Optimal. Leaf size=100 \[ \frac {1}{12} \sqrt [3]{x^3+x} \left (3 x^3+x\right )+\frac {1}{18} \log \left (\sqrt [3]{x^3+x}-x\right )+\frac {\tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^3+x}+x}\right )}{6 \sqrt {3}}-\frac {1}{36} \log \left (\sqrt [3]{x^3+x} x+\left (x^3+x\right )^{2/3}+x^2\right ) \]
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Rubi [A] time = 0.18, antiderivative size = 194, normalized size of antiderivative = 1.94, number of steps used = 12, number of rules used = 12, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.923, 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 = 50, normalized size = 0.50 \begin {gather*} \frac {x \sqrt [3]{x^3+x} \left (\left (x^2+1\right )^{4/3}-\, _2F_1\left (-\frac {1}{3},\frac {2}{3};\frac {5}{3};-x^2\right )\right )}{4 \sqrt [3]{x^2+1}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.24, size = 100, 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 = 0.58, size = 96, normalized size = 0.96 \begin {gather*} \frac {1}{18} \, \sqrt {3} \arctan \left (-\frac {196 \, \sqrt {3} {\left (x^{3} + x\right )}^{\frac {1}{3}} x - \sqrt {3} {\left (539 \, x^{2} + 507\right )} - 1274 \, \sqrt {3} {\left (x^{3} + x\right )}^{\frac {2}{3}}}{2205 \, x^{2} + 2197}\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 = 0.32, size = 77, normalized size = 0.77 \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 = 2.10, size = 735, normalized size = 7.35 \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 {\RootOf \left (\textit {\_Z}^{2}+2 \textit {\_Z} +4\right ) \ln \left (\frac {-2 \RootOf \left (\textit {\_Z}^{2}+2 \textit {\_Z} +4\right )^{2} x^{4}+11 \RootOf \left (\textit {\_Z}^{2}+2 \textit {\_Z} +4\right ) x^{4}+15 \left (x^{6}+2 x^{4}+x^{2}\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{2}+2 \textit {\_Z} +4\right ) x^{2}+40 x^{4}+15 \RootOf \left (\textit {\_Z}^{2}+2 \textit {\_Z} +4\right ) \left (x^{6}+2 x^{4}+x^{2}\right )^{\frac {2}{3}}+48 \left (x^{6}+2 x^{4}+x^{2}\right )^{\frac {1}{3}} x^{2}+28 \RootOf \left (\textit {\_Z}^{2}+2 \textit {\_Z} +4\right ) x^{2}+48 \left (x^{6}+2 x^{4}+x^{2}\right )^{\frac {2}{3}}+15 \RootOf \left (\textit {\_Z}^{2}+2 \textit {\_Z} +4\right ) \left (x^{6}+2 x^{4}+x^{2}\right )^{\frac {1}{3}}+2 \RootOf \left (\textit {\_Z}^{2}+2 \textit {\_Z} +4\right )^{2}+70 x^{2}+48 \left (x^{6}+2 x^{4}+x^{2}\right )^{\frac {1}{3}}+17 \RootOf \left (\textit {\_Z}^{2}+2 \textit {\_Z} +4\right )+30}{x^{2}+1}\right )}{36}-\frac {\ln \left (-\frac {2 \RootOf \left (\textit {\_Z}^{2}+2 \textit {\_Z} +4\right )^{2} x^{4}+19 \RootOf \left (\textit {\_Z}^{2}+2 \textit {\_Z} +4\right ) x^{4}+15 \left (x^{6}+2 x^{4}+x^{2}\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{2}+2 \textit {\_Z} +4\right ) x^{2}-10 x^{4}+15 \RootOf \left (\textit {\_Z}^{2}+2 \textit {\_Z} +4\right ) \left (x^{6}+2 x^{4}+x^{2}\right )^{\frac {2}{3}}-18 \left (x^{6}+2 x^{4}+x^{2}\right )^{\frac {1}{3}} x^{2}+28 \RootOf \left (\textit {\_Z}^{2}+2 \textit {\_Z} +4\right ) x^{2}-18 \left (x^{6}+2 x^{4}+x^{2}\right )^{\frac {2}{3}}+15 \RootOf \left (\textit {\_Z}^{2}+2 \textit {\_Z} +4\right ) \left (x^{6}+2 x^{4}+x^{2}\right )^{\frac {1}{3}}-2 \RootOf \left (\textit {\_Z}^{2}+2 \textit {\_Z} +4\right )^{2}-14 x^{2}-18 \left (x^{6}+2 x^{4}+x^{2}\right )^{\frac {1}{3}}+9 \RootOf \left (\textit {\_Z}^{2}+2 \textit {\_Z} +4\right )-4}{x^{2}+1}\right ) \RootOf \left (\textit {\_Z}^{2}+2 \textit {\_Z} +4\right )}{36}-\frac {\ln \left (-\frac {2 \RootOf \left (\textit {\_Z}^{2}+2 \textit {\_Z} +4\right )^{2} x^{4}+19 \RootOf \left (\textit {\_Z}^{2}+2 \textit {\_Z} +4\right ) x^{4}+15 \left (x^{6}+2 x^{4}+x^{2}\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{2}+2 \textit {\_Z} +4\right ) x^{2}-10 x^{4}+15 \RootOf \left (\textit {\_Z}^{2}+2 \textit {\_Z} +4\right ) \left (x^{6}+2 x^{4}+x^{2}\right )^{\frac {2}{3}}-18 \left (x^{6}+2 x^{4}+x^{2}\right )^{\frac {1}{3}} x^{2}+28 \RootOf \left (\textit {\_Z}^{2}+2 \textit {\_Z} +4\right ) x^{2}-18 \left (x^{6}+2 x^{4}+x^{2}\right )^{\frac {2}{3}}+15 \RootOf \left (\textit {\_Z}^{2}+2 \textit {\_Z} +4\right ) \left (x^{6}+2 x^{4}+x^{2}\right )^{\frac {1}{3}}-2 \RootOf \left (\textit {\_Z}^{2}+2 \textit {\_Z} +4\right )^{2}-14 x^{2}-18 \left (x^{6}+2 x^{4}+x^{2}\right )^{\frac {1}{3}}+9 \RootOf \left (\textit {\_Z}^{2}+2 \textit {\_Z} +4\right )-4}{x^{2}+1}\right )}{18}\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^{2} + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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