Optimal. Leaf size=108 \[ \frac {1}{6} \sqrt [3]{x^3+x^2} (3 x+1)+\frac {1}{9} \log \left (\sqrt [3]{x^3+x^2}-x\right )-\frac {1}{18} \log \left (x^2+\sqrt [3]{x^3+x^2} x+\left (x^3+x^2\right )^{2/3}\right )+\frac {\tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^3+x^2}+x}\right )}{3 \sqrt {3}} \]
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Rubi [A] time = 0.09, antiderivative size = 164, normalized size of antiderivative = 1.52, number of steps used = 4, number of rules used = 4, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {2004, 2024, 2032, 59} \begin {gather*} \frac {1}{2} \sqrt [3]{x^3+x^2} x+\frac {1}{6} \sqrt [3]{x^3+x^2}+\frac {(x+1)^{2/3} x^{4/3} \log (x+1)}{18 \left (x^3+x^2\right )^{2/3}}+\frac {(x+1)^{2/3} x^{4/3} \log \left (\frac {\sqrt [3]{x}}{\sqrt [3]{x+1}}-1\right )}{6 \left (x^3+x^2\right )^{2/3}}+\frac {(x+1)^{2/3} x^{4/3} \tan ^{-1}\left (\frac {2 \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{x+1}}+\frac {1}{\sqrt {3}}\right )}{3 \sqrt {3} \left (x^3+x^2\right )^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 59
Rule 2004
Rule 2024
Rule 2032
Rubi steps
\begin {align*} \int \sqrt [3]{x^2+x^3} \, dx &=\frac {1}{2} x \sqrt [3]{x^2+x^3}+\frac {1}{6} \int \frac {x^2}{\left (x^2+x^3\right )^{2/3}} \, dx\\ &=\frac {1}{6} \sqrt [3]{x^2+x^3}+\frac {1}{2} x \sqrt [3]{x^2+x^3}-\frac {1}{9} \int \frac {x}{\left (x^2+x^3\right )^{2/3}} \, dx\\ &=\frac {1}{6} \sqrt [3]{x^2+x^3}+\frac {1}{2} x \sqrt [3]{x^2+x^3}-\frac {\left (x^{4/3} (1+x)^{2/3}\right ) \int \frac {1}{\sqrt [3]{x} (1+x)^{2/3}} \, dx}{9 \left (x^2+x^3\right )^{2/3}}\\ &=\frac {1}{6} \sqrt [3]{x^2+x^3}+\frac {1}{2} x \sqrt [3]{x^2+x^3}+\frac {x^{4/3} (1+x)^{2/3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{1+x}}\right )}{3 \sqrt {3} \left (x^2+x^3\right )^{2/3}}+\frac {x^{4/3} (1+x)^{2/3} \log (1+x)}{18 \left (x^2+x^3\right )^{2/3}}+\frac {x^{4/3} (1+x)^{2/3} \log \left (-1+\frac {\sqrt [3]{x}}{\sqrt [3]{1+x}}\right )}{6 \left (x^2+x^3\right )^{2/3}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 36, normalized size = 0.33 \begin {gather*} \frac {3 x \sqrt [3]{x^2 (x+1)} \, _2F_1\left (-\frac {1}{3},\frac {5}{3};\frac {8}{3};-x\right )}{5 \sqrt [3]{x+1}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.19, size = 108, normalized size = 1.00 \begin {gather*} \frac {1}{6} (1+3 x) \sqrt [3]{x^2+x^3}+\frac {\tan ^{-1}\left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{x^2+x^3}}\right )}{3 \sqrt {3}}+\frac {1}{9} \log \left (-x+\sqrt [3]{x^2+x^3}\right )-\frac {1}{18} \log \left (x^2+x \sqrt [3]{x^2+x^3}+\left (x^2+x^3\right )^{2/3}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.63, size = 100, normalized size = 0.93 \begin {gather*} -\frac {1}{9} \, \sqrt {3} \arctan \left (\frac {\sqrt {3} x + 2 \, \sqrt {3} {\left (x^{3} + x^{2}\right )}^{\frac {1}{3}}}{3 \, x}\right ) + \frac {1}{6} \, {\left (x^{3} + x^{2}\right )}^{\frac {1}{3}} {\left (3 \, x + 1\right )} + \frac {1}{9} \, \log \left (-\frac {x - {\left (x^{3} + x^{2}\right )}^{\frac {1}{3}}}{x}\right ) - \frac {1}{18} \, \log \left (\frac {x^{2} + {\left (x^{3} + x^{2}\right )}^{\frac {1}{3}} x + {\left (x^{3} + x^{2}\right )}^{\frac {2}{3}}}{x^{2}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 77, normalized size = 0.71 \begin {gather*} \frac {1}{6} \, {\left ({\left (\frac {1}{x} + 1\right )}^{\frac {4}{3}} + 2 \, {\left (\frac {1}{x} + 1\right )}^{\frac {1}{3}}\right )} x^{2} - \frac {1}{9} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (\frac {1}{x} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) - \frac {1}{18} \, \log \left ({\left (\frac {1}{x} + 1\right )}^{\frac {2}{3}} + {\left (\frac {1}{x} + 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {1}{9} \, \log \left ({\left | {\left (\frac {1}{x} + 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 = 1.04, size = 15, normalized size = 0.14
method | result | size |
meijerg | \(\frac {3 x^{\frac {5}{3}} \hypergeom \left (\left [-\frac {1}{3}, \frac {5}{3}\right ], \left [\frac {8}{3}\right ], -x \right )}{5}\) | \(15\) |
risch | \(\frac {\left (1+3 x \right ) \left (x^{2} \left (1+x \right )\right )^{\frac {1}{3}}}{6}+\frac {\left (\frac {\ln \left (-\frac {x^{2} \RootOf \left (\textit {\_Z}^{2}+2 \textit {\_Z} +4\right )^{2}+48 \RootOf \left (\textit {\_Z}^{2}+2 \textit {\_Z} +4\right ) \left (x^{3}+2 x^{2}+x \right )^{\frac {2}{3}}-30 \RootOf \left (\textit {\_Z}^{2}+2 \textit {\_Z} +4\right ) \left (x^{3}+2 x^{2}+x \right )^{\frac {1}{3}} x -16 \RootOf \left (\textit {\_Z}^{2}+2 \textit {\_Z} +4\right ) x^{2}-\RootOf \left (\textit {\_Z}^{2}+2 \textit {\_Z} +4\right )^{2}-30 \RootOf \left (\textit {\_Z}^{2}+2 \textit {\_Z} +4\right ) \left (x^{3}+2 x^{2}+x \right )^{\frac {1}{3}}-14 \RootOf \left (\textit {\_Z}^{2}+2 \textit {\_Z} +4\right ) x +36 \left (x^{3}+2 x^{2}+x \right )^{\frac {2}{3}}-96 \left (x^{3}+2 x^{2}+x \right )^{\frac {1}{3}} x +64 x^{2}+2 \RootOf \left (\textit {\_Z}^{2}+2 \textit {\_Z} +4\right )-96 \left (x^{3}+2 x^{2}+x \right )^{\frac {1}{3}}+112 x +48}{1+x}\right )}{9}+\frac {\RootOf \left (\textit {\_Z}^{2}+2 \textit {\_Z} +4\right ) \ln \left (-\frac {2 x^{2} \RootOf \left (\textit {\_Z}^{2}+2 \textit {\_Z} +4\right )^{2}-24 \RootOf \left (\textit {\_Z}^{2}+2 \textit {\_Z} +4\right ) \left (x^{3}+2 x^{2}+x \right )^{\frac {2}{3}}+9 \RootOf \left (\textit {\_Z}^{2}+2 \textit {\_Z} +4\right ) \left (x^{3}+2 x^{2}+x \right )^{\frac {1}{3}} x +19 \RootOf \left (\textit {\_Z}^{2}+2 \textit {\_Z} +4\right ) x^{2}-2 \RootOf \left (\textit {\_Z}^{2}+2 \textit {\_Z} +4\right )^{2}+9 \RootOf \left (\textit {\_Z}^{2}+2 \textit {\_Z} +4\right ) \left (x^{3}+2 x^{2}+x \right )^{\frac {1}{3}}+28 \RootOf \left (\textit {\_Z}^{2}+2 \textit {\_Z} +4\right ) x -30 \left (x^{3}+2 x^{2}+x \right )^{\frac {2}{3}}+48 \left (x^{3}+2 x^{2}+x \right )^{\frac {1}{3}} x -10 x^{2}+9 \RootOf \left (\textit {\_Z}^{2}+2 \textit {\_Z} +4\right )+48 \left (x^{3}+2 x^{2}+x \right )^{\frac {1}{3}}-14 x -4}{1+x}\right )}{18}\right ) \left (x^{2} \left (1+x \right )\right )^{\frac {1}{3}} \left (\left (1+x \right )^{2} x \right )^{\frac {1}{3}}}{x \left (1+x \right )}\) | \(452\) |
trager | \(\left (\frac {1}{6}+\frac {x}{2}\right ) \left (x^{3}+x^{2}\right )^{\frac {1}{3}}+\frac {\RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \ln \left (\frac {-36 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{2}+45 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}+x^{2}\right )^{\frac {2}{3}}+45 \left (x^{3}+x^{2}\right )^{\frac {1}{3}} \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x +36 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x +33 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{2}+24 \left (x^{3}+x^{2}\right )^{\frac {2}{3}}+24 x \left (x^{3}+x^{2}\right )^{\frac {1}{3}}+51 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x +20 x^{2}+15 x}{x}\right )}{3}-\frac {\ln \left (-\frac {36 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{2}+45 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}+x^{2}\right )^{\frac {2}{3}}+45 \left (x^{3}+x^{2}\right )^{\frac {1}{3}} \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x -36 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x +57 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{2}-9 \left (x^{3}+x^{2}\right )^{\frac {2}{3}}-9 x \left (x^{3}+x^{2}\right )^{\frac {1}{3}}+27 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x -5 x^{2}-2 x}{x}\right )}{9}-\frac {\ln \left (-\frac {36 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{2}+45 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}+x^{2}\right )^{\frac {2}{3}}+45 \left (x^{3}+x^{2}\right )^{\frac {1}{3}} \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x -36 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x +57 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{2}-9 \left (x^{3}+x^{2}\right )^{\frac {2}{3}}-9 x \left (x^{3}+x^{2}\right )^{\frac {1}{3}}+27 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x -5 x^{2}-2 x}{x}\right ) \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )}{3}\) | \(485\) |
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^{2}\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 = 1.13, size = 25, normalized size = 0.23 \begin {gather*} \frac {3\,x\,{\left (x^3+x^2\right )}^{1/3}\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{3},\frac {5}{3};\ \frac {8}{3};\ -x\right )}{5\,{\left (x+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^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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