Integrand size = 18, antiderivative size = 16 \[ \int \frac {\left (1+x^5\right )^{2/3} \left (6+x^5\right )}{x^{11}} \, dx=-\frac {3 \left (1+x^5\right )^{5/3}}{5 x^{10}} \]
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Time = 0.01 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {457, 75} \[ \int \frac {\left (1+x^5\right )^{2/3} \left (6+x^5\right )}{x^{11}} \, dx=-\frac {3 \left (x^5+1\right )^{5/3}}{5 x^{10}} \]
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Rule 75
Rule 457
Rubi steps \begin{align*} \text {integral}& = \frac {1}{5} \text {Subst}\left (\int \frac {(1+x)^{2/3} (6+x)}{x^3} \, dx,x,x^5\right ) \\ & = -\frac {3 \left (1+x^5\right )^{5/3}}{5 x^{10}} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {\left (1+x^5\right )^{2/3} \left (6+x^5\right )}{x^{11}} \, dx=-\frac {3 \left (1+x^5\right )^{5/3}}{5 x^{10}} \]
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Time = 1.06 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.81
method | result | size |
trager | \(-\frac {3 \left (x^{5}+1\right )^{\frac {5}{3}}}{5 x^{10}}\) | \(13\) |
pseudoelliptic | \(-\frac {3 \left (x^{5}+1\right )^{\frac {5}{3}}}{5 x^{10}}\) | \(13\) |
risch | \(-\frac {3 \left (x^{10}+2 x^{5}+1\right )}{5 x^{10} \left (x^{5}+1\right )^{\frac {1}{3}}}\) | \(23\) |
gosper | \(-\frac {3 \left (x^{4}-x^{3}+x^{2}-x +1\right ) \left (1+x \right ) \left (x^{5}+1\right )^{\frac {2}{3}}}{5 x^{10}}\) | \(32\) |
meijerg | \(-\frac {\sqrt {3}\, \Gamma \left (\frac {2}{3}\right ) \left (\frac {\pi \sqrt {3}\, x^{5} \operatorname {hypergeom}\left (\left [1, 1, \frac {4}{3}\right ], \left [2, 3\right ], -x^{5}\right )}{9 \Gamma \left (\frac {2}{3}\right )}-\frac {2 \left (-\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \left (3\right )}{2}-1+5 \ln \left (x \right )\right ) \pi \sqrt {3}}{3 \Gamma \left (\frac {2}{3}\right )}+\frac {\pi \sqrt {3}}{\Gamma \left (\frac {2}{3}\right ) x^{5}}\right )}{15 \pi }-\frac {2 \sqrt {3}\, \Gamma \left (\frac {2}{3}\right ) \left (-\frac {4 \pi \sqrt {3}\, x^{5} \operatorname {hypergeom}\left (\left [1, 1, \frac {7}{3}\right ], \left [2, 4\right ], -x^{5}\right )}{81 \Gamma \left (\frac {2}{3}\right )}+\frac {\left (\frac {3}{2}-\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \left (3\right )}{2}+5 \ln \left (x \right )\right ) \pi \sqrt {3}}{9 \Gamma \left (\frac {2}{3}\right )}+\frac {\pi \sqrt {3}}{2 \Gamma \left (\frac {2}{3}\right ) x^{10}}+\frac {2 \pi \sqrt {3}}{3 \Gamma \left (\frac {2}{3}\right ) x^{5}}\right )}{5 \pi }\) | \(166\) |
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Time = 0.25 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \frac {\left (1+x^5\right )^{2/3} \left (6+x^5\right )}{x^{11}} \, dx=-\frac {3 \, {\left (x^{5} + 1\right )}^{\frac {5}{3}}}{5 \, x^{10}} \]
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Result contains complex when optimal does not.
Time = 77.78 (sec) , antiderivative size = 70, normalized size of antiderivative = 4.38 \[ \int \frac {\left (1+x^5\right )^{2/3} \left (6+x^5\right )}{x^{11}} \, dx=- \frac {\Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, \frac {1}{3} \\ \frac {4}{3} \end {matrix}\middle | {\frac {e^{i \pi }}{x^{5}}} \right )}}{5 x^{\frac {5}{3}} \Gamma \left (\frac {4}{3}\right )} - \frac {6 \Gamma \left (\frac {4}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, \frac {4}{3} \\ \frac {7}{3} \end {matrix}\middle | {\frac {e^{i \pi }}{x^{5}}} \right )}}{5 x^{\frac {20}{3}} \Gamma \left (\frac {7}{3}\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 50 vs. \(2 (12) = 24\).
Time = 0.29 (sec) , antiderivative size = 50, normalized size of antiderivative = 3.12 \[ \int \frac {\left (1+x^5\right )^{2/3} \left (6+x^5\right )}{x^{11}} \, dx=\frac {2 \, {\left (x^{5} + 1\right )}^{\frac {5}{3}} + {\left (x^{5} + 1\right )}^{\frac {2}{3}}}{5 \, {\left (2 \, x^{5} - {\left (x^{5} + 1\right )}^{2} + 1\right )}} - \frac {{\left (x^{5} + 1\right )}^{\frac {2}{3}}}{5 \, x^{5}} \]
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Time = 0.27 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \frac {\left (1+x^5\right )^{2/3} \left (6+x^5\right )}{x^{11}} \, dx=-\frac {3 \, {\left (x^{5} + 1\right )}^{\frac {5}{3}}}{5 \, x^{10}} \]
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Time = 5.01 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \frac {\left (1+x^5\right )^{2/3} \left (6+x^5\right )}{x^{11}} \, dx=-\frac {3\,{\left (x^5+1\right )}^{5/3}}{5\,x^{10}} \]
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