Optimal. Leaf size=250 \[ -\frac {1}{2} x^2 \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {5}{3};x^3\right )+\frac {\log \left (\frac {2^{2/3} (1-x)^2}{\left (1-x^3\right )^{2/3}}-\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1\right )}{3 \sqrt [3]{2}}-\frac {1}{3} 2^{2/3} \log \left (\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1\right )-\frac {\log \left (2^{2/3} \sqrt [3]{1-x^3}+x-1\right )}{2 \sqrt [3]{2}}+\frac {2^{2/3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {\tan ^{-1}\left (\frac {\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt {3}}+\frac {\log \left ((1-x) (x+1)^2\right )}{6 \sqrt [3]{2}} \]
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Rubi [C] time = 0.01, antiderivative size = 26, normalized size of antiderivative = 0.10, number of steps used = 1, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {510} \[ \frac {1}{2} x^2 F_1\left (\frac {2}{3};-\frac {2}{3},1;\frac {5}{3};x^3,-x^3\right ) \]
Warning: Unable to verify antiderivative.
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Rule 510
Rubi steps
\begin {align*} \int \frac {x \left (1-x^3\right )^{2/3}}{1+x^3} \, dx &=\frac {1}{2} x^2 F_1\left (\frac {2}{3};-\frac {2}{3},1;\frac {5}{3};x^3,-x^3\right )\\ \end {align*}
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Mathematica [C] time = 0.01, size = 26, normalized size = 0.10 \[ \frac {1}{2} x^2 F_1\left (\frac {2}{3};-\frac {2}{3},1;\frac {5}{3};x^3,-x^3\right ) \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 4.31, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (-x^{3} + 1\right )}^{\frac {2}{3}} x}{x^{3} + 1}, x\right ) \]
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 \[ \int \frac {{\left (-x^{3} + 1\right )}^{\frac {2}{3}} x}{x^{3} + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.12, size = 0, normalized size = 0.00 \[ \int \frac {\left (-x^{3}+1\right )^{\frac {2}{3}} x}{x^{3}+1}\, dx \]
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 \[ \int \frac {{\left (-x^{3} + 1\right )}^{\frac {2}{3}} x}{x^{3} + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x\,{\left (1-x^3\right )}^{2/3}}{x^3+1} \,d x \]
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 \[ \int \frac {x \left (- \left (x - 1\right ) \left (x^{2} + x + 1\right )\right )^{\frac {2}{3}}}{\left (x + 1\right ) \left (x^{2} - x + 1\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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