Integrand size = 19, antiderivative size = 23 \[ \int \frac {1}{(-1+x) \sqrt [3]{-x^2+x^3}} \, dx=-\frac {3 \left (-x^2+x^3\right )^{2/3}}{(-1+x) x} \]
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Time = 0.02 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.70, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2081, 37} \[ \int \frac {1}{(-1+x) \sqrt [3]{-x^2+x^3}} \, dx=-\frac {3 x}{\sqrt [3]{x^3-x^2}} \]
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Rule 37
Rule 2081
Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt [3]{-1+x} x^{2/3}\right ) \int \frac {1}{(-1+x)^{4/3} x^{2/3}} \, dx}{\sqrt [3]{-x^2+x^3}} \\ & = -\frac {3 x}{\sqrt [3]{-x^2+x^3}} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.61 \[ \int \frac {1}{(-1+x) \sqrt [3]{-x^2+x^3}} \, dx=-\frac {3 x}{\sqrt [3]{(-1+x) x^2}} \]
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Time = 1.01 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.57
method | result | size |
risch | \(-\frac {3 x}{\left (\left (x -1\right ) x^{2}\right )^{\frac {1}{3}}}\) | \(13\) |
pseudoelliptic | \(-\frac {3 x}{\left (\left (x -1\right ) x^{2}\right )^{\frac {1}{3}}}\) | \(13\) |
gosper | \(-\frac {3 x}{\left (x^{3}-x^{2}\right )^{\frac {1}{3}}}\) | \(15\) |
trager | \(-\frac {3 \left (x^{3}-x^{2}\right )^{\frac {2}{3}}}{\left (x -1\right ) x}\) | \(22\) |
meijerg | \(-\frac {3 \left (-\operatorname {signum}\left (x -1\right )\right )^{\frac {1}{3}} x^{\frac {1}{3}}}{\operatorname {signum}\left (x -1\right )^{\frac {1}{3}} \left (1-x \right )^{\frac {1}{3}}}\) | \(27\) |
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none
Time = 0.24 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96 \[ \int \frac {1}{(-1+x) \sqrt [3]{-x^2+x^3}} \, dx=-\frac {3 \, {\left (x^{3} - x^{2}\right )}^{\frac {2}{3}}}{x^{2} - x} \]
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\[ \int \frac {1}{(-1+x) \sqrt [3]{-x^2+x^3}} \, dx=\int \frac {1}{\sqrt [3]{x^{2} \left (x - 1\right )} \left (x - 1\right )}\, dx \]
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\[ \int \frac {1}{(-1+x) \sqrt [3]{-x^2+x^3}} \, dx=\int { \frac {1}{{\left (x^{3} - x^{2}\right )}^{\frac {1}{3}} {\left (x - 1\right )}} \,d x } \]
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none
Time = 0.27 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.48 \[ \int \frac {1}{(-1+x) \sqrt [3]{-x^2+x^3}} \, dx=-\frac {3}{{\left (-\frac {1}{x} + 1\right )}^{\frac {1}{3}}} \]
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Time = 5.03 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91 \[ \int \frac {1}{(-1+x) \sqrt [3]{-x^2+x^3}} \, dx=-\frac {3\,{\left (x^3-x^2\right )}^{2/3}}{x\,\left (x-1\right )} \]
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