Integrand size = 18, antiderivative size = 28 \[ \int \frac {\left (1+x^3\right )^{2/3} \left (2+x^3\right )}{x^9} \, dx=\frac {\left (1+x^3\right )^{2/3} \left (-5-6 x^3-x^6\right )}{20 x^8} \]
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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} \[ \int \frac {\left (1+x^3\right )^{2/3} \left (2+x^3\right )}{x^9} \, dx=-\frac {\left (x^3+1\right )^{5/3}}{4 x^8}-\frac {\left (x^3+1\right )^{5/3}}{20 x^5} \]
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Rule 270
Rule 464
Rubi steps \begin{align*} \text {integral}& = -\frac {\left (1+x^3\right )^{5/3}}{4 x^8}+\frac {1}{4} \int \frac {\left (1+x^3\right )^{2/3}}{x^6} \, dx \\ & = -\frac {\left (1+x^3\right )^{5/3}}{4 x^8}-\frac {\left (1+x^3\right )^{5/3}}{20 x^5} \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {\left (1+x^3\right )^{2/3} \left (2+x^3\right )}{x^9} \, dx=\frac {\left (1+x^3\right )^{2/3} \left (-5-6 x^3-x^6\right )}{20 x^8} \]
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Time = 0.82 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.64
method | result | size |
pseudoelliptic | \(-\frac {\left (x^{3}+5\right ) \left (x^{3}+1\right )^{\frac {5}{3}}}{20 x^{8}}\) | \(18\) |
trager | \(-\frac {\left (x^{6}+6 x^{3}+5\right ) \left (x^{3}+1\right )^{\frac {2}{3}}}{20 x^{8}}\) | \(23\) |
risch | \(-\frac {x^{9}+7 x^{6}+11 x^{3}+5}{20 x^{8} \left (x^{3}+1\right )^{\frac {1}{3}}}\) | \(28\) |
gosper | \(-\frac {\left (1+x \right ) \left (x^{2}-x +1\right ) \left (x^{3}+5\right ) \left (x^{3}+1\right )^{\frac {2}{3}}}{20 x^{8}}\) | \(29\) |
meijerg | \(-\frac {\left (x^{3}+1\right )^{\frac {5}{3}}}{5 x^{5}}-\frac {\left (-\frac {3}{5} x^{6}+\frac {2}{5} x^{3}+1\right ) \left (x^{3}+1\right )^{\frac {2}{3}}}{4 x^{8}}\) | \(38\) |
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none
Time = 0.25 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.79 \[ \int \frac {\left (1+x^3\right )^{2/3} \left (2+x^3\right )}{x^9} \, dx=-\frac {{\left (x^{6} + 6 \, x^{3} + 5\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}}}{20 \, x^{8}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 139 vs. \(2 (24) = 48\).
Time = 1.25 (sec) , antiderivative size = 139, normalized size of antiderivative = 4.96 \[ \int \frac {\left (1+x^3\right )^{2/3} \left (2+x^3\right )}{x^9} \, dx=\frac {\left (1 + \frac {1}{x^{3}}\right )^{\frac {2}{3}} \Gamma \left (- \frac {5}{3}\right )}{3 \Gamma \left (- \frac {2}{3}\right )} + \frac {2 \left (x^{3} + 1\right )^{\frac {2}{3}} \Gamma \left (- \frac {8}{3}\right )}{3 x^{2} \Gamma \left (- \frac {2}{3}\right )} + \frac {\left (1 + \frac {1}{x^{3}}\right )^{\frac {2}{3}} \Gamma \left (- \frac {5}{3}\right )}{3 x^{3} \Gamma \left (- \frac {2}{3}\right )} - \frac {4 \left (x^{3} + 1\right )^{\frac {2}{3}} \Gamma \left (- \frac {8}{3}\right )}{9 x^{5} \Gamma \left (- \frac {2}{3}\right )} - \frac {10 \left (x^{3} + 1\right )^{\frac {2}{3}} \Gamma \left (- \frac {8}{3}\right )}{9 x^{8} \Gamma \left (- \frac {2}{3}\right )} \]
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Time = 0.19 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.89 \[ \int \frac {\left (1+x^3\right )^{2/3} \left (2+x^3\right )}{x^9} \, dx=\frac {{\left (x^{3} + 1\right )}^{\frac {5}{3}}}{5 \, x^{5}} - \frac {{\left (x^{3} + 1\right )}^{\frac {8}{3}}}{4 \, x^{8}} \]
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\[ \int \frac {\left (1+x^3\right )^{2/3} \left (2+x^3\right )}{x^9} \, dx=\int { \frac {{\left (x^{3} + 2\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}}}{x^{9}} \,d x } \]
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Time = 5.25 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.79 \[ \int \frac {\left (1+x^3\right )^{2/3} \left (2+x^3\right )}{x^9} \, dx=-\frac {{\left (x^3+1\right )}^{2/3}\,\left (x^6+6\,x^3+5\right )}{20\,x^8} \]
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