Integrand size = 15, antiderivative size = 18 \[ \int \frac {1}{x \sqrt [3]{x^2+x^3}} \, dx=-\frac {3 \left (x^2+x^3\right )^{2/3}}{2 x^2} \]
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Time = 0.01 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, 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 = 1.00 \[ \int \frac {1}{x \sqrt [3]{x^2+x^3}} \, dx=-\frac {3 (1+x)}{2 \sqrt [3]{x^2 (1+x)}} \]
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Time = 0.76 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.61
method | result | size |
meijerg | \(-\frac {3 \left (1+x \right )^{\frac {2}{3}}}{2 x^{\frac {2}{3}}}\) | \(11\) |
gosper | \(-\frac {3 \left (1+x \right )}{2 \left (x^{3}+x^{2}\right )^{\frac {1}{3}}}\) | \(15\) |
trager | \(-\frac {3 \left (x^{3}+x^{2}\right )^{\frac {2}{3}}}{2 x^{2}}\) | \(15\) |
risch | \(-\frac {3 \left (1+x \right )}{2 \left (x^{2} \left (1+x \right )\right )^{\frac {1}{3}}}\) | \(15\) |
pseudoelliptic | \(-\frac {3 \left (x^{2} \left (1+x \right )\right )^{\frac {2}{3}}}{2 x^{2}}\) | \(15\) |
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none
Time = 0.24 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.78 \[ \int \frac {1}{x \sqrt [3]{x^2+x^3}} \, dx=-\frac {3 \, {\left (x^{3} + x^{2}\right )}^{\frac {2}{3}}}{2 \, x^{2}} \]
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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 = 0.29 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.50 \[ \int \frac {1}{x \sqrt [3]{x^2+x^3}} \, dx=-\frac {3}{2} \, {\left (\frac {1}{x} + 1\right )}^{\frac {2}{3}} \]
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Time = 4.96 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.78 \[ \int \frac {1}{x \sqrt [3]{x^2+x^3}} \, dx=-\frac {3\,{\left (x^3+x^2\right )}^{2/3}}{2\,x^2} \]
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