Integrand size = 15, antiderivative size = 110 \[ \int \frac {x}{\sqrt [3]{-x^2+x^3}} \, dx=\frac {\left (-x^2+x^3\right )^{2/3}}{x}+\frac {\arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-x^2+x^3}}\right )}{\sqrt {3}}-\frac {1}{3} \log \left (-x+\sqrt [3]{-x^2+x^3}\right )+\frac {1}{6} \log \left (x^2+x \sqrt [3]{-x^2+x^3}+\left (-x^2+x^3\right )^{2/3}\right ) \]
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Time = 0.03 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.38, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2049, 2036, 61} \[ \int \frac {x}{\sqrt [3]{-x^2+x^3}} \, dx=-\frac {\sqrt [3]{x-1} x^{2/3} \arctan \left (\frac {2 \sqrt [3]{x-1}}{\sqrt {3} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{x^3-x^2}}+\frac {\left (x^3-x^2\right )^{2/3}}{x}-\frac {\sqrt [3]{x-1} x^{2/3} \log \left (\frac {\sqrt [3]{x-1}}{\sqrt [3]{x}}-1\right )}{2 \sqrt [3]{x^3-x^2}}-\frac {\sqrt [3]{x-1} x^{2/3} \log (x)}{6 \sqrt [3]{x^3-x^2}} \]
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Rule 61
Rule 2036
Rule 2049
Rubi steps \begin{align*} \text {integral}& = \frac {\left (-x^2+x^3\right )^{2/3}}{x}+\frac {1}{3} \int \frac {1}{\sqrt [3]{-x^2+x^3}} \, dx \\ & = \frac {\left (-x^2+x^3\right )^{2/3}}{x}+\frac {\left (\sqrt [3]{-1+x} x^{2/3}\right ) \int \frac {1}{\sqrt [3]{-1+x} x^{2/3}} \, dx}{3 \sqrt [3]{-x^2+x^3}} \\ & = \frac {\left (-x^2+x^3\right )^{2/3}}{x}-\frac {\sqrt [3]{-1+x} x^{2/3} \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{-1+x}}{\sqrt {3} \sqrt [3]{x}}\right )}{\sqrt {3} \sqrt [3]{-x^2+x^3}}-\frac {\sqrt [3]{-1+x} x^{2/3} \log \left (-1+\frac {\sqrt [3]{-1+x}}{\sqrt [3]{x}}\right )}{2 \sqrt [3]{-x^2+x^3}}-\frac {\sqrt [3]{-1+x} x^{2/3} \log (x)}{6 \sqrt [3]{-x^2+x^3}} \\ \end{align*}
Time = 0.20 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.25 \[ \int \frac {x}{\sqrt [3]{-x^2+x^3}} \, dx=\frac {x^{2/3} \left (-6 \sqrt [3]{x}+6 x^{4/3}+2 \sqrt {3} \sqrt [3]{-1+x} \arctan \left (\frac {\sqrt {3} \sqrt [3]{x}}{2 \sqrt [3]{-1+x}+\sqrt [3]{x}}\right )-2 \sqrt [3]{-1+x} \log \left (\sqrt [3]{-1+x}-\sqrt [3]{x}\right )+\sqrt [3]{-1+x} \log \left ((-1+x)^{2/3}+\sqrt [3]{-1+x} \sqrt [3]{x}+x^{2/3}\right )\right )}{6 \sqrt [3]{(-1+x) x^2}} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.39 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.25
method | result | size |
meijerg | \(\frac {3 \left (-\operatorname {signum}\left (-1+x \right )\right )^{\frac {1}{3}} x^{\frac {4}{3}} \operatorname {hypergeom}\left (\left [\frac {1}{3}, \frac {4}{3}\right ], \left [\frac {7}{3}\right ], x\right )}{4 \operatorname {signum}\left (-1+x \right )^{\frac {1}{3}}}\) | \(27\) |
risch | \(\frac {x \left (-1+x \right )}{\left (\left (-1+x \right ) x^{2}\right )^{\frac {1}{3}}}+\frac {\left (-\operatorname {signum}\left (-1+x \right )\right )^{\frac {1}{3}} x^{\frac {1}{3}} \operatorname {hypergeom}\left (\left [\frac {1}{3}, \frac {1}{3}\right ], \left [\frac {4}{3}\right ], x\right )}{\operatorname {signum}\left (-1+x \right )^{\frac {1}{3}}}\) | \(41\) |
pseudoelliptic | \(-\frac {x \left (x \sqrt {3}\, \arctan \left (\frac {\left (2 \left (\left (-1+x \right ) x^{2}\right )^{\frac {1}{3}}+x \right ) \sqrt {3}}{3 x}\right )+x \ln \left (\frac {\left (\left (-1+x \right ) x^{2}\right )^{\frac {1}{3}}-x}{x}\right )-\frac {x \ln \left (\frac {\left (\left (-1+x \right ) x^{2}\right )^{\frac {2}{3}}+\left (\left (-1+x \right ) x^{2}\right )^{\frac {1}{3}} x +x^{2}}{x^{2}}\right )}{2}-3 \left (\left (-1+x \right ) x^{2}\right )^{\frac {2}{3}}\right )}{3 \left (-\left (\left (-1+x \right ) x^{2}\right )^{\frac {1}{3}}+x \right ) \left (\left (\left (-1+x \right ) x^{2}\right )^{\frac {2}{3}}+x \left (x +\left (\left (-1+x \right ) x^{2}\right )^{\frac {1}{3}}\right )\right )}\) | \(135\) |
trager | \(\frac {\left (x^{3}-x^{2}\right )^{\frac {2}{3}}}{x}+\frac {\ln \left (-\frac {-45 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-6 \textit {\_Z} +4\right )^{2} x^{2}+144 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-6 \textit {\_Z} +4\right ) \left (x^{3}-x^{2}\right )^{\frac {2}{3}}+144 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-6 \textit {\_Z} +4\right ) \left (x^{3}-x^{2}\right )^{\frac {1}{3}} x +90 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-6 \textit {\_Z} +4\right )^{2} x +174 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-6 \textit {\_Z} +4\right ) x^{2}-60 \left (x^{3}-x^{2}\right )^{\frac {2}{3}}-60 x \left (x^{3}-x^{2}\right )^{\frac {1}{3}}-138 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-6 \textit {\_Z} +4\right ) x -80 x^{2}+48 x}{x}\right )}{3}-\frac {\ln \left (-\frac {-45 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-6 \textit {\_Z} +4\right )^{2} x^{2}+144 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-6 \textit {\_Z} +4\right ) \left (x^{3}-x^{2}\right )^{\frac {2}{3}}+144 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-6 \textit {\_Z} +4\right ) \left (x^{3}-x^{2}\right )^{\frac {1}{3}} x +90 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-6 \textit {\_Z} +4\right )^{2} x +174 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-6 \textit {\_Z} +4\right ) x^{2}-60 \left (x^{3}-x^{2}\right )^{\frac {2}{3}}-60 x \left (x^{3}-x^{2}\right )^{\frac {1}{3}}-138 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-6 \textit {\_Z} +4\right ) x -80 x^{2}+48 x}{x}\right ) \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-6 \textit {\_Z} +4\right )}{2}+\frac {\operatorname {RootOf}\left (9 \textit {\_Z}^{2}-6 \textit {\_Z} +4\right ) \ln \left (\frac {45 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-6 \textit {\_Z} +4\right )^{2} x^{2}+144 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-6 \textit {\_Z} +4\right ) \left (x^{3}-x^{2}\right )^{\frac {2}{3}}+144 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-6 \textit {\_Z} +4\right ) \left (x^{3}-x^{2}\right )^{\frac {1}{3}} x -90 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-6 \textit {\_Z} +4\right )^{2} x +114 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-6 \textit {\_Z} +4\right ) x^{2}-36 \left (x^{3}-x^{2}\right )^{\frac {2}{3}}-36 x \left (x^{3}-x^{2}\right )^{\frac {1}{3}}-18 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-6 \textit {\_Z} +4\right ) x -16 x^{2}+4 x}{x}\right )}{2}\) | \(509\) |
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Time = 0.25 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.03 \[ \int \frac {x}{\sqrt [3]{-x^2+x^3}} \, dx=-\frac {2 \, \sqrt {3} x \arctan \left (\frac {\sqrt {3} x + 2 \, \sqrt {3} {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}}{3 \, x}\right ) + 2 \, x \log \left (-\frac {x - {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}}{x}\right ) - x \log \left (\frac {x^{2} + {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}} x + {\left (x^{3} - x^{2}\right )}^{\frac {2}{3}}}{x^{2}}\right ) - 6 \, {\left (x^{3} - x^{2}\right )}^{\frac {2}{3}}}{6 \, x} \]
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\[ \int \frac {x}{\sqrt [3]{-x^2+x^3}} \, dx=\int \frac {x}{\sqrt [3]{x^{2} \left (x - 1\right )}}\, dx \]
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\[ \int \frac {x}{\sqrt [3]{-x^2+x^3}} \, dx=\int { \frac {x}{{\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}} \,d x } \]
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Time = 0.28 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.67 \[ \int \frac {x}{\sqrt [3]{-x^2+x^3}} \, dx=x {\left (-\frac {1}{x} + 1\right )}^{\frac {2}{3}} - \frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) + \frac {1}{6} \, \log \left ({\left (-\frac {1}{x} + 1\right )}^{\frac {2}{3}} + {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{3}} + 1\right ) - \frac {1}{3} \, \log \left ({\left | {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{3}} - 1 \right |}\right ) \]
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Timed out. \[ \int \frac {x}{\sqrt [3]{-x^2+x^3}} \, dx=\int \frac {x}{{\left (x^3-x^2\right )}^{1/3}} \,d x \]
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