Integrand size = 45, antiderivative size = 31 \[ \int \frac {1375-3150 x+550 x^2-660 x^3+55 x^4-6 x^5-6000 x \log (x)}{1875+750 x^2+75 x^4} \, dx=-\left (1-\frac {x}{5}\right )^2+\frac {x}{3}-\frac {8 x \log (x)}{\frac {5}{x}+x} \]
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Time = 0.14 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.71, number of steps used = 19, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.244, Rules used = {28, 6874, 205, 209, 267, 294, 272, 45, 327, 2373, 266} \[ \int \frac {1375-3150 x+550 x^2-660 x^3+55 x^4-6 x^5-6000 x \log (x)}{1875+750 x^2+75 x^4} \, dx=-\frac {x^2}{25}-\frac {11 x}{6 \left (x^2+5\right )}-\frac {8 x^2 \log (x)}{x^2+5}-\frac {11 x^3}{30 \left (x^2+5\right )}+\frac {11 x}{10} \]
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Rule 28
Rule 45
Rule 205
Rule 209
Rule 266
Rule 267
Rule 272
Rule 294
Rule 327
Rule 2373
Rule 6874
Rubi steps \begin{align*} \text {integral}& = 75 \int \frac {1375-3150 x+550 x^2-660 x^3+55 x^4-6 x^5-6000 x \log (x)}{\left (375+75 x^2\right )^2} \, dx \\ & = 75 \int \left (\frac {11}{45 \left (5+x^2\right )^2}-\frac {14 x}{25 \left (5+x^2\right )^2}+\frac {22 x^2}{225 \left (5+x^2\right )^2}-\frac {44 x^3}{375 \left (5+x^2\right )^2}+\frac {11 x^4}{1125 \left (5+x^2\right )^2}-\frac {2 x^5}{1875 \left (5+x^2\right )^2}-\frac {16 x \log (x)}{15 \left (5+x^2\right )^2}\right ) \, dx \\ & = -\left (\frac {2}{25} \int \frac {x^5}{\left (5+x^2\right )^2} \, dx\right )+\frac {11}{15} \int \frac {x^4}{\left (5+x^2\right )^2} \, dx+\frac {22}{3} \int \frac {x^2}{\left (5+x^2\right )^2} \, dx-\frac {44}{5} \int \frac {x^3}{\left (5+x^2\right )^2} \, dx+\frac {55}{3} \int \frac {1}{\left (5+x^2\right )^2} \, dx-42 \int \frac {x}{\left (5+x^2\right )^2} \, dx-80 \int \frac {x \log (x)}{\left (5+x^2\right )^2} \, dx \\ & = \frac {21}{5+x^2}-\frac {11 x}{6 \left (5+x^2\right )}-\frac {11 x^3}{30 \left (5+x^2\right )}-\frac {8 x^2 \log (x)}{5+x^2}-\frac {1}{25} \text {Subst}\left (\int \frac {x^2}{(5+x)^2} \, dx,x,x^2\right )+\frac {11}{10} \int \frac {x^2}{5+x^2} \, dx+\frac {11}{6} \int \frac {1}{5+x^2} \, dx+\frac {11}{3} \int \frac {1}{5+x^2} \, dx-\frac {22}{5} \text {Subst}\left (\int \frac {x}{(5+x)^2} \, dx,x,x^2\right )+8 \int \frac {x}{5+x^2} \, dx \\ & = \frac {11 x}{10}+\frac {21}{5+x^2}-\frac {11 x}{6 \left (5+x^2\right )}-\frac {11 x^3}{30 \left (5+x^2\right )}+\frac {11 \arctan \left (\frac {x}{\sqrt {5}}\right )}{2 \sqrt {5}}-\frac {8 x^2 \log (x)}{5+x^2}+4 \log \left (5+x^2\right )-\frac {1}{25} \text {Subst}\left (\int \left (1+\frac {25}{(5+x)^2}-\frac {10}{5+x}\right ) \, dx,x,x^2\right )-\frac {22}{5} \text {Subst}\left (\int \left (-\frac {5}{(5+x)^2}+\frac {1}{5+x}\right ) \, dx,x,x^2\right )-\frac {11}{2} \int \frac {1}{5+x^2} \, dx \\ & = \frac {11 x}{10}-\frac {x^2}{25}-\frac {11 x}{6 \left (5+x^2\right )}-\frac {11 x^3}{30 \left (5+x^2\right )}-\frac {8 x^2 \log (x)}{5+x^2} \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.71 \[ \int \frac {1375-3150 x+550 x^2-660 x^3+55 x^4-6 x^5-6000 x \log (x)}{1875+750 x^2+75 x^4} \, dx=\frac {1}{75} x \left (55-3 x-\frac {600 x \log (x)}{5+x^2}\right ) \]
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Time = 0.06 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.77
method | result | size |
default | \(-\frac {x^{2}}{25}+\frac {11 x}{15}-\frac {8 \ln \left (x \right ) x^{2}}{x^{2}+5}\) | \(24\) |
parts | \(-\frac {x^{2}}{25}+\frac {11 x}{15}-\frac {8 \ln \left (x \right ) x^{2}}{x^{2}+5}\) | \(24\) |
risch | \(\frac {40 \ln \left (x \right )}{x^{2}+5}-\frac {x^{2}}{25}+\frac {11 x}{15}-8 \ln \left (x \right )\) | \(25\) |
norman | \(\frac {\frac {11 x}{3}+\frac {11 x^{3}}{15}-\frac {x^{4}}{25}-8 x^{2} \ln \left (x \right )+1}{x^{2}+5}\) | \(31\) |
parallelrisch | \(\frac {-3 x^{4}+55 x^{3}-600 x^{2} \ln \left (x \right )+75+275 x}{75 x^{2}+375}\) | \(32\) |
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Time = 0.26 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.13 \[ \int \frac {1375-3150 x+550 x^2-660 x^3+55 x^4-6 x^5-6000 x \log (x)}{1875+750 x^2+75 x^4} \, dx=-\frac {3 \, x^{4} - 55 \, x^{3} + 600 \, x^{2} \log \left (x\right ) + 15 \, x^{2} - 275 \, x}{75 \, {\left (x^{2} + 5\right )}} \]
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Time = 0.07 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.77 \[ \int \frac {1375-3150 x+550 x^2-660 x^3+55 x^4-6 x^5-6000 x \log (x)}{1875+750 x^2+75 x^4} \, dx=- \frac {x^{2}}{25} + \frac {11 x}{15} - 8 \log {\left (x \right )} + \frac {40 \log {\left (x \right )}}{x^{2} + 5} \]
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Time = 0.31 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.84 \[ \int \frac {1375-3150 x+550 x^2-660 x^3+55 x^4-6 x^5-6000 x \log (x)}{1875+750 x^2+75 x^4} \, dx=-\frac {1}{25} \, x^{2} + \frac {11}{15} \, x + \frac {40 \, \log \left (x\right )}{x^{2} + 5} - 4 \, \log \left (x^{2}\right ) \]
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Time = 0.27 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.77 \[ \int \frac {1375-3150 x+550 x^2-660 x^3+55 x^4-6 x^5-6000 x \log (x)}{1875+750 x^2+75 x^4} \, dx=-\frac {1}{25} \, x^{2} + \frac {11}{15} \, x + \frac {40 \, \log \left (x\right )}{x^{2} + 5} - 8 \, \log \left (x\right ) \]
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Time = 12.01 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.77 \[ \int \frac {1375-3150 x+550 x^2-660 x^3+55 x^4-6 x^5-6000 x \log (x)}{1875+750 x^2+75 x^4} \, dx=\frac {11\,x}{15}-8\,\ln \left (x\right )-\frac {x^2}{25}+\frac {40\,\ln \left (x\right )}{x^2+5} \]
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