Integrand size = 73, antiderivative size = 23 \[ \int \frac {-1500+525 x+45 x^2-33 x^3+3 x^4+\left (-375 x+205 x^2-50 x^3+3 x^4\right ) \log (x)}{\left (-1500 x+525 x^2+45 x^3-33 x^4+3 x^5\right ) \log (x)} \, dx=7+\frac {x^2}{6 (5-x)^2}+\log ((4+x) \log (x)) \]
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Time = 0.13 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.30, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.055, Rules used = {6820, 1634, 2339, 29} \[ \int \frac {-1500+525 x+45 x^2-33 x^3+3 x^4+\left (-375 x+205 x^2-50 x^3+3 x^4\right ) \log (x)}{\left (-1500 x+525 x^2+45 x^3-33 x^4+3 x^5\right ) \log (x)} \, dx=-\frac {5}{3 (5-x)}+\frac {25}{6 (5-x)^2}+\log (x+4)+\log (\log (x)) \]
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Rule 29
Rule 1634
Rule 2339
Rule 6820
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {-375+205 x-50 x^2+3 x^3}{3 (-5+x)^3 (4+x)}+\frac {1}{x \log (x)}\right ) \, dx \\ & = \frac {1}{3} \int \frac {-375+205 x-50 x^2+3 x^3}{(-5+x)^3 (4+x)} \, dx+\int \frac {1}{x \log (x)} \, dx \\ & = \frac {1}{3} \int \left (-\frac {25}{(-5+x)^3}-\frac {5}{(-5+x)^2}+\frac {3}{4+x}\right ) \, dx+\text {Subst}\left (\int \frac {1}{x} \, dx,x,\log (x)\right ) \\ & = \frac {25}{6 (5-x)^2}-\frac {5}{3 (5-x)}+\log (4+x)+\log (\log (x)) \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.13 \[ \int \frac {-1500+525 x+45 x^2-33 x^3+3 x^4+\left (-375 x+205 x^2-50 x^3+3 x^4\right ) \log (x)}{\left (-1500 x+525 x^2+45 x^3-33 x^4+3 x^5\right ) \log (x)} \, dx=\frac {25}{6 (-5+x)^2}+\frac {5}{3 (-5+x)}+\log (4+x)+\log (\log (x)) \]
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Time = 0.37 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.87
method | result | size |
norman | \(\frac {\frac {5 x}{3}-\frac {25}{6}}{\left (-5+x \right )^{2}}+\ln \left (\ln \left (x \right )\right )+\ln \left (4+x \right )\) | \(20\) |
default | \(\ln \left (4+x \right )+\frac {25}{6 \left (-5+x \right )^{2}}+\frac {5}{3 \left (-5+x \right )}+\ln \left (\ln \left (x \right )\right )\) | \(23\) |
parts | \(\ln \left (4+x \right )+\frac {25}{6 \left (-5+x \right )^{2}}+\frac {5}{3 \left (-5+x \right )}+\ln \left (\ln \left (x \right )\right )\) | \(23\) |
risch | \(\frac {6 x^{2} \ln \left (4+x \right )-60 \ln \left (4+x \right ) x +150 \ln \left (4+x \right )+10 x -25}{6 x^{2}-60 x +150}+\ln \left (\ln \left (x \right )\right )\) | \(44\) |
parallelrisch | \(\frac {6 x^{2} \ln \left (\ln \left (x \right )\right )+6 x^{2} \ln \left (4+x \right )-25-60 x \ln \left (\ln \left (x \right )\right )-60 \ln \left (4+x \right ) x +150 \ln \left (\ln \left (x \right )\right )+150 \ln \left (4+x \right )+10 x}{6 x^{2}-60 x +150}\) | \(59\) |
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Leaf count of result is larger than twice the leaf count of optimal. 44 vs. \(2 (19) = 38\).
Time = 0.27 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.91 \[ \int \frac {-1500+525 x+45 x^2-33 x^3+3 x^4+\left (-375 x+205 x^2-50 x^3+3 x^4\right ) \log (x)}{\left (-1500 x+525 x^2+45 x^3-33 x^4+3 x^5\right ) \log (x)} \, dx=\frac {6 \, {\left (x^{2} - 10 \, x + 25\right )} \log \left (x + 4\right ) + 6 \, {\left (x^{2} - 10 \, x + 25\right )} \log \left (\log \left (x\right )\right ) + 10 \, x - 25}{6 \, {\left (x^{2} - 10 \, x + 25\right )}} \]
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Time = 0.10 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.04 \[ \int \frac {-1500+525 x+45 x^2-33 x^3+3 x^4+\left (-375 x+205 x^2-50 x^3+3 x^4\right ) \log (x)}{\left (-1500 x+525 x^2+45 x^3-33 x^4+3 x^5\right ) \log (x)} \, dx=\frac {10 x - 25}{6 x^{2} - 60 x + 150} + \log {\left (x + 4 \right )} + \log {\left (\log {\left (x \right )} \right )} \]
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Time = 0.22 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {-1500+525 x+45 x^2-33 x^3+3 x^4+\left (-375 x+205 x^2-50 x^3+3 x^4\right ) \log (x)}{\left (-1500 x+525 x^2+45 x^3-33 x^4+3 x^5\right ) \log (x)} \, dx=\frac {5 \, {\left (2 \, x - 5\right )}}{6 \, {\left (x^{2} - 10 \, x + 25\right )}} + \log \left (x + 4\right ) + \log \left (\log \left (x\right )\right ) \]
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Time = 0.25 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {-1500+525 x+45 x^2-33 x^3+3 x^4+\left (-375 x+205 x^2-50 x^3+3 x^4\right ) \log (x)}{\left (-1500 x+525 x^2+45 x^3-33 x^4+3 x^5\right ) \log (x)} \, dx=\frac {5 \, {\left (2 \, x - 5\right )}}{6 \, {\left (x^{2} - 10 \, x + 25\right )}} + \log \left (x + 4\right ) + \log \left (\log \left (x\right )\right ) \]
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Time = 9.25 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.83 \[ \int \frac {-1500+525 x+45 x^2-33 x^3+3 x^4+\left (-375 x+205 x^2-50 x^3+3 x^4\right ) \log (x)}{\left (-1500 x+525 x^2+45 x^3-33 x^4+3 x^5\right ) \log (x)} \, dx=\ln \left (x+4\right )+\ln \left (\ln \left (x\right )\right )+\frac {\frac {5\,x}{3}-\frac {25}{6}}{{\left (x-5\right )}^2} \]
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