Integrand size = 18, antiderivative size = 16 \[ \int \frac {\left (-6+x^5\right ) \left (-1+x^5\right )^{2/3}}{x^{11}} \, dx=-\frac {3 \left (-1+x^5\right )^{5/3}}{5 x^{10}} \]
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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} \[ \int \frac {\left (-6+x^5\right ) \left (-1+x^5\right )^{2/3}}{x^{11}} \, dx=-\frac {3 \left (x^5-1\right )^{5/3}}{5 x^{10}} \]
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Rule 75
Rule 457
Rubi steps \begin{align*} \text {integral}& = \frac {1}{5} \text {Subst}\left (\int \frac {(-6+x) (-1+x)^{2/3}}{x^3} \, dx,x,x^5\right ) \\ & = -\frac {3 \left (-1+x^5\right )^{5/3}}{5 x^{10}} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-6+x^5\right ) \left (-1+x^5\right )^{2/3}}{x^{11}} \, dx=-\frac {3 \left (-1+x^5\right )^{5/3}}{5 x^{10}} \]
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Time = 1.08 (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 | \(-\frac {3 \left (x^{4}+x^{3}+x^{2}+x +1\right ) \left (x -1\right ) \left (x^{5}-1\right )^{\frac {2}{3}}}{5 x^{10}}\) | \(28\) |
meijerg | \(\frac {2 \sqrt {3}\, \Gamma \left (\frac {2}{3}\right ) \operatorname {signum}\left (x^{5}-1\right )^{\frac {2}{3}} \left (\frac {4 \pi \sqrt {3}\, x^{5} \operatorname {hypergeom}\left (\left [1, 1, \frac {7}{3}\right ], \left [2, 4\right ], x^{5}\right )}{81 \Gamma \left (\frac {2}{3}\right )}+\frac {\left (\frac {3}{2}-\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \left (3\right )}{2}+5 \ln \left (x \right )+i \pi \right ) \pi \sqrt {3}}{9 \Gamma \left (\frac {2}{3}\right )}+\frac {\pi \sqrt {3}}{2 \Gamma \left (\frac {2}{3}\right ) x^{10}}-\frac {2 \pi \sqrt {3}}{3 \Gamma \left (\frac {2}{3}\right ) x^{5}}\right )}{5 \pi {\left (-\operatorname {signum}\left (x^{5}-1\right )\right )}^{\frac {2}{3}}}+\frac {\sqrt {3}\, \Gamma \left (\frac {2}{3}\right ) \operatorname {signum}\left (x^{5}-1\right )^{\frac {2}{3}} \left (-\frac {\pi \sqrt {3}\, x^{5} \operatorname {hypergeom}\left (\left [1, 1, \frac {4}{3}\right ], \left [2, 3\right ], x^{5}\right )}{9 \Gamma \left (\frac {2}{3}\right )}-\frac {2 \left (-\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \left (3\right )}{2}-1+5 \ln \left (x \right )+i \pi \right ) \pi \sqrt {3}}{3 \Gamma \left (\frac {2}{3}\right )}-\frac {\pi \sqrt {3}}{\Gamma \left (\frac {2}{3}\right ) x^{5}}\right )}{15 \pi {\left (-\operatorname {signum}\left (x^{5}-1\right )\right )}^{\frac {2}{3}}}\) | \(207\) |
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Time = 0.24 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \frac {\left (-6+x^5\right ) \left (-1+x^5\right )^{2/3}}{x^{11}} \, dx=-\frac {3 \, {\left (x^{5} - 1\right )}^{\frac {5}{3}}}{5 \, x^{10}} \]
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Result contains complex when optimal does not.
Time = 78.62 (sec) , antiderivative size = 71, normalized size of antiderivative = 4.44 \[ \int \frac {\left (-6+x^5\right ) \left (-1+x^5\right )^{2/3}}{x^{11}} \, dx=- \frac {\Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, \frac {1}{3} \\ \frac {4}{3} \end {matrix}\middle | {\frac {e^{2 i \pi }}{x^{5}}} \right )}}{5 x^{\frac {5}{3}} \Gamma \left (\frac {4}{3}\right )} + \frac {6 \Gamma \left (\frac {4}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, \frac {4}{3} \\ \frac {7}{3} \end {matrix}\middle | {\frac {e^{2 i \pi }}{x^{5}}} \right )}}{5 x^{\frac {20}{3}} \Gamma \left (\frac {7}{3}\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 50 vs. \(2 (12) = 24\).
Time = 0.27 (sec) , antiderivative size = 50, normalized size of antiderivative = 3.12 \[ \int \frac {\left (-6+x^5\right ) \left (-1+x^5\right )^{2/3}}{x^{11}} \, dx=-\frac {2 \, {\left (x^{5} - 1\right )}^{\frac {5}{3}} - {\left (x^{5} - 1\right )}^{\frac {2}{3}}}{5 \, {\left (2 \, x^{5} + {\left (x^{5} - 1\right )}^{2} - 1\right )}} - \frac {{\left (x^{5} - 1\right )}^{\frac {2}{3}}}{5 \, x^{5}} \]
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Time = 0.27 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \frac {\left (-6+x^5\right ) \left (-1+x^5\right )^{2/3}}{x^{11}} \, dx=-\frac {3 \, {\left (x^{5} - 1\right )}^{\frac {5}{3}}}{5 \, x^{10}} \]
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Time = 5.04 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \frac {\left (-6+x^5\right ) \left (-1+x^5\right )^{2/3}}{x^{11}} \, dx=-\frac {3\,{\left (x^5-1\right )}^{5/3}}{5\,x^{10}} \]
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