Integrand size = 18, antiderivative size = 28 \[ \int \frac {\left (-1+x^3\right ) \sqrt [3]{1+x^3}}{x^8} \, dx=\frac {\sqrt [3]{1+x^3} \left (2-3 x^3-5 x^6\right )}{14 x^7} \]
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Time = 0.01 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.18, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {464, 270} \[ \int \frac {\left (-1+x^3\right ) \sqrt [3]{1+x^3}}{x^8} \, dx=\frac {\left (x^3+1\right )^{4/3}}{7 x^7}-\frac {5 \left (x^3+1\right )^{4/3}}{14 x^4} \]
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Rule 270
Rule 464
Rubi steps \begin{align*} \text {integral}& = \frac {\left (1+x^3\right )^{4/3}}{7 x^7}+\frac {10}{7} \int \frac {\sqrt [3]{1+x^3}}{x^5} \, dx \\ & = \frac {\left (1+x^3\right )^{4/3}}{7 x^7}-\frac {5 \left (1+x^3\right )^{4/3}}{14 x^4} \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.82 \[ \int \frac {\left (-1+x^3\right ) \sqrt [3]{1+x^3}}{x^8} \, dx=\frac {\left (2-5 x^3\right ) \left (1+x^3\right )^{4/3}}{14 x^7} \]
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Time = 0.81 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.71
method | result | size |
pseudoelliptic | \(-\frac {\left (x^{3}+1\right )^{\frac {4}{3}} \left (5 x^{3}-2\right )}{14 x^{7}}\) | \(20\) |
trager | \(-\frac {\left (5 x^{6}+3 x^{3}-2\right ) \left (x^{3}+1\right )^{\frac {1}{3}}}{14 x^{7}}\) | \(25\) |
risch | \(-\frac {5 x^{9}+8 x^{6}+x^{3}-2}{14 \left (x^{3}+1\right )^{\frac {2}{3}} x^{7}}\) | \(28\) |
gosper | \(-\frac {\left (x^{2}-x +1\right ) \left (1+x \right ) \left (5 x^{3}-2\right ) \left (x^{3}+1\right )^{\frac {1}{3}}}{14 x^{7}}\) | \(31\) |
meijerg | \(\frac {\left (-\frac {3}{4} x^{6}+\frac {1}{4} x^{3}+1\right ) \left (x^{3}+1\right )^{\frac {1}{3}}}{7 x^{7}}-\frac {\left (x^{3}+1\right )^{\frac {4}{3}}}{4 x^{4}}\) | \(38\) |
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Time = 0.24 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.86 \[ \int \frac {\left (-1+x^3\right ) \sqrt [3]{1+x^3}}{x^8} \, dx=-\frac {{\left (5 \, x^{6} + 3 \, x^{3} - 2\right )} {\left (x^{3} + 1\right )}^{\frac {1}{3}}}{14 \, x^{7}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 134 vs. \(2 (24) = 48\).
Time = 1.06 (sec) , antiderivative size = 134, normalized size of antiderivative = 4.79 \[ \int \frac {\left (-1+x^3\right ) \sqrt [3]{1+x^3}}{x^8} \, dx=\frac {\sqrt [3]{1 + \frac {1}{x^{3}}} \Gamma \left (- \frac {4}{3}\right )}{3 \Gamma \left (- \frac {1}{3}\right )} - \frac {\sqrt [3]{x^{3} + 1} \Gamma \left (- \frac {7}{3}\right )}{3 x \Gamma \left (- \frac {1}{3}\right )} + \frac {\sqrt [3]{1 + \frac {1}{x^{3}}} \Gamma \left (- \frac {4}{3}\right )}{3 x^{3} \Gamma \left (- \frac {1}{3}\right )} + \frac {\sqrt [3]{x^{3} + 1} \Gamma \left (- \frac {7}{3}\right )}{9 x^{4} \Gamma \left (- \frac {1}{3}\right )} + \frac {4 \sqrt [3]{x^{3} + 1} \Gamma \left (- \frac {7}{3}\right )}{9 x^{7} \Gamma \left (- \frac {1}{3}\right )} \]
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Time = 0.19 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.89 \[ \int \frac {\left (-1+x^3\right ) \sqrt [3]{1+x^3}}{x^8} \, dx=-\frac {{\left (x^{3} + 1\right )}^{\frac {4}{3}}}{2 \, x^{4}} + \frac {{\left (x^{3} + 1\right )}^{\frac {7}{3}}}{7 \, x^{7}} \]
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\[ \int \frac {\left (-1+x^3\right ) \sqrt [3]{1+x^3}}{x^8} \, dx=\int { \frac {{\left (x^{3} + 1\right )}^{\frac {1}{3}} {\left (x^{3} - 1\right )}}{x^{8}} \,d x } \]
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Time = 0.19 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.39 \[ \int \frac {\left (-1+x^3\right ) \sqrt [3]{1+x^3}}{x^8} \, dx=-\frac {3\,x^3\,{\left (x^3+1\right )}^{1/3}-2\,{\left (x^3+1\right )}^{1/3}+5\,x^6\,{\left (x^3+1\right )}^{1/3}}{14\,x^7} \]
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