Integrand size = 115, antiderivative size = 30 \[ \int \frac {-250 x^2-25 x^3-125 x^5+\left (-29600 x-8380 x^2-745 x^3+7380 x^4+1725 x^5+100 x^6\right ) \log \left (\frac {37+4 x}{8+x}\right )}{\left (29600+12820 x+2076 x^2+29749 x^3+9864 x^4+1090 x^5+7440 x^6+1725 x^7+100 x^8\right ) \log ^2\left (\frac {37+4 x}{8+x}\right )} \, dx=\frac {x}{\left (-\frac {1}{5}-\frac {2}{x}-x^2\right ) \log \left (4+\frac {5}{8+x}\right )} \]
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\[ \int \frac {-250 x^2-25 x^3-125 x^5+\left (-29600 x-8380 x^2-745 x^3+7380 x^4+1725 x^5+100 x^6\right ) \log \left (\frac {37+4 x}{8+x}\right )}{\left (29600+12820 x+2076 x^2+29749 x^3+9864 x^4+1090 x^5+7440 x^6+1725 x^7+100 x^8\right ) \log ^2\left (\frac {37+4 x}{8+x}\right )} \, dx=\int \frac {-250 x^2-25 x^3-125 x^5+\left (-29600 x-8380 x^2-745 x^3+7380 x^4+1725 x^5+100 x^6\right ) \log \left (\frac {37+4 x}{8+x}\right )}{\left (29600+12820 x+2076 x^2+29749 x^3+9864 x^4+1090 x^5+7440 x^6+1725 x^7+100 x^8\right ) \log ^2\left (\frac {37+4 x}{8+x}\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {5 x \left (-5 x \left (10+x+5 x^3\right )+\left (-5920-1676 x-149 x^2+1476 x^3+345 x^4+20 x^5\right ) \log \left (\frac {37+4 x}{8+x}\right )\right )}{\left (296+69 x+4 x^2\right ) \left (10+x+5 x^3\right )^2 \log ^2\left (\frac {37+4 x}{8+x}\right )} \, dx \\ & = 5 \int \frac {x \left (-5 x \left (10+x+5 x^3\right )+\left (-5920-1676 x-149 x^2+1476 x^3+345 x^4+20 x^5\right ) \log \left (\frac {37+4 x}{8+x}\right )\right )}{\left (296+69 x+4 x^2\right ) \left (10+x+5 x^3\right )^2 \log ^2\left (\frac {37+4 x}{8+x}\right )} \, dx \\ & = 5 \int \left (-\frac {5 x^2}{(8+x) (37+4 x) \left (10+x+5 x^3\right ) \log ^2\left (\frac {37+4 x}{8+x}\right )}+\frac {x \left (-20-x+5 x^3\right )}{\left (10+x+5 x^3\right )^2 \log \left (\frac {37+4 x}{8+x}\right )}\right ) \, dx \\ & = 5 \int \frac {x \left (-20-x+5 x^3\right )}{\left (10+x+5 x^3\right )^2 \log \left (\frac {37+4 x}{8+x}\right )} \, dx-25 \int \frac {x^2}{(8+x) (37+4 x) \left (10+x+5 x^3\right ) \log ^2\left (\frac {37+4 x}{8+x}\right )} \, dx \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.97 \[ \int \frac {-250 x^2-25 x^3-125 x^5+\left (-29600 x-8380 x^2-745 x^3+7380 x^4+1725 x^5+100 x^6\right ) \log \left (\frac {37+4 x}{8+x}\right )}{\left (29600+12820 x+2076 x^2+29749 x^3+9864 x^4+1090 x^5+7440 x^6+1725 x^7+100 x^8\right ) \log ^2\left (\frac {37+4 x}{8+x}\right )} \, dx=-\frac {5 x^2}{\left (10+x+5 x^3\right ) \log \left (\frac {37+4 x}{8+x}\right )} \]
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Time = 13.32 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00
method | result | size |
risch | \(-\frac {5 x^{2}}{\left (5 x^{3}+x +10\right ) \ln \left (\frac {4 x +37}{x +8}\right )}\) | \(30\) |
parallelrisch | \(-\frac {5 x^{2}}{\left (5 x^{3}+x +10\right ) \ln \left (\frac {4 x +37}{x +8}\right )}\) | \(30\) |
derivativedivides | \(\frac {160}{1279 \ln \left (4+\frac {5}{x +8}\right )}+\frac {\frac {257200 \left (4+\frac {5}{x +8}\right )^{2}}{1279}-\frac {4015200}{1279}-\frac {11888625}{1279 \left (x +8\right )}}{\left (2558 \left (4+\frac {5}{x +8}\right )^{3}-35501 \left (4+\frac {5}{x +8}\right )^{2}+403679+\frac {821120}{x +8}\right ) \ln \left (4+\frac {5}{x +8}\right )}\) | \(89\) |
default | \(\frac {160}{1279 \ln \left (4+\frac {5}{x +8}\right )}+\frac {\frac {257200 \left (4+\frac {5}{x +8}\right )^{2}}{1279}-\frac {4015200}{1279}-\frac {11888625}{1279 \left (x +8\right )}}{\left (2558 \left (4+\frac {5}{x +8}\right )^{3}-35501 \left (4+\frac {5}{x +8}\right )^{2}+403679+\frac {821120}{x +8}\right ) \ln \left (4+\frac {5}{x +8}\right )}\) | \(89\) |
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Time = 0.26 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.97 \[ \int \frac {-250 x^2-25 x^3-125 x^5+\left (-29600 x-8380 x^2-745 x^3+7380 x^4+1725 x^5+100 x^6\right ) \log \left (\frac {37+4 x}{8+x}\right )}{\left (29600+12820 x+2076 x^2+29749 x^3+9864 x^4+1090 x^5+7440 x^6+1725 x^7+100 x^8\right ) \log ^2\left (\frac {37+4 x}{8+x}\right )} \, dx=-\frac {5 \, x^{2}}{{\left (5 \, x^{3} + x + 10\right )} \log \left (\frac {4 \, x + 37}{x + 8}\right )} \]
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Time = 0.08 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.80 \[ \int \frac {-250 x^2-25 x^3-125 x^5+\left (-29600 x-8380 x^2-745 x^3+7380 x^4+1725 x^5+100 x^6\right ) \log \left (\frac {37+4 x}{8+x}\right )}{\left (29600+12820 x+2076 x^2+29749 x^3+9864 x^4+1090 x^5+7440 x^6+1725 x^7+100 x^8\right ) \log ^2\left (\frac {37+4 x}{8+x}\right )} \, dx=- \frac {5 x^{2}}{\left (5 x^{3} + x + 10\right ) \log {\left (\frac {4 x + 37}{x + 8} \right )}} \]
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Time = 0.27 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.23 \[ \int \frac {-250 x^2-25 x^3-125 x^5+\left (-29600 x-8380 x^2-745 x^3+7380 x^4+1725 x^5+100 x^6\right ) \log \left (\frac {37+4 x}{8+x}\right )}{\left (29600+12820 x+2076 x^2+29749 x^3+9864 x^4+1090 x^5+7440 x^6+1725 x^7+100 x^8\right ) \log ^2\left (\frac {37+4 x}{8+x}\right )} \, dx=-\frac {5 \, x^{2}}{{\left (5 \, x^{3} + x + 10\right )} \log \left (4 \, x + 37\right ) - {\left (5 \, x^{3} + x + 10\right )} \log \left (x + 8\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 137 vs. \(2 (29) = 58\).
Time = 0.38 (sec) , antiderivative size = 137, normalized size of antiderivative = 4.57 \[ \int \frac {-250 x^2-25 x^3-125 x^5+\left (-29600 x-8380 x^2-745 x^3+7380 x^4+1725 x^5+100 x^6\right ) \log \left (\frac {37+4 x}{8+x}\right )}{\left (29600+12820 x+2076 x^2+29749 x^3+9864 x^4+1090 x^5+7440 x^6+1725 x^7+100 x^8\right ) \log ^2\left (\frac {37+4 x}{8+x}\right )} \, dx=\frac {5 \, {\left (\frac {64 \, {\left (4 \, x + 37\right )}^{3}}{{\left (x + 8\right )}^{3}} - \frac {848 \, {\left (4 \, x + 37\right )}^{2}}{{\left (x + 8\right )}^{2}} + \frac {3737 \, {\left (4 \, x + 37\right )}}{x + 8} - 5476\right )}}{\frac {2558 \, {\left (4 \, x + 37\right )}^{3} \log \left (\frac {4 \, x + 37}{x + 8}\right )}{{\left (x + 8\right )}^{3}} - \frac {35501 \, {\left (4 \, x + 37\right )}^{2} \log \left (\frac {4 \, x + 37}{x + 8}\right )}{{\left (x + 8\right )}^{2}} + \frac {164224 \, {\left (4 \, x + 37\right )} \log \left (\frac {4 \, x + 37}{x + 8}\right )}{x + 8} - 253217 \, \log \left (\frac {4 \, x + 37}{x + 8}\right )} \]
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Time = 14.25 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.70 \[ \int \frac {-250 x^2-25 x^3-125 x^5+\left (-29600 x-8380 x^2-745 x^3+7380 x^4+1725 x^5+100 x^6\right ) \log \left (\frac {37+4 x}{8+x}\right )}{\left (29600+12820 x+2076 x^2+29749 x^3+9864 x^4+1090 x^5+7440 x^6+1725 x^7+100 x^8\right ) \log ^2\left (\frac {37+4 x}{8+x}\right )} \, dx=\frac {4\,x^3+\frac {4\,x}{5}+8}{5\,x^3+x+10}-\frac {5\,x^2}{\ln \left (\frac {4\,x+37}{x+8}\right )\,\left (5\,x^3+x+10\right )} \]
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