Integrand size = 13, antiderivative size = 16 \[ \int \frac {1}{x^2 \sqrt [3]{x+x^3}} \, dx=-\frac {3 \left (x+x^3\right )^{2/3}}{4 x^2} \]
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Time = 0.01 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {2039} \[ \int \frac {1}{x^2 \sqrt [3]{x+x^3}} \, dx=-\frac {3 \left (x^3+x\right )^{2/3}}{4 x^2} \]
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Rule 2039
Rubi steps \begin{align*} \text {integral}& = -\frac {3 \left (x+x^3\right )^{2/3}}{4 x^2} \\ \end{align*}
Time = 0.59 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^2 \sqrt [3]{x+x^3}} \, dx=-\frac {3 \left (x+x^3\right )^{2/3}}{4 x^2} \]
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Time = 0.81 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.81
method | result | size |
trager | \(-\frac {3 \left (x^{3}+x \right )^{\frac {2}{3}}}{4 x^{2}}\) | \(13\) |
meijerg | \(-\frac {3 \left (x^{2}+1\right )^{\frac {2}{3}}}{4 x^{\frac {4}{3}}}\) | \(13\) |
pseudoelliptic | \(-\frac {3 {\left (\left (x^{2}+1\right ) x \right )}^{\frac {2}{3}}}{4 x^{2}}\) | \(15\) |
gosper | \(-\frac {3 \left (x^{2}+1\right )}{4 x \left (x^{3}+x \right )^{\frac {1}{3}}}\) | \(18\) |
risch | \(-\frac {3 \left (x^{2}+1\right )}{4 x {\left (\left (x^{2}+1\right ) x \right )}^{\frac {1}{3}}}\) | \(20\) |
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none
Time = 0.24 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \frac {1}{x^2 \sqrt [3]{x+x^3}} \, dx=-\frac {3 \, {\left (x^{3} + x\right )}^{\frac {2}{3}}}{4 \, x^{2}} \]
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\[ \int \frac {1}{x^2 \sqrt [3]{x+x^3}} \, dx=\int \frac {1}{x^{2} \sqrt [3]{x \left (x^{2} + 1\right )}}\, dx \]
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none
Time = 0.27 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.06 \[ \int \frac {1}{x^2 \sqrt [3]{x+x^3}} \, dx=-\frac {3 \, {\left (x^{3} + x\right )}}{4 \, {\left (x^{2} + 1\right )}^{\frac {1}{3}} x^{\frac {7}{3}}} \]
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none
Time = 0.29 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.56 \[ \int \frac {1}{x^2 \sqrt [3]{x+x^3}} \, dx=-\frac {3}{4} \, {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {2}{3}} \]
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Time = 4.94 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \frac {1}{x^2 \sqrt [3]{x+x^3}} \, dx=-\frac {3\,{\left (x^3+x\right )}^{2/3}}{4\,x^2} \]
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