Integrand size = 19, antiderivative size = 17 \[ \int \frac {8+5 x^{10}}{2 x-x^{11}} \, dx=4 \log (x)-\frac {9}{10} \log \left (2-x^{10}\right ) \]
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Time = 0.02 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {1607, 457, 78} \[ \int \frac {8+5 x^{10}}{2 x-x^{11}} \, dx=4 \log (x)-\frac {9}{10} \log \left (2-x^{10}\right ) \]
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Rule 78
Rule 457
Rule 1607
Rubi steps \begin{align*} \text {integral}& = \int \frac {8+5 x^{10}}{x \left (2-x^{10}\right )} \, dx \\ & = \frac {1}{10} \text {Subst}\left (\int \frac {8+5 x}{(2-x) x} \, dx,x,x^{10}\right ) \\ & = \frac {1}{10} \text {Subst}\left (\int \left (-\frac {9}{-2+x}+\frac {4}{x}\right ) \, dx,x,x^{10}\right ) \\ & = 4 \log (x)-\frac {9}{10} \log \left (2-x^{10}\right ) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00 \[ \int \frac {8+5 x^{10}}{2 x-x^{11}} \, dx=4 \log (x)-\frac {9}{10} \log \left (2-x^{10}\right ) \]
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Time = 1.87 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.82
method | result | size |
default | \(4 \ln \left (x \right )-\frac {9 \ln \left (x^{10}-2\right )}{10}\) | \(14\) |
norman | \(4 \ln \left (x \right )-\frac {9 \ln \left (x^{10}-2\right )}{10}\) | \(14\) |
risch | \(4 \ln \left (x \right )-\frac {9 \ln \left (x^{10}-2\right )}{10}\) | \(14\) |
parallelrisch | \(4 \ln \left (x \right )-\frac {9 \ln \left (x^{10}-2\right )}{10}\) | \(14\) |
meijerg | \(4 \ln \left (x \right )-\frac {2 \ln \left (2\right )}{5}+\frac {2 i \pi }{5}-\frac {9 \ln \left (1-\frac {x^{10}}{2}\right )}{10}\) | \(24\) |
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Time = 0.25 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.76 \[ \int \frac {8+5 x^{10}}{2 x-x^{11}} \, dx=-\frac {9}{10} \, \log \left (x^{10} - 2\right ) + 4 \, \log \left (x\right ) \]
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Time = 0.06 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.82 \[ \int \frac {8+5 x^{10}}{2 x-x^{11}} \, dx=4 \log {\left (x \right )} - \frac {9 \log {\left (x^{10} - 2 \right )}}{10} \]
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Time = 0.28 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.76 \[ \int \frac {8+5 x^{10}}{2 x-x^{11}} \, dx=-\frac {9}{10} \, \log \left (x^{10} - 2\right ) + 4 \, \log \left (x\right ) \]
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Time = 0.26 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.94 \[ \int \frac {8+5 x^{10}}{2 x-x^{11}} \, dx=\frac {2}{5} \, \log \left (x^{10}\right ) - \frac {9}{10} \, \log \left ({\left | x^{10} - 2 \right |}\right ) \]
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Time = 9.16 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.76 \[ \int \frac {8+5 x^{10}}{2 x-x^{11}} \, dx=4\,\ln \left (x\right )-\frac {9\,\ln \left (x^{10}-2\right )}{10} \]
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