Integrand size = 17, antiderivative size = 25 \[ \int \frac {1}{x^2 \sqrt [3]{-x^2+x^3}} \, dx=\frac {3 (2+3 x) \left (-x^2+x^3\right )^{2/3}}{10 x^3} \]
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Time = 0.04 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.64, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {2041, 2039} \[ \int \frac {1}{x^2 \sqrt [3]{-x^2+x^3}} \, dx=\frac {9 \left (x^3-x^2\right )^{2/3}}{10 x^2}+\frac {3 \left (x^3-x^2\right )^{2/3}}{5 x^3} \]
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Rule 2039
Rule 2041
Rubi steps \begin{align*} \text {integral}& = \frac {3 \left (-x^2+x^3\right )^{2/3}}{5 x^3}+\frac {3}{5} \int \frac {1}{x \sqrt [3]{-x^2+x^3}} \, dx \\ & = \frac {3 \left (-x^2+x^3\right )^{2/3}}{5 x^3}+\frac {9 \left (-x^2+x^3\right )^{2/3}}{10 x^2} \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92 \[ \int \frac {1}{x^2 \sqrt [3]{-x^2+x^3}} \, dx=\frac {3 \left ((-1+x) x^2\right )^{2/3} (2+3 x)}{10 x^3} \]
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Time = 0.86 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.80
method | result | size |
pseudoelliptic | \(\frac {3 \left (2+3 x \right ) \left (\left (x -1\right ) x^{2}\right )^{\frac {2}{3}}}{10 x^{3}}\) | \(20\) |
trager | \(\frac {3 \left (2+3 x \right ) \left (x^{3}-x^{2}\right )^{\frac {2}{3}}}{10 x^{3}}\) | \(22\) |
gosper | \(\frac {3 \left (x -1\right ) \left (2+3 x \right )}{10 x \left (x^{3}-x^{2}\right )^{\frac {1}{3}}}\) | \(25\) |
risch | \(\frac {-\frac {3}{10} x -\frac {3}{5}+\frac {9}{10} x^{2}}{x \left (\left (x -1\right ) x^{2}\right )^{\frac {1}{3}}}\) | \(25\) |
meijerg | \(-\frac {3 \left (-\operatorname {signum}\left (x -1\right )\right )^{\frac {1}{3}} \left (1+\frac {3 x}{2}\right ) \left (1-x \right )^{\frac {2}{3}}}{5 \operatorname {signum}\left (x -1\right )^{\frac {1}{3}} x^{\frac {5}{3}}}\) | \(32\) |
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Time = 0.25 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.84 \[ \int \frac {1}{x^2 \sqrt [3]{-x^2+x^3}} \, dx=\frac {3 \, {\left (x^{3} - x^{2}\right )}^{\frac {2}{3}} {\left (3 \, x + 2\right )}}{10 \, x^{3}} \]
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\[ \int \frac {1}{x^2 \sqrt [3]{-x^2+x^3}} \, dx=\int \frac {1}{x^{2} \sqrt [3]{x^{2} \left (x - 1\right )}}\, dx \]
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\[ \int \frac {1}{x^2 \sqrt [3]{-x^2+x^3}} \, dx=\int { \frac {1}{{\left (x^{3} - x^{2}\right )}^{\frac {1}{3}} x^{2}} \,d x } \]
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Time = 0.27 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92 \[ \int \frac {1}{x^2 \sqrt [3]{-x^2+x^3}} \, dx=-\frac {3}{5} \, {\left (-\frac {1}{x} + 1\right )}^{\frac {5}{3}} + \frac {3}{2} \, {\left (-\frac {1}{x} + 1\right )}^{\frac {2}{3}} \]
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Time = 4.80 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.32 \[ \int \frac {1}{x^2 \sqrt [3]{-x^2+x^3}} \, dx=\frac {9\,x\,{\left (x^3-x^2\right )}^{2/3}+6\,{\left (x^3-x^2\right )}^{2/3}}{10\,x^3} \]
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