Integrand size = 72, antiderivative size = 22 \[ \int \frac {425-275 x-125 x^2-25 x^3+\left (300+600 x+300 x^2\right ) \log (5-x)+\left (-125-225 x-75 x^2+25 x^3\right ) \log ^2(5-x)}{-5-9 x-3 x^2+x^3} \, dx=25 (1+x) \left (-1+\frac {2}{1+x}+\log (5-x)\right )^2 \]
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Leaf count is larger than twice the leaf count of optimal. \(57\) vs. \(2(22)=44\).
Time = 0.13 (sec) , antiderivative size = 57, normalized size of antiderivative = 2.59, number of steps used = 15, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {6874, 46, 78, 90, 2437, 2338, 2436, 2333, 2332} \[ \int \frac {425-275 x-125 x^2-25 x^3+\left (300+600 x+300 x^2\right ) \log (5-x)+\left (-125-225 x-75 x^2+25 x^3\right ) \log ^2(5-x)}{-5-9 x-3 x^2+x^3} \, dx=25 x+\frac {100}{x+1}-25 (5-x) \log ^2(5-x)+150 \log ^2(5-x)+50 (5-x) \log (5-x)-200 \log (5-x) \]
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Rule 46
Rule 78
Rule 90
Rule 2332
Rule 2333
Rule 2338
Rule 2436
Rule 2437
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {425}{(-5+x) (1+x)^2}-\frac {275 x}{(-5+x) (1+x)^2}-\frac {125 x^2}{(-5+x) (1+x)^2}-\frac {25 x^3}{(-5+x) (1+x)^2}+\frac {300 \log (5-x)}{-5+x}+25 \log ^2(5-x)\right ) \, dx \\ & = -\left (25 \int \frac {x^3}{(-5+x) (1+x)^2} \, dx\right )+25 \int \log ^2(5-x) \, dx-125 \int \frac {x^2}{(-5+x) (1+x)^2} \, dx-275 \int \frac {x}{(-5+x) (1+x)^2} \, dx+300 \int \frac {\log (5-x)}{-5+x} \, dx+425 \int \frac {1}{(-5+x) (1+x)^2} \, dx \\ & = -\left (25 \int \left (1+\frac {125}{36 (-5+x)}+\frac {1}{6 (1+x)^2}-\frac {17}{36 (1+x)}\right ) \, dx\right )-25 \text {Subst}\left (\int \log ^2(x) \, dx,x,5-x\right )-125 \int \left (\frac {25}{36 (-5+x)}-\frac {1}{6 (1+x)^2}+\frac {11}{36 (1+x)}\right ) \, dx-275 \int \left (\frac {5}{36 (-5+x)}+\frac {1}{6 (1+x)^2}-\frac {5}{36 (1+x)}\right ) \, dx+300 \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,5-x\right )+425 \int \left (\frac {1}{36 (-5+x)}-\frac {1}{6 (1+x)^2}-\frac {1}{36 (1+x)}\right ) \, dx \\ & = -25 x+\frac {100}{1+x}-200 \log (5-x)+150 \log ^2(5-x)-25 (5-x) \log ^2(5-x)+50 \text {Subst}(\int \log (x) \, dx,x,5-x) \\ & = 25 x+\frac {100}{1+x}-200 \log (5-x)+50 (5-x) \log (5-x)+150 \log ^2(5-x)-25 (5-x) \log ^2(5-x) \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.82 \[ \int \frac {425-275 x-125 x^2-25 x^3+\left (300+600 x+300 x^2\right ) \log (5-x)+\left (-125-225 x-75 x^2+25 x^3\right ) \log ^2(5-x)}{-5-9 x-3 x^2+x^3} \, dx=\frac {25 \left (4+x+x^2-2 \left (-1+x^2\right ) \log (5-x)+(1+x)^2 \log ^2(5-x)\right )}{1+x} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(50\) vs. \(2(22)=44\).
Time = 0.14 (sec) , antiderivative size = 51, normalized size of antiderivative = 2.32
method | result | size |
risch | \(\left (25 x +25\right ) \ln \left (5-x \right )^{2}-50 \ln \left (5-x \right ) x +\frac {50 x \ln \left (-5+x \right )+25 x^{2}+50 \ln \left (-5+x \right )+25 x +100}{1+x}\) | \(51\) |
parts | \(-25 \ln \left (5-x \right )^{2} \left (5-x \right )+50 \left (5-x \right ) \ln \left (5-x \right )-250+25 x +150 \ln \left (5-x \right )^{2}+\frac {100}{1+x}-200 \ln \left (-5+x \right )\) | \(57\) |
derivativedivides | \(-25 \ln \left (5-x \right )^{2} \left (5-x \right )+50 \left (5-x \right ) \ln \left (5-x \right )-125+25 x +150 \ln \left (5-x \right )^{2}-200 \ln \left (5-x \right )-\frac {100}{-1-x}\) | \(61\) |
default | \(-25 \ln \left (5-x \right )^{2} \left (5-x \right )+50 \left (5-x \right ) \ln \left (5-x \right )-125+25 x +150 \ln \left (5-x \right )^{2}-200 \ln \left (5-x \right )-\frac {100}{-1-x}\) | \(61\) |
norman | \(\frac {50 \ln \left (5-x \right )+25 x^{2}+25 \ln \left (5-x \right )^{2}-50 x^{2} \ln \left (5-x \right )+50 \ln \left (5-x \right )^{2} x +25 \ln \left (5-x \right )^{2} x^{2}+75}{1+x}\) | \(67\) |
parallelrisch | \(\frac {25 \ln \left (5-x \right )^{2} x^{2}+325-50 x^{2} \ln \left (5-x \right )+50 \ln \left (5-x \right )^{2} x +25 x^{2}+25 \ln \left (5-x \right )^{2}+250 x +50 \ln \left (5-x \right )}{1+x}\) | \(70\) |
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Time = 0.25 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.95 \[ \int \frac {425-275 x-125 x^2-25 x^3+\left (300+600 x+300 x^2\right ) \log (5-x)+\left (-125-225 x-75 x^2+25 x^3\right ) \log ^2(5-x)}{-5-9 x-3 x^2+x^3} \, dx=\frac {25 \, {\left ({\left (x^{2} + 2 \, x + 1\right )} \log \left (-x + 5\right )^{2} + x^{2} - 2 \, {\left (x^{2} - 1\right )} \log \left (-x + 5\right ) + x + 4\right )}}{x + 1} \]
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Time = 0.10 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.55 \[ \int \frac {425-275 x-125 x^2-25 x^3+\left (300+600 x+300 x^2\right ) \log (5-x)+\left (-125-225 x-75 x^2+25 x^3\right ) \log ^2(5-x)}{-5-9 x-3 x^2+x^3} \, dx=- 50 x \log {\left (5 - x \right )} + 25 x + \left (25 x + 25\right ) \log {\left (5 - x \right )}^{2} + 50 \log {\left (x - 5 \right )} + \frac {100}{x + 1} \]
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Leaf count of result is larger than twice the leaf count of optimal. 67 vs. \(2 (22) = 44\).
Time = 0.22 (sec) , antiderivative size = 67, normalized size of antiderivative = 3.05 \[ \int \frac {425-275 x-125 x^2-25 x^3+\left (300+600 x+300 x^2\right ) \log (5-x)+\left (-125-225 x-75 x^2+25 x^3\right ) \log ^2(5-x)}{-5-9 x-3 x^2+x^3} \, dx=\frac {25 \, {\left (36 \, {\left (x^{2} + 2 \, x + 1\right )} \log \left (-x + 5\right )^{2} + 36 \, x^{2} - {\left (72 \, x^{2} + 17 \, x - 55\right )} \log \left (-x + 5\right ) + 36 \, x + 42\right )}}{36 \, {\left (x + 1\right )}} + \frac {425}{6 \, {\left (x + 1\right )}} + \frac {425}{36} \, \log \left (x - 5\right ) \]
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Time = 0.29 (sec) , antiderivative size = 44, normalized size of antiderivative = 2.00 \[ \int \frac {425-275 x-125 x^2-25 x^3+\left (300+600 x+300 x^2\right ) \log (5-x)+\left (-125-225 x-75 x^2+25 x^3\right ) \log ^2(5-x)}{-5-9 x-3 x^2+x^3} \, dx=25 \, {\left (x + 1\right )} \log \left (-x + 5\right )^{2} - 50 \, {\left (x - 5\right )} \log \left (-x + 5\right ) + 25 \, x + \frac {100}{x + 1} - 200 \, \log \left (-x + 5\right ) - 125 \]
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Time = 0.22 (sec) , antiderivative size = 46, normalized size of antiderivative = 2.09 \[ \int \frac {425-275 x-125 x^2-25 x^3+\left (300+600 x+300 x^2\right ) \log (5-x)+\left (-125-225 x-75 x^2+25 x^3\right ) \log ^2(5-x)}{-5-9 x-3 x^2+x^3} \, dx=50\,\ln \left (x-5\right )+25\,{\ln \left (5-x\right )}^2+\frac {100}{x+1}+x\,\left (25\,{\ln \left (5-x\right )}^2-50\,\ln \left (5-x\right )+25\right ) \]
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