Optimal. Leaf size=118 \[ \frac {1}{324} \sqrt [3]{x^3+x^2} \left (81 x^3+9 x^2-12 x+20\right )+\frac {10}{243} \log \left (\sqrt [3]{x^3+x^2}-x\right )-\frac {5}{243} \log \left (x^2+\sqrt [3]{x^3+x^2} x+\left (x^3+x^2\right )^{2/3}\right )+\frac {10 \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^3+x^2}+x}\right )}{81 \sqrt {3}} \]
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Rubi [A] time = 0.19, antiderivative size = 200, normalized size of antiderivative = 1.69, number of steps used = 6, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {2021, 2024, 2032, 59} \begin {gather*} \frac {1}{4} \sqrt [3]{x^3+x^2} x^3+\frac {1}{36} \sqrt [3]{x^3+x^2} x^2-\frac {1}{27} \sqrt [3]{x^3+x^2} x+\frac {5}{81} \sqrt [3]{x^3+x^2}+\frac {5 (x+1)^{2/3} x^{4/3} \log (x+1)}{243 \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 )}{81 \left (x^3+x^2\right )^{2/3}}+\frac {10 (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 )}{81 \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^2 \sqrt [3]{x^2+x^3} \, dx &=\frac {1}{4} x^3 \sqrt [3]{x^2+x^3}+\frac {1}{12} \int \frac {x^4}{\left (x^2+x^3\right )^{2/3}} \, dx\\ &=\frac {1}{36} x^2 \sqrt [3]{x^2+x^3}+\frac {1}{4} x^3 \sqrt [3]{x^2+x^3}-\frac {2}{27} \int \frac {x^3}{\left (x^2+x^3\right )^{2/3}} \, dx\\ &=-\frac {1}{27} x \sqrt [3]{x^2+x^3}+\frac {1}{36} x^2 \sqrt [3]{x^2+x^3}+\frac {1}{4} x^3 \sqrt [3]{x^2+x^3}+\frac {5}{81} \int \frac {x^2}{\left (x^2+x^3\right )^{2/3}} \, dx\\ &=\frac {5}{81} \sqrt [3]{x^2+x^3}-\frac {1}{27} x \sqrt [3]{x^2+x^3}+\frac {1}{36} x^2 \sqrt [3]{x^2+x^3}+\frac {1}{4} x^3 \sqrt [3]{x^2+x^3}-\frac {10}{243} \int \frac {x}{\left (x^2+x^3\right )^{2/3}} \, dx\\ &=\frac {5}{81} \sqrt [3]{x^2+x^3}-\frac {1}{27} x \sqrt [3]{x^2+x^3}+\frac {1}{36} x^2 \sqrt [3]{x^2+x^3}+\frac {1}{4} x^3 \sqrt [3]{x^2+x^3}-\frac {\left (10 x^{4/3} (1+x)^{2/3}\right ) \int \frac {1}{\sqrt [3]{x} (1+x)^{2/3}} \, dx}{243 \left (x^2+x^3\right )^{2/3}}\\ &=\frac {5}{81} \sqrt [3]{x^2+x^3}-\frac {1}{27} x \sqrt [3]{x^2+x^3}+\frac {1}{36} x^2 \sqrt [3]{x^2+x^3}+\frac {1}{4} x^3 \sqrt [3]{x^2+x^3}+\frac {10 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 )}{81 \sqrt {3} \left (x^2+x^3\right )^{2/3}}+\frac {5 x^{4/3} (1+x)^{2/3} \log (1+x)}{243 \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 )}{81 \left (x^2+x^3\right )^{2/3}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 38, normalized size = 0.32 \begin {gather*} \frac {3 x^3 \sqrt [3]{x^2 (x+1)} \, _2F_1\left (-\frac {1}{3},\frac {11}{3};\frac {14}{3};-x\right )}{11 \sqrt [3]{x+1}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.35, size = 118, normalized size = 1.00 \begin {gather*} \frac {1}{324} \sqrt [3]{x^2+x^3} \left (20-12 x+9 x^2+81 x^3\right )+\frac {10 \tan ^{-1}\left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{x^2+x^3}}\right )}{81 \sqrt {3}}+\frac {10}{243} \log \left (-x+\sqrt [3]{x^2+x^3}\right )-\frac {5}{243} \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 = 0.45, size = 110, normalized size = 0.93 \begin {gather*} -\frac {10}{243} \, \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}{324} \, {\left (81 \, x^{3} + 9 \, x^{2} - 12 \, x + 20\right )} {\left (x^{3} + x^{2}\right )}^{\frac {1}{3}} + \frac {10}{243} \, \log \left (-\frac {x - {\left (x^{3} + x^{2}\right )}^{\frac {1}{3}}}{x}\right ) - \frac {5}{243} \, \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 = 0.19, size = 97, normalized size = 0.82 \begin {gather*} \frac {1}{324} \, {\left (20 \, {\left (\frac {1}{x} + 1\right )}^{\frac {10}{3}} - 72 \, {\left (\frac {1}{x} + 1\right )}^{\frac {7}{3}} + 93 \, {\left (\frac {1}{x} + 1\right )}^{\frac {4}{3}} + 40 \, {\left (\frac {1}{x} + 1\right )}^{\frac {1}{3}}\right )} x^{4} - \frac {10}{243} \, \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}{243} \, \log \left ({\left (\frac {1}{x} + 1\right )}^{\frac {2}{3}} + {\left (\frac {1}{x} + 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {10}{243} \, \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 = 0.84, size = 15, normalized size = 0.13
method | result | size |
meijerg | \(\frac {3 x^{\frac {11}{3}} \hypergeom \left (\left [-\frac {1}{3}, \frac {11}{3}\right ], \left [\frac {14}{3}\right ], -x \right )}{11}\) | \(15\) |
trager | \(\left (\frac {1}{4} x^{3}+\frac {1}{36} x^{2}-\frac {1}{27} x +\frac {5}{81}\right ) \left (x^{3}+x^{2}\right )^{\frac {1}{3}}-\frac {10 \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 )}{243}-\frac {10 \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 )}{81}+\frac {10 \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 )}{81}\) | \(496\) |
risch | \(\frac {\left (81 x^{3}+9 x^{2}-12 x +20\right ) \left (x^{2} \left (1+x \right )\right )^{\frac {1}{3}}}{324}+\frac {\left (\frac {5 \RootOf \left (\textit {\_Z}^{2}+2 \textit {\_Z} +4\right ) \ln \left (-\frac {-5 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}}+48 \RootOf \left (\textit {\_Z}^{2}+2 \textit {\_Z} +4\right ) \left (x^{3}+2 x^{2}+x \right )^{\frac {1}{3}} x +38 \RootOf \left (\textit {\_Z}^{2}+2 \textit {\_Z} +4\right ) x^{2}+36 \left (x^{3}+2 x^{2}+x \right )^{\frac {2}{3}}+48 \RootOf \left (\textit {\_Z}^{2}+2 \textit {\_Z} +4\right ) \left (x^{3}+2 x^{2}+x \right )^{\frac {1}{3}}+36 \left (x^{3}+2 x^{2}+x \right )^{\frac {1}{3}} x +5 \RootOf \left (\textit {\_Z}^{2}+2 \textit {\_Z} +4\right )^{2}+70 \RootOf \left (\textit {\_Z}^{2}+2 \textit {\_Z} +4\right ) x +16 x^{2}+36 \left (x^{3}+2 x^{2}+x \right )^{\frac {1}{3}}+32 \RootOf \left (\textit {\_Z}^{2}+2 \textit {\_Z} +4\right )+28 x +12}{1+x}\right )}{243}-\frac {5 \ln \left (\frac {5 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}}+48 \RootOf \left (\textit {\_Z}^{2}+2 \textit {\_Z} +4\right ) \left (x^{3}+2 x^{2}+x \right )^{\frac {1}{3}} x +58 \RootOf \left (\textit {\_Z}^{2}+2 \textit {\_Z} +4\right ) x^{2}+60 \left (x^{3}+2 x^{2}+x \right )^{\frac {2}{3}}+48 \RootOf \left (\textit {\_Z}^{2}+2 \textit {\_Z} +4\right ) \left (x^{3}+2 x^{2}+x \right )^{\frac {1}{3}}+60 \left (x^{3}+2 x^{2}+x \right )^{\frac {1}{3}} x -5 \RootOf \left (\textit {\_Z}^{2}+2 \textit {\_Z} +4\right )^{2}+70 \RootOf \left (\textit {\_Z}^{2}+2 \textit {\_Z} +4\right ) x +80 x^{2}+60 \left (x^{3}+2 x^{2}+x \right )^{\frac {1}{3}}+12 \RootOf \left (\textit {\_Z}^{2}+2 \textit {\_Z} +4\right )+112 x +32}{1+x}\right ) \RootOf \left (\textit {\_Z}^{2}+2 \textit {\_Z} +4\right )}{243}-\frac {10 \ln \left (\frac {5 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}}+48 \RootOf \left (\textit {\_Z}^{2}+2 \textit {\_Z} +4\right ) \left (x^{3}+2 x^{2}+x \right )^{\frac {1}{3}} x +58 \RootOf \left (\textit {\_Z}^{2}+2 \textit {\_Z} +4\right ) x^{2}+60 \left (x^{3}+2 x^{2}+x \right )^{\frac {2}{3}}+48 \RootOf \left (\textit {\_Z}^{2}+2 \textit {\_Z} +4\right ) \left (x^{3}+2 x^{2}+x \right )^{\frac {1}{3}}+60 \left (x^{3}+2 x^{2}+x \right )^{\frac {1}{3}} x -5 \RootOf \left (\textit {\_Z}^{2}+2 \textit {\_Z} +4\right )^{2}+70 \RootOf \left (\textit {\_Z}^{2}+2 \textit {\_Z} +4\right ) x +80 x^{2}+60 \left (x^{3}+2 x^{2}+x \right )^{\frac {1}{3}}+12 \RootOf \left (\textit {\_Z}^{2}+2 \textit {\_Z} +4\right )+112 x +32}{1+x}\right )}{243}\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 )}\) | \(669\) |
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^{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^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^{2} \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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