x^2-x+1\right )^{2/3}} \]

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 = 5.04434, size = 37, normalized size = 0.82 \[ \frac{x \sqrt [3]{x + 1} \sqrt [3]{x^{2} - x + 1}{{}_{2}F_{1}\left (\begin{matrix} \frac{2}{3}, \frac{1}{3} \\ \frac{4}{3} \end{matrix}\middle |{- x^{3}} \right )}}{\sqrt [3]{x^{3} + 1}} \]

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 = 1.30578, size = 148, normalized size = 3.29 \[ -\frac{\left (-2 i x+\sqrt{3}+i\right ) \sqrt [3]{x+1} \left (-\frac{\left (\sqrt{3}-3 i\right ) (x+1)+6 i}{\left (\sqrt{3}+3 i\right ) (x+1)-6 i}\right )^{2/3} \left (\left (\sqrt{3}+3 i\right ) (x+1)-6 i\right )^2 \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{4}{3};\frac{2 \sqrt{3} (x+1)}{\left (3 i+\sqrt{3}\right ) (x+1)-6 i}\right )}{4 \left (\sqrt{3}+3 i\right ) \left (x^2-x+1\right )^{5/3}} \]

Antiderivative was successfully verified.

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

[Out]

-((I + Sqrt[3] - (2*I)*x)*(1 + x)^(1/3)*(-((6*I + (-3*I + Sqrt[3])*(1 + x))/(-6*
I + (3*I + Sqrt[3])*(1 + x))))^(2/3)*(-6*I + (3*I + Sqrt[3])*(1 + x))^2*Hypergeo
metric2F1[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 = 0.122, 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)