Integrand size = 39, antiderivative size = 24 \[ \int \frac {-5700-3020 x-400 x^2+(-76-40 x) \log (2 x)}{361 x^2+190 x^3+25 x^4} \, dx=\frac {16}{x}-\frac {4 \log (2 x)}{x (1-5 (4+x))} \]
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Time = 0.18 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.62, number of steps used = 11, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.205, Rules used = {1608, 27, 6874, 78, 2404, 2341, 2351, 31} \[ \int \frac {-5700-3020 x-400 x^2+(-76-40 x) \log (2 x)}{361 x^2+190 x^3+25 x^4} \, dx=\frac {16}{x}-\frac {20 \log (x)}{361}+\frac {4 \log (2 x)}{19 x}+\frac {100 x \log (2 x)}{361 (5 x+19)} \]
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Rule 27
Rule 31
Rule 78
Rule 1608
Rule 2341
Rule 2351
Rule 2404
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \frac {-5700-3020 x-400 x^2+(-76-40 x) \log (2 x)}{x^2 \left (361+190 x+25 x^2\right )} \, dx \\ & = \int \frac {-5700-3020 x-400 x^2+(-76-40 x) \log (2 x)}{x^2 (19+5 x)^2} \, dx \\ & = \int \left (-\frac {20 (15+4 x)}{x^2 (19+5 x)}-\frac {4 (19+10 x) \log (2 x)}{x^2 (19+5 x)^2}\right ) \, dx \\ & = -\left (4 \int \frac {(19+10 x) \log (2 x)}{x^2 (19+5 x)^2} \, dx\right )-20 \int \frac {15+4 x}{x^2 (19+5 x)} \, dx \\ & = -\left (4 \int \left (\frac {\log (2 x)}{19 x^2}-\frac {25 \log (2 x)}{19 (19+5 x)^2}\right ) \, dx\right )-20 \int \left (\frac {15}{19 x^2}+\frac {1}{361 x}-\frac {5}{361 (19+5 x)}\right ) \, dx \\ & = \frac {300}{19 x}-\frac {20 \log (x)}{361}+\frac {20}{361} \log (19+5 x)-\frac {4}{19} \int \frac {\log (2 x)}{x^2} \, dx+\frac {100}{19} \int \frac {\log (2 x)}{(19+5 x)^2} \, dx \\ & = \frac {16}{x}-\frac {20 \log (x)}{361}+\frac {4 \log (2 x)}{19 x}+\frac {100 x \log (2 x)}{361 (19+5 x)}+\frac {20}{361} \log (19+5 x)-\frac {100}{361} \int \frac {1}{19+5 x} \, dx \\ & = \frac {16}{x}-\frac {20 \log (x)}{361}+\frac {4 \log (2 x)}{19 x}+\frac {100 x \log (2 x)}{361 (19+5 x)} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int \frac {-5700-3020 x-400 x^2+(-76-40 x) \log (2 x)}{361 x^2+190 x^3+25 x^4} \, dx=\frac {4 (76+20 x+\log (2 x))}{19 x+5 x^2} \]
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Time = 0.53 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.96
method | result | size |
norman | \(\frac {304+80 x +4 \ln \left (2 x \right )}{x \left (5 x +19\right )}\) | \(23\) |
risch | \(\frac {4 \ln \left (2 x \right )}{x \left (5 x +19\right )}+\frac {16}{x}\) | \(23\) |
parallelrisch | \(\frac {1520+400 x +20 \ln \left (2 x \right )}{5 x \left (5 x +19\right )}\) | \(24\) |
derivativedivides | \(\frac {16}{x}-\frac {20 \ln \left (2 x \right )}{361}+\frac {4 \ln \left (2 x \right )}{19 x}+\frac {200 \ln \left (2 x \right ) x}{361 \left (10 x +38\right )}\) | \(36\) |
default | \(\frac {16}{x}-\frac {20 \ln \left (2 x \right )}{361}+\frac {4 \ln \left (2 x \right )}{19 x}+\frac {200 \ln \left (2 x \right ) x}{361 \left (10 x +38\right )}\) | \(36\) |
parts | \(\frac {20 \ln \left (5 x +19\right )}{361}+\frac {16}{x}-\frac {20 \ln \left (x \right )}{361}-\frac {20 \ln \left (10 x +38\right )}{361}+\frac {200 \ln \left (2 x \right ) x}{361 \left (10 x +38\right )}+\frac {4 \ln \left (2 x \right )}{19 x}\) | \(50\) |
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Time = 0.26 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int \frac {-5700-3020 x-400 x^2+(-76-40 x) \log (2 x)}{361 x^2+190 x^3+25 x^4} \, dx=\frac {4 \, {\left (20 \, x + \log \left (2 \, x\right ) + 76\right )}}{5 \, x^{2} + 19 \, x} \]
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Time = 0.08 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.71 \[ \int \frac {-5700-3020 x-400 x^2+(-76-40 x) \log (2 x)}{361 x^2+190 x^3+25 x^4} \, dx=\frac {4 \log {\left (2 x \right )}}{5 x^{2} + 19 x} + \frac {16}{x} \]
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Leaf count of result is larger than twice the leaf count of optimal. 54 vs. \(2 (22) = 44\).
Time = 0.32 (sec) , antiderivative size = 54, normalized size of antiderivative = 2.25 \[ \int \frac {-5700-3020 x-400 x^2+(-76-40 x) \log (2 x)}{361 x^2+190 x^3+25 x^4} \, dx=\frac {300 \, {\left (10 \, x + 19\right )}}{19 \, {\left (5 \, x^{2} + 19 \, x\right )}} + \frac {4 \, {\left (5 \, x + 19 \, \log \left (2\right ) + 19 \, \log \left (x\right ) + 19\right )}}{19 \, {\left (5 \, x^{2} + 19 \, x\right )}} - \frac {1500}{19 \, {\left (5 \, x + 19\right )}} \]
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Time = 0.28 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.12 \[ \int \frac {-5700-3020 x-400 x^2+(-76-40 x) \log (2 x)}{361 x^2+190 x^3+25 x^4} \, dx=-\frac {4}{19} \, {\left (\frac {5}{5 \, x + 19} - \frac {1}{x}\right )} \log \left (2 \, x\right ) + \frac {16}{x} \]
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Time = 16.28 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.04 \[ \int \frac {-5700-3020 x-400 x^2+(-76-40 x) \log (2 x)}{361 x^2+190 x^3+25 x^4} \, dx=\frac {4\,\left (19\,\ln \left (2\,x\right )-100\,x^2+1444\right )}{19\,x\,\left (5\,x+19\right )} \]
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