Optimal. Leaf size=106 \[ \sqrt [3]{x^3-x^2}+\frac {1}{3} \log \left (\sqrt [3]{x^3-x^2}-x\right )-\frac {1}{6} \log \left (x^2+\sqrt [3]{x^3-x^2} x+\left (x^3-x^2\right )^{2/3}\right )+\frac {\tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^3-x^2}+x}\right )}{\sqrt {3}} \]
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Rubi [A] time = 0.07, antiderivative size = 149, normalized size of antiderivative = 1.41, number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {2021, 2032, 59} \begin {gather*} \sqrt [3]{x^3-x^2}+\frac {(x-1)^{2/3} x^{4/3} \log \left (\frac {\sqrt [3]{x}}{\sqrt [3]{x-1}}-1\right )}{2 \left (x^3-x^2\right )^{2/3}}+\frac {(x-1)^{2/3} x^{4/3} \log (x-1)}{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 )}{\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 2032
Rubi steps
\begin {align*} \int \frac {\sqrt [3]{-x^2+x^3}}{x} \, dx &=\sqrt [3]{-x^2+x^3}-\frac {1}{3} \int \frac {x}{\left (-x^2+x^3\right )^{2/3}} \, dx\\ &=\sqrt [3]{-x^2+x^3}-\frac {\left ((-1+x)^{2/3} x^{4/3}\right ) \int \frac {1}{(-1+x)^{2/3} \sqrt [3]{x}} \, dx}{3 \left (-x^2+x^3\right )^{2/3}}\\ &=\sqrt [3]{-x^2+x^3}+\frac {(-1+x)^{2/3} x^{4/3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{-1+x}}\right )}{\sqrt {3} \left (-x^2+x^3\right )^{2/3}}+\frac {(-1+x)^{2/3} x^{4/3} \log \left (-1+\frac {\sqrt [3]{x}}{\sqrt [3]{-1+x}}\right )}{2 \left (-x^2+x^3\right )^{2/3}}+\frac {(-1+x)^{2/3} x^{4/3} \log (-1+x)}{6 \left (-x^2+x^3\right )^{2/3}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 35, normalized size = 0.33 \begin {gather*} \frac {3 \left ((x-1) x^2\right )^{4/3} \, _2F_1\left (\frac {1}{3},\frac {4}{3};\frac {7}{3};1-x\right )}{4 x^{8/3}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.21, size = 106, normalized size = 1.00 \begin {gather*} \sqrt [3]{x^3-x^2}+\frac {1}{3} \log \left (\sqrt [3]{x^3-x^2}-x\right )-\frac {1}{6} \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 )}{\sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 103, normalized size = 0.97 \begin {gather*} -\frac {1}{3} \, \sqrt {3} \arctan \left (\frac {\sqrt {3} x + 2 \, \sqrt {3} {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}}{3 \, x}\right ) + {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}} + \frac {1}{3} \, \log \left (-\frac {x - {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}}{x}\right ) - \frac {1}{6} \, \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 = 0.14, size = 74, normalized size = 0.70 \begin {gather*} -\frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) + x {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{3}} - \frac {1}{6} \, \log \left ({\left (-\frac {1}{x} + 1\right )}^{\frac {2}{3}} + {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {1}{3} \, \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 = 0.50, size = 434, normalized size = 4.09 \begin {gather*} \left (\left (-1+x \right ) x^{2}\right )^{\frac {1}{3}}+\frac {\left (\frac {\ln \left (-\frac {\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{2}+15 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}-2 x^{2}+x \right )^{\frac {2}{3}}-24 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}-2 x^{2}+x \right )^{\frac {1}{3}} x -3 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x +10 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}-9 \left (x^{3}-2 x^{2}+x \right )^{\frac {2}{3}}+24 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}-2 x^{2}+x \right )^{\frac {1}{3}}-15 \left (x^{3}-2 x^{2}+x \right )^{\frac {1}{3}} x +2 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2}-23 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x +25 x^{2}+15 \left (x^{3}-2 x^{2}+x \right )^{\frac {1}{3}}+13 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-40 x +15}{-1+x}\right )}{3}+\frac {\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \ln \left (\frac {5 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{2}+15 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}-2 x^{2}+x \right )^{\frac {2}{3}}+9 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}-2 x^{2}+x \right )^{\frac {1}{3}} x -15 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x -19 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}+24 \left (x^{3}-2 x^{2}+x \right )^{\frac {2}{3}}-9 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}-2 x^{2}+x \right )^{\frac {1}{3}}-15 \left (x^{3}-2 x^{2}+x \right )^{\frac {1}{3}} x +10 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2}+22 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x -4 x^{2}+15 \left (x^{3}-2 x^{2}+x \right )^{\frac {1}{3}}-3 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+5 x -1}{-1+x}\right )}{3}\right ) \left (\left (-1+x \right ) x^{2}\right )^{\frac {1}{3}} \left (\left (-1+x \right )^{2} x \right )^{\frac {1}{3}}}{x \left (-1+x \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 \frac {{\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 \frac {{\left (x^3-x^2\right )}^{1/3}}{x} \,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 \frac {\sqrt [3]{x^{2} \left (x - 1\right )}}{x}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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