Optimal. Leaf size=94 \[ \frac {1}{2} \sqrt [3]{x^3+x} x-\frac {1}{6} \log \left (\sqrt [3]{x^3+x}-x\right )-\frac {\tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^3+x}+x}\right )}{2 \sqrt {3}}+\frac {1}{12} \log \left (\sqrt [3]{x^3+x} x+\left (x^3+x\right )^{2/3}+x^2\right ) \]
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Rubi [A] time = 0.13, antiderivative size = 178, normalized size of antiderivative = 1.89, number of steps used = 11, number of rules used = 11, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.222, Rules used = {2004, 2032, 329, 275, 331, 292, 31, 634, 618, 204, 628} \begin {gather*} \frac {1}{2} \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 )}{6 \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 )}{12 \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 )}{2 \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 2004
Rule 2032
Rubi steps
\begin {align*} \int \sqrt [3]{x+x^3} \, dx &=\frac {1}{2} x \sqrt [3]{x+x^3}+\frac {1}{3} \int \frac {x}{\left (x+x^3\right )^{2/3}} \, dx\\ &=\frac {1}{2} x \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}{3 \left (x+x^3\right )^{2/3}}\\ &=\frac {1}{2} x \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 )}{\left (x+x^3\right )^{2/3}}\\ &=\frac {1}{2} x \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 )}{2 \left (x+x^3\right )^{2/3}}\\ &=\frac {1}{2} x \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 )}{2 \left (x+x^3\right )^{2/3}}\\ &=\frac {1}{2} x \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 )}{6 \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 )}{6 \left (x+x^3\right )^{2/3}}\\ &=\frac {1}{2} x \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 )}{6 \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 )}{12 \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 )}{4 \left (x+x^3\right )^{2/3}}\\ &=\frac {1}{2} x \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 )}{6 \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 )}{12 \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 )}{2 \left (x+x^3\right )^{2/3}}\\ &=\frac {1}{2} x \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 )}{2 \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 )}{6 \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 )}{12 \left (x+x^3\right )^{2/3}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 38, normalized size = 0.40 \begin {gather*} \frac {3 x \sqrt [3]{x^3+x} \, _2F_1\left (-\frac {1}{3},\frac {2}{3};\frac {5}{3};-x^2\right )}{4 \sqrt [3]{x^2+1}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.17, size = 94, normalized size = 1.00 \begin {gather*} \frac {1}{2} x \sqrt [3]{x+x^3}-\frac {\tan ^{-1}\left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{x+x^3}}\right )}{2 \sqrt {3}}-\frac {1}{6} \log \left (-x+\sqrt [3]{x+x^3}\right )+\frac {1}{12} \log \left (x^2+x \sqrt [3]{x+x^3}+\left (x+x^3\right )^{2/3}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.63, size = 90, normalized size = 0.96 \begin {gather*} -\frac {1}{6} \, \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}{2} \, {\left (x^{3} + x\right )}^{\frac {1}{3}} x - \frac {1}{12} \, \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 = 0.36, size = 67, normalized size = 0.71 \begin {gather*} \frac {1}{2} \, x^{2} {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} + \frac {1}{6} \, \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}{12} \, \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}{6} \, \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 = 4.24, size = 17, normalized size = 0.18
method | result | size |
meijerg | \(\frac {3 x^{\frac {4}{3}} \hypergeom \left (\left [-\frac {1}{3}, \frac {2}{3}\right ], \left [\frac {5}{3}\right ], -x^{2}\right )}{4}\) | \(17\) |
trager | \(\frac {x \left (x^{3}+x \right )^{\frac {1}{3}}}{2}-\frac {\ln \left (-36 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{2}-72 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (x^{3}+x \right )^{\frac {2}{3}}+27 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (x^{3}+x \right )^{\frac {1}{3}} x +57 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{2}+36 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2}+15 \left (x^{3}+x \right )^{\frac {2}{3}}-24 x \left (x^{3}+x \right )^{\frac {1}{3}}+5 x^{2}+27 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )+2\right )}{6}+\frac {\ln \left (-9 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{2}+72 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (x^{3}+x \right )^{\frac {2}{3}}-45 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (x^{3}+x \right )^{\frac {1}{3}} x -24 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{2}+9 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2}-9 \left (x^{3}+x \right )^{\frac {2}{3}}+24 x \left (x^{3}+x \right )^{\frac {1}{3}}-16 x^{2}+3 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )-12\right )}{6}-\frac {\ln \left (-9 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{2}+72 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (x^{3}+x \right )^{\frac {2}{3}}-45 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (x^{3}+x \right )^{\frac {1}{3}} x -24 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{2}+9 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2}-9 \left (x^{3}+x \right )^{\frac {2}{3}}+24 x \left (x^{3}+x \right )^{\frac {1}{3}}-16 x^{2}+3 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )-12\right ) \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )}{2}\) | \(419\) |
risch | \(\frac {x \left (x \left (x^{2}+1\right )\right )^{\frac {1}{3}}}{2}+\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 )}{12}-\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 )}{12}+\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 )}{6}\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 )}\) | \(727\) |
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}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.80, size = 27, normalized size = 0.29 \begin {gather*} \frac {3\,x\,{\left (x^3+x\right )}^{1/3}\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{3},\frac {2}{3};\ \frac {5}{3};\ -x^2\right )}{4\,{\left (x^2+1\right )}^{1/3}} \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 \sqrt [3]{x^{3} + x}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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