Optimal. Leaf size=168 \[ -\frac{x^2 F_1\left (\frac{2}{3};1,\frac{1}{3};\frac{5}{3};x^3,-\frac{x^3}{2}\right )}{2 \sqrt [3]{2}}+\frac{\log \left (1-x^3\right )}{6 \sqrt [3]{3}}+\frac{\log \left (\sqrt [3]{3}-\sqrt [3]{x^3+2}\right )}{2 \sqrt [3]{3}}-\frac{\log \left (\sqrt [3]{3} x-\sqrt [3]{x^3+2}\right )}{\sqrt [3]{3}}+\frac{2 \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{3} x}{\sqrt [3]{x^3+2}}+1}{\sqrt{3}}\right )}{3^{5/6}}+\frac{\tan ^{-1}\left (\frac{2 \sqrt [3]{x^3+2}+\sqrt [3]{3}}{3^{5/6}}\right )}{3^{5/6}} \]
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Rubi [F] time = 0.204262, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{2+x}{\left (1+x+x^2\right ) \sqrt [3]{2+x^3}} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \frac{2+x}{\left (1+x+x^2\right ) \sqrt [3]{2+x^3}} \, dx &=\int \left (\frac{1-i \sqrt{3}}{\left (1-i \sqrt{3}+2 x\right ) \sqrt [3]{2+x^3}}+\frac{1+i \sqrt{3}}{\left (1+i \sqrt{3}+2 x\right ) \sqrt [3]{2+x^3}}\right ) \, dx\\ &=\left (1-i \sqrt{3}\right ) \int \frac{1}{\left (1-i \sqrt{3}+2 x\right ) \sqrt [3]{2+x^3}} \, dx+\left (1+i \sqrt{3}\right ) \int \frac{1}{\left (1+i \sqrt{3}+2 x\right ) \sqrt [3]{2+x^3}} \, dx\\ \end{align*}
Mathematica [F] time = 0.137935, size = 0, normalized size = 0. \[ \int \frac{2+x}{\left (1+x+x^2\right ) \sqrt [3]{2+x^3}} \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 0.115, size = 0, normalized size = 0. \begin{align*} \int{\frac{2+x}{{x}^{2}+x+1}{\frac{1}{\sqrt [3]{{x}^{3}+2}}}}\, 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{x + 2}{{\left (x^{3} + 2\right )}^{\frac{1}{3}}{\left (x^{2} + x + 1\right )}}\,{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^{3} + 2\right )}^{\frac{2}{3}}{\left (x + 2\right )}}{x^{5} + x^{4} + x^{3} + 2 \, x^{2} + 2 \, x + 2}, 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{x + 2}{\sqrt [3]{x^{3} + 2} \left (x^{2} + x + 1\right )}\, 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{x + 2}{{\left (x^{3} + 2\right )}^{\frac{1}{3}}{\left (x^{2} + x + 1\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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