Integrand size = 13, antiderivative size = 16 \[ \int \frac {\sqrt [3]{1+x^3}}{x^5} \, dx=-\frac {\left (1+x^3\right )^{4/3}}{4 x^4} \]
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Time = 0.00 (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 = {270} \[ \int \frac {\sqrt [3]{1+x^3}}{x^5} \, dx=-\frac {\left (x^3+1\right )^{4/3}}{4 x^4} \]
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Rule 270
Rubi steps \begin{align*} \text {integral}& = -\frac {\left (1+x^3\right )^{4/3}}{4 x^4} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt [3]{1+x^3}}{x^5} \, dx=-\frac {\left (1+x^3\right )^{4/3}}{4 x^4} \]
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Time = 0.80 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.81
method | result | size |
trager | \(-\frac {\left (x^{3}+1\right )^{\frac {4}{3}}}{4 x^{4}}\) | \(13\) |
meijerg | \(-\frac {\left (x^{3}+1\right )^{\frac {4}{3}}}{4 x^{4}}\) | \(13\) |
pseudoelliptic | \(-\frac {\left (x^{3}+1\right )^{\frac {4}{3}}}{4 x^{4}}\) | \(13\) |
risch | \(-\frac {x^{6}+2 x^{3}+1}{4 \left (x^{3}+1\right )^{\frac {2}{3}} x^{4}}\) | \(23\) |
gosper | \(-\frac {\left (1+x \right ) \left (x^{2}-x +1\right ) \left (x^{3}+1\right )^{\frac {1}{3}}}{4 x^{4}}\) | \(24\) |
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none
Time = 0.24 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \frac {\sqrt [3]{1+x^3}}{x^5} \, dx=-\frac {{\left (x^{3} + 1\right )}^{\frac {4}{3}}}{4 \, x^{4}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (14) = 28\).
Time = 0.38 (sec) , antiderivative size = 53, normalized size of antiderivative = 3.31 \[ \int \frac {\sqrt [3]{1+x^3}}{x^5} \, dx=\frac {\sqrt [3]{1 + \frac {1}{x^{3}}} \Gamma \left (- \frac {4}{3}\right )}{3 \Gamma \left (- \frac {1}{3}\right )} + \frac {\sqrt [3]{1 + \frac {1}{x^{3}}} \Gamma \left (- \frac {4}{3}\right )}{3 x^{3} \Gamma \left (- \frac {1}{3}\right )} \]
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none
Time = 0.18 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \frac {\sqrt [3]{1+x^3}}{x^5} \, dx=-\frac {{\left (x^{3} + 1\right )}^{\frac {4}{3}}}{4 \, x^{4}} \]
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\[ \int \frac {\sqrt [3]{1+x^3}}{x^5} \, dx=\int { \frac {{\left (x^{3} + 1\right )}^{\frac {1}{3}}}{x^{5}} \,d x } \]
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Time = 5.13 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.50 \[ \int \frac {\sqrt [3]{1+x^3}}{x^5} \, dx=-\frac {{\left (x^3+1\right )}^{1/3}+x^3\,{\left (x^3+1\right )}^{1/3}}{4\,x^4} \]
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