Integrand size = 96, antiderivative size = 26 \[ \int \frac {2+2 x+5 x^3-10 x^4+15 x^5-20 x^6+\left (10 x^2-20 x^3+30 x^4-40 x^5\right ) \log (x)+\left (5 x-10 x^2+15 x^3-20 x^4\right ) \log ^2(x)}{5 x^3+10 x^2 \log (x)+5 x \log ^2(x)} \, dx=-3+x-x^2+x^3-x^4-\frac {2}{5 (x+\log (x))} \]
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Time = 0.31 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.96, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {6873, 12, 6874, 6818} \[ \int \frac {2+2 x+5 x^3-10 x^4+15 x^5-20 x^6+\left (10 x^2-20 x^3+30 x^4-40 x^5\right ) \log (x)+\left (5 x-10 x^2+15 x^3-20 x^4\right ) \log ^2(x)}{5 x^3+10 x^2 \log (x)+5 x \log ^2(x)} \, dx=-x^4+x^3-x^2+x-\frac {2}{5 (x+\log (x))} \]
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Rule 12
Rule 6818
Rule 6873
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \frac {2+2 x+5 x^3-10 x^4+15 x^5-20 x^6+\left (10 x^2-20 x^3+30 x^4-40 x^5\right ) \log (x)+\left (5 x-10 x^2+15 x^3-20 x^4\right ) \log ^2(x)}{5 x (x+\log (x))^2} \, dx \\ & = \frac {1}{5} \int \frac {2+2 x+5 x^3-10 x^4+15 x^5-20 x^6+\left (10 x^2-20 x^3+30 x^4-40 x^5\right ) \log (x)+\left (5 x-10 x^2+15 x^3-20 x^4\right ) \log ^2(x)}{x (x+\log (x))^2} \, dx \\ & = \frac {1}{5} \int \left (-5 \left (-1+2 x-3 x^2+4 x^3\right )+\frac {2 (1+x)}{x (x+\log (x))^2}\right ) \, dx \\ & = \frac {2}{5} \int \frac {1+x}{x (x+\log (x))^2} \, dx-\int \left (-1+2 x-3 x^2+4 x^3\right ) \, dx \\ & = x-x^2+x^3-x^4-\frac {2}{5 (x+\log (x))} \\ \end{align*}
Time = 0.27 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.19 \[ \int \frac {2+2 x+5 x^3-10 x^4+15 x^5-20 x^6+\left (10 x^2-20 x^3+30 x^4-40 x^5\right ) \log (x)+\left (5 x-10 x^2+15 x^3-20 x^4\right ) \log ^2(x)}{5 x^3+10 x^2 \log (x)+5 x \log ^2(x)} \, dx=\frac {1}{5} \left (5 x-5 x^2+5 x^3-5 x^4-\frac {2}{x+\log (x)}\right ) \]
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Time = 1.05 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92
method | result | size |
risch | \(-x^{4}+x^{3}-x^{2}+x -\frac {2}{5 \left (x +\ln \left (x \right )\right )}\) | \(24\) |
norman | \(\frac {-\frac {2}{5}+x^{2}+x^{4}+x^{3} \ln \left (x \right )+x \ln \left (x \right )-x^{3}-x^{5}-x^{2} \ln \left (x \right )-x^{4} \ln \left (x \right )}{x +\ln \left (x \right )}\) | \(50\) |
default | \(-\frac {2-5 x^{2}+5 x^{3}-5 x^{4}+5 x^{5}-5 x \ln \left (x \right )+5 x^{2} \ln \left (x \right )-5 x^{3} \ln \left (x \right )+5 x^{4} \ln \left (x \right )}{5 \left (x +\ln \left (x \right )\right )}\) | \(57\) |
parallelrisch | \(\frac {-5 x^{5}-5 x^{4} \ln \left (x \right )+5 x^{4}+5 x^{3} \ln \left (x \right )-5 x^{3}-5 x^{2} \ln \left (x \right )+5 x^{2}+5 x \ln \left (x \right )-2}{5 x +5 \ln \left (x \right )}\) | \(57\) |
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Leaf count of result is larger than twice the leaf count of optimal. 49 vs. \(2 (24) = 48\).
Time = 0.27 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.88 \[ \int \frac {2+2 x+5 x^3-10 x^4+15 x^5-20 x^6+\left (10 x^2-20 x^3+30 x^4-40 x^5\right ) \log (x)+\left (5 x-10 x^2+15 x^3-20 x^4\right ) \log ^2(x)}{5 x^3+10 x^2 \log (x)+5 x \log ^2(x)} \, dx=-\frac {5 \, x^{5} - 5 \, x^{4} + 5 \, x^{3} - 5 \, x^{2} + 5 \, {\left (x^{4} - x^{3} + x^{2} - x\right )} \log \left (x\right ) + 2}{5 \, {\left (x + \log \left (x\right )\right )}} \]
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Time = 0.05 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.77 \[ \int \frac {2+2 x+5 x^3-10 x^4+15 x^5-20 x^6+\left (10 x^2-20 x^3+30 x^4-40 x^5\right ) \log (x)+\left (5 x-10 x^2+15 x^3-20 x^4\right ) \log ^2(x)}{5 x^3+10 x^2 \log (x)+5 x \log ^2(x)} \, dx=- x^{4} + x^{3} - x^{2} + x - \frac {2}{5 x + 5 \log {\left (x \right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 49 vs. \(2 (24) = 48\).
Time = 0.23 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.88 \[ \int \frac {2+2 x+5 x^3-10 x^4+15 x^5-20 x^6+\left (10 x^2-20 x^3+30 x^4-40 x^5\right ) \log (x)+\left (5 x-10 x^2+15 x^3-20 x^4\right ) \log ^2(x)}{5 x^3+10 x^2 \log (x)+5 x \log ^2(x)} \, dx=-\frac {5 \, x^{5} - 5 \, x^{4} + 5 \, x^{3} - 5 \, x^{2} + 5 \, {\left (x^{4} - x^{3} + x^{2} - x\right )} \log \left (x\right ) + 2}{5 \, {\left (x + \log \left (x\right )\right )}} \]
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none
Time = 0.28 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.88 \[ \int \frac {2+2 x+5 x^3-10 x^4+15 x^5-20 x^6+\left (10 x^2-20 x^3+30 x^4-40 x^5\right ) \log (x)+\left (5 x-10 x^2+15 x^3-20 x^4\right ) \log ^2(x)}{5 x^3+10 x^2 \log (x)+5 x \log ^2(x)} \, dx=-x^{4} + x^{3} - x^{2} + x - \frac {2}{5 \, {\left (x + \log \left (x\right )\right )}} \]
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Time = 15.83 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.04 \[ \int \frac {2+2 x+5 x^3-10 x^4+15 x^5-20 x^6+\left (10 x^2-20 x^3+30 x^4-40 x^5\right ) \log (x)+\left (5 x-10 x^2+15 x^3-20 x^4\right ) \log ^2(x)}{5 x^3+10 x^2 \log (x)+5 x \log ^2(x)} \, dx=x-\frac {2}{5\,\left (x+\ln \left (x\right )\right )}-x^2+x^3-x^4 \]
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