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 = 0.20, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, 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*}
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Mathematica [F] time = 0.14, size = 0, normalized size = 0.00 \[ \int \frac {2+x}{\left (1+x+x^2\right ) \sqrt [3]{2+x^3}} \, dx \]
Verification is Not applicable to the result.
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fricas [F] time = 18.87, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (x^{3} + 2\right )}^{\frac {2}{3}} {\left (x + 2\right )}}{x^{5} + x^{4} + x^{3} + 2 \, x^{2} + 2 \, x + 2}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x + 2}{{\left (x^{3} + 2\right )}^{\frac {1}{3}} {\left (x^{2} + x + 1\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 4.24, size = 0, normalized size = 0.00 \[ \int \frac {x +2}{\left (x^{2}+x +1\right ) \left (x^{3}+2\right )^{\frac {1}{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x + 2}{{\left (x^{3} + 2\right )}^{\frac {1}{3}} {\left (x^{2} + x + 1\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x+2}{{\left (x^3+2\right )}^{1/3}\,\left (x^2+x+1\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x + 2}{\sqrt [3]{x^{3} + 2} \left (x^{2} + x + 1\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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