Optimal. Leaf size=73 \[ -\frac {1}{6} \log \left (-\sqrt [3]{1-x^3}-x\right )-\frac {\tan ^{-1}\left (\frac {1-\frac {2 x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{3 \sqrt {3}}+\frac {1}{3} \sqrt [3]{1-x^3} x^2 \]
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Rubi [A] time = 0.04, antiderivative size = 107, normalized size of antiderivative = 1.47, number of steps used = 8, number of rules used = 8, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.615, Rules used = {279, 331, 292, 31, 634, 618, 204, 628} \[ \frac {1}{3} \sqrt [3]{1-x^3} x^2+\frac {1}{18} \log \left (\frac {x^2}{\left (1-x^3\right )^{2/3}}-\frac {x}{\sqrt [3]{1-x^3}}+1\right )-\frac {1}{9} \log \left (\frac {x}{\sqrt [3]{1-x^3}}+1\right )-\frac {\tan ^{-1}\left (\frac {1-\frac {2 x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{3 \sqrt {3}} \]
Antiderivative was successfully verified.
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Rule 31
Rule 204
Rule 279
Rule 292
Rule 331
Rule 618
Rule 628
Rule 634
Rubi steps
\begin {align*} \int x \sqrt [3]{1-x^3} \, dx &=\frac {1}{3} x^2 \sqrt [3]{1-x^3}+\frac {1}{3} \int \frac {x}{\left (1-x^3\right )^{2/3}} \, dx\\ &=\frac {1}{3} x^2 \sqrt [3]{1-x^3}+\frac {1}{3} \operatorname {Subst}\left (\int \frac {x}{1+x^3} \, dx,x,\frac {x}{\sqrt [3]{1-x^3}}\right )\\ &=\frac {1}{3} x^2 \sqrt [3]{1-x^3}-\frac {1}{9} \operatorname {Subst}\left (\int \frac {1}{1+x} \, dx,x,\frac {x}{\sqrt [3]{1-x^3}}\right )+\frac {1}{9} \operatorname {Subst}\left (\int \frac {1+x}{1-x+x^2} \, dx,x,\frac {x}{\sqrt [3]{1-x^3}}\right )\\ &=\frac {1}{3} x^2 \sqrt [3]{1-x^3}-\frac {1}{9} \log \left (1+\frac {x}{\sqrt [3]{1-x^3}}\right )+\frac {1}{18} \operatorname {Subst}\left (\int \frac {-1+2 x}{1-x+x^2} \, dx,x,\frac {x}{\sqrt [3]{1-x^3}}\right )+\frac {1}{6} \operatorname {Subst}\left (\int \frac {1}{1-x+x^2} \, dx,x,\frac {x}{\sqrt [3]{1-x^3}}\right )\\ &=\frac {1}{3} x^2 \sqrt [3]{1-x^3}+\frac {1}{18} \log \left (1+\frac {x^2}{\left (1-x^3\right )^{2/3}}-\frac {x}{\sqrt [3]{1-x^3}}\right )-\frac {1}{9} \log \left (1+\frac {x}{\sqrt [3]{1-x^3}}\right )-\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+\frac {2 x}{\sqrt [3]{1-x^3}}\right )\\ &=\frac {1}{3} x^2 \sqrt [3]{1-x^3}-\frac {\tan ^{-1}\left (\frac {1-\frac {2 x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{3 \sqrt {3}}+\frac {1}{18} \log \left (1+\frac {x^2}{\left (1-x^3\right )^{2/3}}-\frac {x}{\sqrt [3]{1-x^3}}\right )-\frac {1}{9} \log \left (1+\frac {x}{\sqrt [3]{1-x^3}}\right )\\ \end {align*}
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Mathematica [C] time = 0.00, size = 20, normalized size = 0.27 \[ \frac {1}{2} x^2 \, _2F_1\left (-\frac {1}{3},\frac {2}{3};\frac {5}{3};x^3\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 1.03, size = 96, normalized size = 1.32 \[ \frac {1}{3} \, {\left (-x^{3} + 1\right )}^{\frac {1}{3}} x^{2} - \frac {1}{9} \, \sqrt {3} \arctan \left (-\frac {\sqrt {3} x - 2 \, \sqrt {3} {\left (-x^{3} + 1\right )}^{\frac {1}{3}}}{3 \, x}\right ) - \frac {1}{9} \, \log \left (\frac {x + {\left (-x^{3} + 1\right )}^{\frac {1}{3}}}{x}\right ) + \frac {1}{18} \, \log \left (\frac {x^{2} - {\left (-x^{3} + 1\right )}^{\frac {1}{3}} x + {\left (-x^{3} + 1\right )}^{\frac {2}{3}}}{x^{2}}\right ) \]
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 \[ \int {\left (-x^{3} + 1\right )}^{\frac {1}{3}} x\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.10, size = 69, normalized size = 0.95 \[ \frac {\left (x^{3}-1\right )^{\frac {2}{3}} \left (-\mathrm {signum}\left (x^{3}-1\right )\right )^{\frac {2}{3}} x^{2} \hypergeom \left (\left [\frac {2}{3}, \frac {2}{3}\right ], \left [\frac {5}{3}\right ], x^{3}\right )}{6 \left (-x^{3}+1\right )^{\frac {2}{3}} \mathrm {signum}\left (x^{3}-1\right )^{\frac {2}{3}}}-\frac {\left (x^{3}-1\right ) x^{2}}{3 \left (-x^{3}+1\right )^{\frac {2}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.25, size = 105, normalized size = 1.44 \[ -\frac {1}{9} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (\frac {2 \, {\left (-x^{3} + 1\right )}^{\frac {1}{3}}}{x} - 1\right )}\right ) - \frac {{\left (-x^{3} + 1\right )}^{\frac {1}{3}}}{3 \, x {\left (\frac {x^{3} - 1}{x^{3}} - 1\right )}} - \frac {1}{9} \, \log \left (\frac {{\left (-x^{3} + 1\right )}^{\frac {1}{3}}}{x} + 1\right ) + \frac {1}{18} \, \log \left (-\frac {{\left (-x^{3} + 1\right )}^{\frac {1}{3}}}{x} + \frac {{\left (-x^{3} + 1\right )}^{\frac {2}{3}}}{x^{2}} + 1\right ) \]
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 \[ \int x\,{\left (1-x^3\right )}^{1/3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 1.02, size = 32, normalized size = 0.44 \[ \frac {x^{2} \Gamma \left (\frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, \frac {2}{3} \\ \frac {5}{3} \end {matrix}\middle | {x^{3} e^{2 i \pi }} \right )}}{3 \Gamma \left (\frac {5}{3}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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