Integrand size = 13, antiderivative size = 118 \[ \int \sqrt [3]{-x^2+x^3} \, dx=\frac {1}{6} (-1+3 x) \sqrt [3]{-x^2+x^3}+\frac {\arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-x^2+x^3}}\right )}{3 \sqrt {3}}+\frac {1}{9} \log \left (-x+\sqrt [3]{-x^2+x^3}\right )-\frac {1}{18} \log \left (x^2+x \sqrt [3]{-x^2+x^3}+\left (-x^2+x^3\right )^{2/3}\right ) \]
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Time = 0.05 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.47, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {2029, 2049, 2057, 61} \[ \int \sqrt [3]{-x^2+x^3} \, dx=\frac {(x-1)^{2/3} x^{4/3} \arctan \left (\frac {2 \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{x-1}}+\frac {1}{\sqrt {3}}\right )}{3 \sqrt {3} \left (x^3-x^2\right )^{2/3}}+\frac {1}{2} \sqrt [3]{x^3-x^2} x-\frac {1}{6} \sqrt [3]{x^3-x^2}+\frac {(x-1)^{2/3} x^{4/3} \log \left (\frac {\sqrt [3]{x}}{\sqrt [3]{x-1}}-1\right )}{6 \left (x^3-x^2\right )^{2/3}}+\frac {(x-1)^{2/3} x^{4/3} \log (x-1)}{18 \left (x^3-x^2\right )^{2/3}} \]
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Rule 61
Rule 2029
Rule 2049
Rule 2057
Rubi steps \begin{align*} \text {integral}& = \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 ((-1+x)^{2/3} x^{4/3}\right ) \int \frac {1}{(-1+x)^{2/3} \sqrt [3]{x}} \, 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 {(-1+x)^{2/3} x^{4/3} \arctan \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 {(-1+x)^{2/3} x^{4/3} \log \left (-1+\frac {\sqrt [3]{x}}{\sqrt [3]{-1+x}}\right )}{6 \left (-x^2+x^3\right )^{2/3}}+\frac {(-1+x)^{2/3} x^{4/3} \log (-1+x)}{18 \left (-x^2+x^3\right )^{2/3}} \\ \end{align*}
Time = 0.21 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.18 \[ \int \sqrt [3]{-x^2+x^3} \, dx=\frac {(-1+x)^{2/3} x^{4/3} \left (-3 \sqrt [3]{-1+x} x^{2/3}+9 \sqrt [3]{-1+x} x^{5/3}+2 \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{x}}{2 \sqrt [3]{-1+x}+\sqrt [3]{x}}\right )+2 \log \left (\sqrt [3]{-1+x}-\sqrt [3]{x}\right )-\log \left ((-1+x)^{2/3}+\sqrt [3]{-1+x} \sqrt [3]{x}+x^{2/3}\right )\right )}{18 \left ((-1+x) x^2\right )^{2/3}} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.58 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.23
method | result | size |
meijerg | \(\frac {3 \operatorname {signum}\left (-1+x \right )^{\frac {1}{3}} x^{\frac {5}{3}} \operatorname {hypergeom}\left (\left [-\frac {1}{3}, \frac {5}{3}\right ], \left [\frac {8}{3}\right ], x\right )}{5 \left (-\operatorname {signum}\left (-1+x \right )\right )^{\frac {1}{3}}}\) | \(27\) |
pseudoelliptic | \(\frac {x^{4} \left (9 \left (\left (-1+x \right ) x^{2}\right )^{\frac {1}{3}} x -2 \sqrt {3}\, \arctan \left (\frac {\left (2 \left (\left (-1+x \right ) x^{2}\right )^{\frac {1}{3}}+x \right ) \sqrt {3}}{3 x}\right )-3 \left (\left (-1+x \right ) x^{2}\right )^{\frac {1}{3}}-\ln \left (\frac {\left (\left (-1+x \right ) x^{2}\right )^{\frac {2}{3}}+\left (\left (-1+x \right ) x^{2}\right )^{\frac {1}{3}} x +x^{2}}{x^{2}}\right )+2 \ln \left (\frac {\left (\left (-1+x \right ) x^{2}\right )^{\frac {1}{3}}-x}{x}\right )\right )}{18 {\left (\left (\left (-1+x \right ) x^{2}\right )^{\frac {2}{3}}+\left (\left (-1+x \right ) x^{2}\right )^{\frac {1}{3}} x +x^{2}\right )}^{2} {\left (\left (\left (-1+x \right ) x^{2}\right )^{\frac {1}{3}}-x \right )}^{2}}\) | \(149\) |
risch | \(\frac {\left (-1+3 x \right ) \left (\left (-1+x \right ) x^{2}\right )^{\frac {1}{3}}}{6}+\frac {\left (\frac {\ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{2}+15 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}-2 x^{2}+x \right )^{\frac {2}{3}}-24 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}-2 x^{2}+x \right )^{\frac {1}{3}} x -3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x +10 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}-9 \left (x^{3}-2 x^{2}+x \right )^{\frac {2}{3}}+24 \operatorname {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 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2}-23 \operatorname {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 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-40 x +15}{-1+x}\right )}{9}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \ln \left (\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{2}+15 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}-2 x^{2}+x \right )^{\frac {2}{3}}+9 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}-2 x^{2}+x \right )^{\frac {1}{3}} x -15 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x -19 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}+24 \left (x^{3}-2 x^{2}+x \right )^{\frac {2}{3}}-9 \operatorname {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 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2}+22 \operatorname {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 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+5 x -1}{-1+x}\right )}{9}\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 )}\) | \(441\) |
trager | \(\left (-\frac {1}{6}+\frac {x}{2}\right ) \left (x^{3}-x^{2}\right )^{\frac {1}{3}}+\frac {\operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \ln \left (\frac {45 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{2}-90 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x +72 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}-x^{2}\right )^{\frac {2}{3}}+72 \left (x^{3}-x^{2}\right )^{\frac {1}{3}} \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x +87 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{2}-69 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x +15 \left (x^{3}-x^{2}\right )^{\frac {2}{3}}+15 x \left (x^{3}-x^{2}\right )^{\frac {1}{3}}+20 x^{2}-12 x}{x}\right )}{3}-\frac {\ln \left (\frac {45 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{2}-90 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x -72 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}-x^{2}\right )^{\frac {2}{3}}-72 \left (x^{3}-x^{2}\right )^{\frac {1}{3}} \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x -57 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{2}+9 \operatorname {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}}-4 x^{2}+x}{x}\right )}{9}-\frac {\ln \left (\frac {45 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{2}-90 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x -72 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}-x^{2}\right )^{\frac {2}{3}}-72 \left (x^{3}-x^{2}\right )^{\frac {1}{3}} \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x -57 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{2}+9 \operatorname {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}}-4 x^{2}+x}{x}\right ) \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )}{3}\) | \(505\) |
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Time = 0.27 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.93 \[ \int \sqrt [3]{-x^2+x^3} \, dx=-\frac {1}{9} \, \sqrt {3} \arctan \left (\frac {\sqrt {3} x + 2 \, \sqrt {3} {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}}{3 \, x}\right ) + \frac {1}{6} \, {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}} {\left (3 \, x - 1\right )} + \frac {1}{9} \, \log \left (-\frac {x - {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}}{x}\right ) - \frac {1}{18} \, \log \left (\frac {x^{2} + {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}} x + {\left (x^{3} - x^{2}\right )}^{\frac {2}{3}}}{x^{2}}\right ) \]
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\[ \int \sqrt [3]{-x^2+x^3} \, dx=\int \sqrt [3]{x^{3} - x^{2}}\, dx \]
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\[ \int \sqrt [3]{-x^2+x^3} \, dx=\int { {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}} \,d x } \]
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Time = 0.28 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.75 \[ \int \sqrt [3]{-x^2+x^3} \, dx=\frac {1}{6} \, {\left ({\left (-\frac {1}{x} + 1\right )}^{\frac {4}{3}} + 2 \, {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{3}}\right )} x^{2} - \frac {1}{9} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) - \frac {1}{18} \, \log \left ({\left (-\frac {1}{x} + 1\right )}^{\frac {2}{3}} + {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {1}{9} \, \log \left ({\left | {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{3}} - 1 \right |}\right ) \]
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Time = 5.95 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.23 \[ \int \sqrt [3]{-x^2+x^3} \, dx=\frac {3\,x\,{\left (x^3-x^2\right )}^{1/3}\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{3},\frac {5}{3};\ \frac {8}{3};\ x\right )}{5\,{\left (1-x\right )}^{1/3}} \]
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