Integrand size = 43, antiderivative size = 13 \[ \int \frac {3920 x-x^2+(-19600+5 x) \log (-3920+x)+\left (3925 x-x^2\right ) \log (x)}{-19600 x+5 x^2} \, dx=\left (-\frac {x}{5}+\log (-3920+x)\right ) \log (x) \]
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Time = 0.16 (sec) , antiderivative size = 26, normalized size of antiderivative = 2.00, number of steps used = 12, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.186, Rules used = {1607, 6820, 2441, 2352, 6874, 2404, 2332, 2353} \[ \int \frac {3920 x-x^2+(-19600+5 x) \log (-3920+x)+\left (3925 x-x^2\right ) \log (x)}{-19600 x+5 x^2} \, dx=\log \left (\frac {x}{3920}\right ) \log (x-3920)+\log (3920) \log (x-3920)-\frac {1}{5} x \log (x) \]
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Rule 1607
Rule 2332
Rule 2352
Rule 2353
Rule 2404
Rule 2441
Rule 6820
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \frac {3920 x-x^2+(-19600+5 x) \log (-3920+x)+\left (3925 x-x^2\right ) \log (x)}{x (-19600+5 x)} \, dx \\ & = \int \left (\frac {\log (-3920+x)}{x}-\frac {-3920+x+(-3925+x) \log (x)}{5 (-3920+x)}\right ) \, dx \\ & = -\left (\frac {1}{5} \int \frac {-3920+x+(-3925+x) \log (x)}{-3920+x} \, dx\right )+\int \frac {\log (-3920+x)}{x} \, dx \\ & = \log (-3920+x) \log \left (\frac {x}{3920}\right )-\frac {1}{5} \int \left (1+\frac {(-3925+x) \log (x)}{-3920+x}\right ) \, dx-\int \frac {\log \left (\frac {x}{3920}\right )}{-3920+x} \, dx \\ & = -\frac {x}{5}+\log (-3920+x) \log \left (\frac {x}{3920}\right )+\operatorname {PolyLog}\left (2,1-\frac {x}{3920}\right )-\frac {1}{5} \int \frac {(-3925+x) \log (x)}{-3920+x} \, dx \\ & = -\frac {x}{5}+\log (-3920+x) \log \left (\frac {x}{3920}\right )+\operatorname {PolyLog}\left (2,1-\frac {x}{3920}\right )-\frac {1}{5} \int \left (\log (x)-\frac {5 \log (x)}{-3920+x}\right ) \, dx \\ & = -\frac {x}{5}+\log (-3920+x) \log \left (\frac {x}{3920}\right )+\operatorname {PolyLog}\left (2,1-\frac {x}{3920}\right )-\frac {1}{5} \int \log (x) \, dx+\int \frac {\log (x)}{-3920+x} \, dx \\ & = \log (3920) \log (-3920+x)+\log (-3920+x) \log \left (\frac {x}{3920}\right )-\frac {1}{5} x \log (x)+\operatorname {PolyLog}\left (2,1-\frac {x}{3920}\right )+\int \frac {\log \left (\frac {x}{3920}\right )}{-3920+x} \, dx \\ & = \log (3920) \log (-3920+x)+\log (-3920+x) \log \left (\frac {x}{3920}\right )-\frac {1}{5} x \log (x) \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 26, normalized size of antiderivative = 2.00 \[ \int \frac {3920 x-x^2+(-19600+5 x) \log (-3920+x)+\left (3925 x-x^2\right ) \log (x)}{-19600 x+5 x^2} \, dx=\log (3920) \log (-3920+x)+\log (-3920+x) \log \left (\frac {x}{3920}\right )-\frac {1}{5} x \log (x) \]
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Time = 0.13 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.08
method | result | size |
norman | \(\ln \left (x \right ) \ln \left (x -3920\right )-\frac {x \ln \left (x \right )}{5}\) | \(14\) |
risch | \(\ln \left (x \right ) \ln \left (x -3920\right )-\frac {x \ln \left (x \right )}{5}\) | \(14\) |
parallelrisch | \(\ln \left (x \right ) \ln \left (x -3920\right )-\frac {x \ln \left (x \right )}{5}\) | \(14\) |
default | \(\ln \left (x -3920\right ) \ln \left (\frac {x}{3920}\right )-\frac {x \ln \left (x \right )}{5}+\left (\ln \left (x \right )-\ln \left (\frac {x}{3920}\right )\right ) \ln \left (-\frac {x}{3920}+1\right )\) | \(32\) |
parts | \(\ln \left (x -3920\right ) \ln \left (\frac {x}{3920}\right )-\frac {x \ln \left (x \right )}{5}+\left (\ln \left (x \right )-\ln \left (\frac {x}{3920}\right )\right ) \ln \left (-\frac {x}{3920}+1\right )\) | \(32\) |
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Time = 0.25 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.92 \[ \int \frac {3920 x-x^2+(-19600+5 x) \log (-3920+x)+\left (3925 x-x^2\right ) \log (x)}{-19600 x+5 x^2} \, dx=-\frac {1}{5} \, {\left (x - 5 \, \log \left (x - 3920\right )\right )} \log \left (x\right ) \]
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Time = 0.11 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.08 \[ \int \frac {3920 x-x^2+(-19600+5 x) \log (-3920+x)+\left (3925 x-x^2\right ) \log (x)}{-19600 x+5 x^2} \, dx=- \frac {x \log {\left (x \right )}}{5} + \log {\left (x \right )} \log {\left (x - 3920 \right )} \]
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Time = 0.21 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00 \[ \int \frac {3920 x-x^2+(-19600+5 x) \log (-3920+x)+\left (3925 x-x^2\right ) \log (x)}{-19600 x+5 x^2} \, dx=-\frac {1}{5} \, x \log \left (x\right ) + \log \left (x - 3920\right ) \log \left (x\right ) \]
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Time = 0.29 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00 \[ \int \frac {3920 x-x^2+(-19600+5 x) \log (-3920+x)+\left (3925 x-x^2\right ) \log (x)}{-19600 x+5 x^2} \, dx=-\frac {1}{5} \, x \log \left (x\right ) + \log \left (x - 3920\right ) \log \left (x\right ) \]
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Time = 8.34 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.92 \[ \int \frac {3920 x-x^2+(-19600+5 x) \log (-3920+x)+\left (3925 x-x^2\right ) \log (x)}{-19600 x+5 x^2} \, dx=-\frac {\ln \left (x\right )\,\left (x-5\,\ln \left (x-3920\right )\right )}{5} \]
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