Optimal. Leaf size=113 \[ \frac {1}{54} \sqrt [3]{x^3+x^2} \left (18 x^2+3 x-5\right )-\frac {5}{81} \log \left (\sqrt [3]{x^3+x^2}-x\right )+\frac {5}{162} \log \left (x^2+\sqrt [3]{x^3+x^2} x+\left (x^3+x^2\right )^{2/3}\right )-\frac {5 \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^3+x^2}+x}\right )}{27 \sqrt {3}} \]
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Rubi [A] time = 0.13, antiderivative size = 182, normalized size of antiderivative = 1.61, number of steps used = 5, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {2021, 2024, 2032, 59} \begin {gather*} \frac {1}{3} \sqrt [3]{x^3+x^2} x^2+\frac {1}{18} \sqrt [3]{x^3+x^2} x-\frac {5}{54} \sqrt [3]{x^3+x^2}-\frac {5 (x+1)^{2/3} x^{4/3} \log (x+1)}{162 \left (x^3+x^2\right )^{2/3}}-\frac {5 (x+1)^{2/3} x^{4/3} \log \left (\frac {\sqrt [3]{x}}{\sqrt [3]{x+1}}-1\right )}{54 \left (x^3+x^2\right )^{2/3}}-\frac {5 (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 )}{27 \sqrt {3} \left (x^3+x^2\right )^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 59
Rule 2021
Rule 2024
Rule 2032
Rubi steps
\begin {align*} \int x \sqrt [3]{x^2+x^3} \, dx &=\frac {1}{3} x^2 \sqrt [3]{x^2+x^3}+\frac {1}{9} \int \frac {x^3}{\left (x^2+x^3\right )^{2/3}} \, dx\\ &=\frac {1}{18} x \sqrt [3]{x^2+x^3}+\frac {1}{3} x^2 \sqrt [3]{x^2+x^3}-\frac {5}{54} \int \frac {x^2}{\left (x^2+x^3\right )^{2/3}} \, dx\\ &=-\frac {5}{54} \sqrt [3]{x^2+x^3}+\frac {1}{18} x \sqrt [3]{x^2+x^3}+\frac {1}{3} x^2 \sqrt [3]{x^2+x^3}+\frac {5}{81} \int \frac {x}{\left (x^2+x^3\right )^{2/3}} \, dx\\ &=-\frac {5}{54} \sqrt [3]{x^2+x^3}+\frac {1}{18} x \sqrt [3]{x^2+x^3}+\frac {1}{3} x^2 \sqrt [3]{x^2+x^3}+\frac {\left (5 x^{4/3} (1+x)^{2/3}\right ) \int \frac {1}{\sqrt [3]{x} (1+x)^{2/3}} \, dx}{81 \left (x^2+x^3\right )^{2/3}}\\ &=-\frac {5}{54} \sqrt [3]{x^2+x^3}+\frac {1}{18} x \sqrt [3]{x^2+x^3}+\frac {1}{3} x^2 \sqrt [3]{x^2+x^3}-\frac {5 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 )}{27 \sqrt {3} \left (x^2+x^3\right )^{2/3}}-\frac {5 x^{4/3} (1+x)^{2/3} \log (1+x)}{162 \left (x^2+x^3\right )^{2/3}}-\frac {5 x^{4/3} (1+x)^{2/3} \log \left (-1+\frac {\sqrt [3]{x}}{\sqrt [3]{1+x}}\right )}{54 \left (x^2+x^3\right )^{2/3}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 35, normalized size = 0.31 \begin {gather*} \frac {3 \left (x^2 (x+1)\right )^{4/3} \, _2F_1\left (-\frac {1}{3},\frac {8}{3};\frac {11}{3};-x\right )}{8 (x+1)^{4/3}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.27, size = 113, normalized size = 1.00 \begin {gather*} \frac {1}{54} \left (-5+3 x+18 x^2\right ) \sqrt [3]{x^2+x^3}-\frac {5 \tan ^{-1}\left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{x^2+x^3}}\right )}{27 \sqrt {3}}-\frac {5}{81} \log \left (-x+\sqrt [3]{x^2+x^3}\right )+\frac {5}{162} \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 = 0.44, size = 105, normalized size = 0.93 \begin {gather*} \frac {5}{81} \, \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}{54} \, {\left (x^{3} + x^{2}\right )}^{\frac {1}{3}} {\left (18 \, x^{2} + 3 \, x - 5\right )} - \frac {5}{81} \, \log \left (-\frac {x - {\left (x^{3} + x^{2}\right )}^{\frac {1}{3}}}{x}\right ) + \frac {5}{162} \, \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 = 88, normalized size = 0.78 \begin {gather*} -\frac {1}{54} \, {\left (5 \, {\left (\frac {1}{x} + 1\right )}^{\frac {7}{3}} - 13 \, {\left (\frac {1}{x} + 1\right )}^{\frac {4}{3}} - 10 \, {\left (\frac {1}{x} + 1\right )}^{\frac {1}{3}}\right )} x^{3} + \frac {5}{81} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (\frac {1}{x} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) + \frac {5}{162} \, \log \left ({\left (\frac {1}{x} + 1\right )}^{\frac {2}{3}} + {\left (\frac {1}{x} + 1\right )}^{\frac {1}{3}} + 1\right ) - \frac {5}{81} \, \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 = 0.90, size = 15, normalized size = 0.13
method | result | size |
meijerg | \(\frac {3 x^{\frac {8}{3}} \hypergeom \left (\left [-\frac {1}{3}, \frac {8}{3}\right ], \left [\frac {11}{3}\right ], -x \right )}{8}\) | \(15\) |
risch | \(\frac {\left (18 x^{2}+3 x -5\right ) \left (x^{2} \left (1+x \right )\right )^{\frac {1}{3}}}{54}+\frac {\left (-\frac {5 \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2} x^{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 )}{81}+\frac {5 \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^{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 )}{162}\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 )}\) | \(457\) |
trager | \(\left (\frac {1}{3} x^{2}+\frac {1}{18} x -\frac {5}{54}\right ) \left (x^{3}+x^{2}\right )^{\frac {1}{3}}+\frac {5 \ln \left (-\frac {36 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{2}-36 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x -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 -57 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{2}-27 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x -9 \left (x^{3}+x^{2}\right )^{\frac {2}{3}}-9 x \left (x^{3}+x^{2}\right )^{\frac {1}{3}}-5 x^{2}-2 x}{x}\right )}{81}-\frac {5 \ln \left (-\frac {36 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{2}-36 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x -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 -57 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{2}-27 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x -9 \left (x^{3}+x^{2}\right )^{\frac {2}{3}}-9 x \left (x^{3}+x^{2}\right )^{\frac {1}{3}}-5 x^{2}-2 x}{x}\right ) \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )}{27}+\frac {5 \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}-36 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x +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 +33 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{2}+51 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x -24 \left (x^{3}+x^{2}\right )^{\frac {2}{3}}-24 x \left (x^{3}+x^{2}\right )^{\frac {1}{3}}-20 x^{2}-15 x}{x}\right )}{27}\) | \(491\) |
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}} x\,{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\,{\left (x^3+x^2\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 \sqrt [3]{x^{2} \left (x + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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