Optimal. Leaf size=112 \[ \frac {1}{12} \sqrt [3]{x^3-x} \left (3 x^3-x\right )+\frac {1}{18} \log \left (\sqrt [3]{x^3-x}-x\right )+\frac {\tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^3-x}+x}\right )}{6 \sqrt {3}}-\frac {1}{36} \log \left (\sqrt [3]{x^3-x} x+\left (x^3-x\right )^{2/3}+x^2\right ) \]
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Rubi [A] time = 0.19, antiderivative size = 204, normalized size of antiderivative = 1.82, number of steps used = 12, number of rules used = 12, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {2021, 2024, 2032, 329, 275, 331, 292, 31, 634, 618, 204, 628} \begin {gather*} \frac {1}{4} \sqrt [3]{x^3-x} x^3-\frac {1}{12} \sqrt [3]{x^3-x} x+\frac {\left (x^2-1\right )^{2/3} x^{2/3} \log \left (1-\frac {x^{2/3}}{\sqrt [3]{x^2-1}}\right )}{18 \left (x^3-x\right )^{2/3}}-\frac {\left (x^2-1\right )^{2/3} x^{2/3} \log \left (\frac {x^{4/3}}{\left (x^2-1\right )^{2/3}}+\frac {x^{2/3}}{\sqrt [3]{x^2-1}}+1\right )}{36 \left (x^3-x\right )^{2/3}}+\frac {\left (x^2-1\right )^{2/3} x^{2/3} \tan ^{-1}\left (\frac {\frac {2 x^{2/3}}{\sqrt [3]{x^2-1}}+1}{\sqrt {3}}\right )}{6 \sqrt {3} \left (x^3-x\right )^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 204
Rule 275
Rule 292
Rule 329
Rule 331
Rule 618
Rule 628
Rule 634
Rule 2021
Rule 2024
Rule 2032
Rubi steps
\begin {align*} \int x^2 \sqrt [3]{-x+x^3} \, dx &=\frac {1}{4} x^3 \sqrt [3]{-x+x^3}-\frac {1}{6} \int \frac {x^3}{\left (-x+x^3\right )^{2/3}} \, dx\\ &=-\frac {1}{12} x \sqrt [3]{-x+x^3}+\frac {1}{4} x^3 \sqrt [3]{-x+x^3}-\frac {1}{9} \int \frac {x}{\left (-x+x^3\right )^{2/3}} \, dx\\ &=-\frac {1}{12} x \sqrt [3]{-x+x^3}+\frac {1}{4} x^3 \sqrt [3]{-x+x^3}-\frac {\left (x^{2/3} \left (-1+x^2\right )^{2/3}\right ) \int \frac {\sqrt [3]{x}}{\left (-1+x^2\right )^{2/3}} \, dx}{9 \left (-x+x^3\right )^{2/3}}\\ &=-\frac {1}{12} x \sqrt [3]{-x+x^3}+\frac {1}{4} x^3 \sqrt [3]{-x+x^3}-\frac {\left (x^{2/3} \left (-1+x^2\right )^{2/3}\right ) \operatorname {Subst}\left (\int \frac {x^3}{\left (-1+x^6\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{3 \left (-x+x^3\right )^{2/3}}\\ &=-\frac {1}{12} x \sqrt [3]{-x+x^3}+\frac {1}{4} x^3 \sqrt [3]{-x+x^3}-\frac {\left (x^{2/3} \left (-1+x^2\right )^{2/3}\right ) \operatorname {Subst}\left (\int \frac {x}{\left (-1+x^3\right )^{2/3}} \, dx,x,x^{2/3}\right )}{6 \left (-x+x^3\right )^{2/3}}\\ &=-\frac {1}{12} x \sqrt [3]{-x+x^3}+\frac {1}{4} x^3 \sqrt [3]{-x+x^3}-\frac {\left (x^{2/3} \left (-1+x^2\right )^{2/3}\right ) \operatorname {Subst}\left (\int \frac {x}{1-x^3} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{-1+x^2}}\right )}{6 \left (-x+x^3\right )^{2/3}}\\ &=-\frac {1}{12} x \sqrt [3]{-x+x^3}+\frac {1}{4} x^3 \sqrt [3]{-x+x^3}-\frac {\left (x^{2/3} \left (-1+x^2\right )^{2/3}\right ) \operatorname {Subst}\left (\int \frac {1}{1-x} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{-1+x^2}}\right )}{18 \left (-x+x^3\right )^{2/3}}+\frac {\left (x^{2/3} \left (-1+x^2\right )^{2/3}\right ) \operatorname {Subst}\left (\int \frac {1-x}{1+x+x^2} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{-1+x^2}}\right )}{18 \left (-x+x^3\right )^{2/3}}\\ &=-\frac {1}{12} x \sqrt [3]{-x+x^3}+\frac {1}{4} x^3 \sqrt [3]{-x+x^3}+\frac {x^{2/3} \left (-1+x^2\right )^{2/3} \log \left (1-\frac {x^{2/3}}{\sqrt [3]{-1+x^2}}\right )}{18 \left (-x+x^3\right )^{2/3}}-\frac {\left (x^{2/3} \left (-1+x^2\right )^{2/3}\right ) \operatorname {Subst}\left (\int \frac {1+2 x}{1+x+x^2} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{-1+x^2}}\right )}{36 \left (-x+x^3\right )^{2/3}}+\frac {\left (x^{2/3} \left (-1+x^2\right )^{2/3}\right ) \operatorname {Subst}\left (\int \frac {1}{1+x+x^2} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{-1+x^2}}\right )}{12 \left (-x+x^3\right )^{2/3}}\\ &=-\frac {1}{12} x \sqrt [3]{-x+x^3}+\frac {1}{4} x^3 \sqrt [3]{-x+x^3}+\frac {x^{2/3} \left (-1+x^2\right )^{2/3} \log \left (1-\frac {x^{2/3}}{\sqrt [3]{-1+x^2}}\right )}{18 \left (-x+x^3\right )^{2/3}}-\frac {x^{2/3} \left (-1+x^2\right )^{2/3} \log \left (1+\frac {x^{4/3}}{\left (-1+x^2\right )^{2/3}}+\frac {x^{2/3}}{\sqrt [3]{-1+x^2}}\right )}{36 \left (-x+x^3\right )^{2/3}}-\frac {\left (x^{2/3} \left (-1+x^2\right )^{2/3}\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 x^{2/3}}{\sqrt [3]{-1+x^2}}\right )}{6 \left (-x+x^3\right )^{2/3}}\\ &=-\frac {1}{12} x \sqrt [3]{-x+x^3}+\frac {1}{4} x^3 \sqrt [3]{-x+x^3}+\frac {x^{2/3} \left (-1+x^2\right )^{2/3} \tan ^{-1}\left (\frac {1+\frac {2 x^{2/3}}{\sqrt [3]{-1+x^2}}}{\sqrt {3}}\right )}{6 \sqrt {3} \left (-x+x^3\right )^{2/3}}+\frac {x^{2/3} \left (-1+x^2\right )^{2/3} \log \left (1-\frac {x^{2/3}}{\sqrt [3]{-1+x^2}}\right )}{18 \left (-x+x^3\right )^{2/3}}-\frac {x^{2/3} \left (-1+x^2\right )^{2/3} \log \left (1+\frac {x^{4/3}}{\left (-1+x^2\right )^{2/3}}+\frac {x^{2/3}}{\sqrt [3]{-1+x^2}}\right )}{36 \left (-x+x^3\right )^{2/3}}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 54, normalized size = 0.48 \begin {gather*} \frac {x \sqrt [3]{x \left (x^2-1\right )} \left (\, _2F_1\left (-\frac {1}{3},\frac {2}{3};\frac {5}{3};x^2\right )-\left (1-x^2\right )^{4/3}\right )}{4 \sqrt [3]{1-x^2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.23, size = 112, normalized size = 1.00 \begin {gather*} \frac {1}{12} \sqrt [3]{x^3-x} \left (3 x^3-x\right )+\frac {1}{18} \log \left (\sqrt [3]{x^3-x}-x\right )+\frac {\tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^3-x}+x}\right )}{6 \sqrt {3}}-\frac {1}{36} \log \left (\sqrt [3]{x^3-x} x+\left (x^3-x\right )^{2/3}+x^2\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.62, size = 107, normalized size = 0.96 \begin {gather*} \frac {1}{18} \, \sqrt {3} \arctan \left (-\frac {44032959556 \, \sqrt {3} {\left (x^{3} - x\right )}^{\frac {1}{3}} x + \sqrt {3} {\left (16754327161 \, x^{2} - 2707204793\right )} - 10524305234 \, \sqrt {3} {\left (x^{3} - x\right )}^{\frac {2}{3}}}{81835897185 \, x^{2} - 1102302937}\right ) + \frac {1}{12} \, {\left (3 \, x^{3} - x\right )} {\left (x^{3} - x\right )}^{\frac {1}{3}} + \frac {1}{36} \, \log \left (-3 \, {\left (x^{3} - x\right )}^{\frac {1}{3}} x + 3 \, {\left (x^{3} - x\right )}^{\frac {2}{3}} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 2.01, size = 89, normalized size = 0.79 \begin {gather*} \frac {1}{12} \, {\left ({\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {4}{3}} + 2 \, {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}}\right )} x^{4} - \frac {1}{18} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) - \frac {1}{36} \, \log \left ({\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {2}{3}} + {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {1}{18} \, \log \left ({\left | {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} - 1 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 2.18, size = 543, normalized size = 4.85 \begin {gather*} \frac {x \left (3 x^{2}-1\right ) \left (x \left (x^{2}-1\right )\right )^{\frac {1}{3}}}{12}+\frac {\left (\frac {\ln \left (-\frac {-35 \RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right )^{2} x^{4}-1956 \RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right ) x^{4}-4104 \left (x^{6}-2 x^{4}+x^{2}\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right ) x^{2}+175 \RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right )^{2} x^{2}+23364 x^{4}+5850 \RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right ) \left (x^{6}-2 x^{4}+x^{2}\right )^{\frac {2}{3}}-35100 \left (x^{6}-2 x^{4}+x^{2}\right )^{\frac {1}{3}} x^{2}+2010 \RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right ) x^{2}+10476 \left (x^{6}-2 x^{4}+x^{2}\right )^{\frac {2}{3}}+4104 \RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right ) \left (x^{6}-2 x^{4}+x^{2}\right )^{\frac {1}{3}}-140 \RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right )^{2}-38232 x^{2}+35100 \left (x^{6}-2 x^{4}+x^{2}\right )^{\frac {1}{3}}-54 \RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right )+14868}{\left (-1+x \right ) \left (1+x \right )}\right )}{18}+\frac {\RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right ) \ln \left (\frac {59 \RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right )^{2} x^{4}-3750 \RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right ) x^{4}-1746 \left (x^{6}-2 x^{4}+x^{2}\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right ) x^{2}-295 \RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right )^{2} x^{2}+12600 x^{4}+5850 \RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right ) \left (x^{6}-2 x^{4}+x^{2}\right )^{\frac {2}{3}}-35100 \left (x^{6}-2 x^{4}+x^{2}\right )^{\frac {1}{3}} x^{2}+5652 \RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right ) x^{2}+24624 \left (x^{6}-2 x^{4}+x^{2}\right )^{\frac {2}{3}}+1746 \RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right ) \left (x^{6}-2 x^{4}+x^{2}\right )^{\frac {1}{3}}+236 \RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right )^{2}-16380 x^{2}+35100 \left (x^{6}-2 x^{4}+x^{2}\right )^{\frac {1}{3}}-1902 \RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right )+3780}{\left (-1+x \right ) \left (1+x \right )}\right )}{108}\right ) \left (x \left (x^{2}-1\right )\right )^{\frac {1}{3}} \left (x^{2} \left (x^{2}-1\right )^{2}\right )^{\frac {1}{3}}}{x \left (x^{2}-1\right )} \end {gather*}
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 \begin {gather*} \int {\left (x^{3} - x\right )}^{\frac {1}{3}} x^{2}\,{d x} \end {gather*}
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 \begin {gather*} \int x^2\,{\left (x^3-x\right )}^{1/3} \,d x \end {gather*}
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 \begin {gather*} \int x^{2} \sqrt [3]{x \left (x - 1\right ) \left (x + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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