Integrand size = 17, antiderivative size = 20 \[ \int \frac {1}{x \sqrt [3]{-x^2+x^3}} \, dx=\frac {3 \left (-x^2+x^3\right )^{2/3}}{2 x^2} \]
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Time = 0.02 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {2039} \[ \int \frac {1}{x \sqrt [3]{-x^2+x^3}} \, dx=\frac {3 \left (x^3-x^2\right )^{2/3}}{2 x^2} \]
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Rule 2039
Rubi steps \begin{align*} \text {integral}& = \frac {3 \left (-x^2+x^3\right )^{2/3}}{2 x^2} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.90 \[ \int \frac {1}{x \sqrt [3]{-x^2+x^3}} \, dx=\frac {3 (-1+x)}{2 \sqrt [3]{(-1+x) x^2}} \]
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Time = 0.78 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.75
method | result | size |
risch | \(\frac {\frac {3 x}{2}-\frac {3}{2}}{\left (\left (x -1\right ) x^{2}\right )^{\frac {1}{3}}}\) | \(15\) |
pseudoelliptic | \(\frac {3 \left (\left (x -1\right ) x^{2}\right )^{\frac {2}{3}}}{2 x^{2}}\) | \(15\) |
gosper | \(\frac {\frac {3 x}{2}-\frac {3}{2}}{\left (x^{3}-x^{2}\right )^{\frac {1}{3}}}\) | \(17\) |
trager | \(\frac {3 \left (x^{3}-x^{2}\right )^{\frac {2}{3}}}{2 x^{2}}\) | \(17\) |
meijerg | \(-\frac {3 \left (-\operatorname {signum}\left (x -1\right )\right )^{\frac {1}{3}} \left (1-x \right )^{\frac {2}{3}}}{2 \operatorname {signum}\left (x -1\right )^{\frac {1}{3}} x^{\frac {2}{3}}}\) | \(27\) |
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none
Time = 0.25 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.80 \[ \int \frac {1}{x \sqrt [3]{-x^2+x^3}} \, dx=\frac {3 \, {\left (x^{3} - x^{2}\right )}^{\frac {2}{3}}}{2 \, x^{2}} \]
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\[ \int \frac {1}{x \sqrt [3]{-x^2+x^3}} \, dx=\int \frac {1}{x \sqrt [3]{x^{2} \left (x - 1\right )}}\, dx \]
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\[ \int \frac {1}{x \sqrt [3]{-x^2+x^3}} \, dx=\int { \frac {1}{{\left (x^{3} - x^{2}\right )}^{\frac {1}{3}} x} \,d x } \]
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none
Time = 0.28 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.55 \[ \int \frac {1}{x \sqrt [3]{-x^2+x^3}} \, dx=\frac {3}{2} \, {\left (-\frac {1}{x} + 1\right )}^{\frac {2}{3}} \]
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Time = 5.18 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.80 \[ \int \frac {1}{x \sqrt [3]{-x^2+x^3}} \, dx=\frac {3\,{\left (x^3-x^2\right )}^{2/3}}{2\,x^2} \]
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