Optimal. Leaf size=45 \[ \frac{x \left (x^3+1\right )^{2/3} \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{4}{3};-x^3\right )}{(x+1)^{2/3} \left (x^2-x+1\right )^{2/3}} \]
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Rubi [A] time = 0.0152849, 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, Rules used = {713, 245} \[ \frac{x \left (x^3+1\right )^{2/3} \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{4}{3};-x^3\right )}{(x+1)^{2/3} \left (x^2-x+1\right )^{2/3}} \]
Antiderivative was successfully verified.
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Rule 713
Rule 245
Rubi steps
\begin{align*} \int \frac{1}{(1+x)^{2/3} \left (1-x+x^2\right )^{2/3}} \, dx &=\frac{\left (1+x^3\right )^{2/3} \int \frac{1}{\left (1+x^3\right )^{2/3}} \, dx}{(1+x)^{2/3} \left (1-x+x^2\right )^{2/3}}\\ &=\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 (1-x+x^2\right )^{2/3}}\\ \end{align*}
Mathematica [C] time = 0.0740744, size = 143, normalized size = 3.18 \[ \frac{3 \left (2 i x+\sqrt{3}-i\right ) \sqrt [3]{x+1} \left (-\frac{\left (\sqrt{3}-3 i\right ) x+\sqrt{3}+3 i}{\left (\sqrt{3}+3 i\right ) x+\sqrt{3}-3 i}\right )^{2/3} \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{4}{3};\frac{4 i \sqrt{3} (x+1)}{\left (3 i+\sqrt{3}\right ) \left (2 i x+\sqrt{3}-i\right )}\right )}{\left (\sqrt{3}-3 i\right ) \left (x^2-x+1\right )^{2/3}} \]
Antiderivative was successfully verified.
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Maple [F] time = 1.603, size = 0, normalized size = 0. \begin{align*} \int{ \left ( 1+x \right ) ^{-{\frac{2}{3}}} \left ({x}^{2}-x+1 \right ) ^{-{\frac{2}{3}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (x^{2} - x + 1\right )}^{\frac{2}{3}}{\left (x + 1\right )}^{\frac{2}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (x^{2} - x + 1\right )}^{\frac{1}{3}}{\left (x + 1\right )}^{\frac{1}{3}}}{x^{3} + 1}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (x + 1\right )^{\frac{2}{3}} \left (x^{2} - x + 1\right )^{\frac{2}{3}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (x^{2} - x + 1\right )}^{\frac{2}{3}}{\left (x + 1\right )}^{\frac{2}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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