Integrand size = 14, antiderivative size = 54 \[ \int x^3 \log \left (\frac {7+5 x}{x^2}\right ) \, dx=\frac {343 x}{500}-\frac {49 x^2}{200}+\frac {7 x^3}{60}+\frac {x^4}{16}-\frac {2401 \log (7+5 x)}{2500}+\frac {1}{4} x^4 \log \left (\frac {7+5 x}{x^2}\right ) \]
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Time = 0.03 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {2581, 30, 45} \[ \int x^3 \log \left (\frac {7+5 x}{x^2}\right ) \, dx=\frac {x^4}{16}+\frac {7 x^3}{60}-\frac {49 x^2}{200}+\frac {1}{4} x^4 \log \left (\frac {5 x+7}{x^2}\right )+\frac {343 x}{500}-\frac {2401 \log (5 x+7)}{2500} \]
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Rule 30
Rule 45
Rule 2581
Rubi steps \begin{align*} \text {integral}& = \frac {1}{4} x^4 \log \left (\frac {7+5 x}{x^2}\right )+\frac {\int x^3 \, dx}{2}-\frac {5}{4} \int \frac {x^4}{7+5 x} \, dx \\ & = \frac {x^4}{8}+\frac {1}{4} x^4 \log \left (\frac {7+5 x}{x^2}\right )-\frac {5}{4} \int \left (-\frac {343}{625}+\frac {49 x}{125}-\frac {7 x^2}{25}+\frac {x^3}{5}+\frac {2401}{625 (7+5 x)}\right ) \, dx \\ & = \frac {343 x}{500}-\frac {49 x^2}{200}+\frac {7 x^3}{60}+\frac {x^4}{16}-\frac {2401 \log (7+5 x)}{2500}+\frac {1}{4} x^4 \log \left (\frac {7+5 x}{x^2}\right ) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.00 \[ \int x^3 \log \left (\frac {7+5 x}{x^2}\right ) \, dx=\frac {343 x}{500}-\frac {49 x^2}{200}+\frac {7 x^3}{60}+\frac {x^4}{16}-\frac {2401 \log (7+5 x)}{2500}+\frac {1}{4} x^4 \log \left (\frac {7+5 x}{x^2}\right ) \]
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Time = 0.35 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.80
method | result | size |
risch | \(\frac {343 x}{500}-\frac {49 x^{2}}{200}+\frac {7 x^{3}}{60}+\frac {x^{4}}{16}-\frac {2401 \ln \left (7+5 x \right )}{2500}+\frac {x^{4} \ln \left (\frac {7+5 x}{x^{2}}\right )}{4}\) | \(43\) |
parts | \(\frac {343 x}{500}-\frac {49 x^{2}}{200}+\frac {7 x^{3}}{60}+\frac {x^{4}}{16}-\frac {2401 \ln \left (7+5 x \right )}{2500}+\frac {x^{4} \ln \left (\frac {7+5 x}{x^{2}}\right )}{4}\) | \(43\) |
parallelrisch | \(\frac {x^{4} \ln \left (\frac {7+5 x}{x^{2}}\right )}{4}+\frac {x^{4}}{16}+\frac {7 x^{3}}{60}-\frac {2401}{2500}-\frac {49 x^{2}}{200}-\frac {2401 \ln \left (x \right )}{1250}+\frac {343 x}{500}-\frac {2401 \ln \left (\frac {7+5 x}{x^{2}}\right )}{2500}\) | \(52\) |
derivativedivides | \(\frac {x^{4} \ln \left (\frac {\frac {7}{x}+5}{x}\right )}{4}+\frac {x^{4}}{16}+\frac {7 x^{3}}{60}-\frac {49 x^{2}}{200}+\frac {343 x}{500}+\frac {2401 \ln \left (\frac {1}{x}\right )}{2500}-\frac {2401 \ln \left (\frac {7}{x}+5\right )}{2500}\) | \(53\) |
default | \(\frac {x^{4} \ln \left (\frac {\frac {7}{x}+5}{x}\right )}{4}+\frac {x^{4}}{16}+\frac {7 x^{3}}{60}-\frac {49 x^{2}}{200}+\frac {343 x}{500}+\frac {2401 \ln \left (\frac {1}{x}\right )}{2500}-\frac {2401 \ln \left (\frac {7}{x}+5\right )}{2500}\) | \(53\) |
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Time = 0.29 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.78 \[ \int x^3 \log \left (\frac {7+5 x}{x^2}\right ) \, dx=\frac {1}{4} \, x^{4} \log \left (\frac {5 \, x + 7}{x^{2}}\right ) + \frac {1}{16} \, x^{4} + \frac {7}{60} \, x^{3} - \frac {49}{200} \, x^{2} + \frac {343}{500} \, x - \frac {2401}{2500} \, \log \left (5 \, x + 7\right ) \]
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Time = 0.06 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.89 \[ \int x^3 \log \left (\frac {7+5 x}{x^2}\right ) \, dx=\frac {x^{4} \log {\left (\frac {5 x + 7}{x^{2}} \right )}}{4} + \frac {x^{4}}{16} + \frac {7 x^{3}}{60} - \frac {49 x^{2}}{200} + \frac {343 x}{500} - \frac {2401 \log {\left (5 x + 7 \right )}}{2500} \]
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Time = 0.19 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.78 \[ \int x^3 \log \left (\frac {7+5 x}{x^2}\right ) \, dx=\frac {1}{4} \, x^{4} \log \left (\frac {5 \, x + 7}{x^{2}}\right ) + \frac {1}{16} \, x^{4} + \frac {7}{60} \, x^{3} - \frac {49}{200} \, x^{2} + \frac {343}{500} \, x - \frac {2401}{2500} \, \log \left (5 \, x + 7\right ) \]
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Time = 0.31 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.80 \[ \int x^3 \log \left (\frac {7+5 x}{x^2}\right ) \, dx=\frac {1}{4} \, x^{4} \log \left (\frac {5 \, x + 7}{x^{2}}\right ) + \frac {1}{16} \, x^{4} + \frac {7}{60} \, x^{3} - \frac {49}{200} \, x^{2} + \frac {343}{500} \, x - \frac {2401}{2500} \, \log \left ({\left | 5 \, x + 7 \right |}\right ) \]
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Time = 1.79 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.04 \[ \int x^3 \log \left (\frac {7+5 x}{x^2}\right ) \, dx=\frac {343\,x}{500}-\frac {2401\,\ln \left (x\,\left (5\,x+7\right )\right )}{3750}-\frac {2401\,\ln \left (\frac {5\,x+7}{x^2}\right )}{7500}+\frac {x^4\,\ln \left (\frac {5\,x+7}{x^2}\right )}{4}-\frac {49\,x^2}{200}+\frac {7\,x^3}{60}+\frac {x^4}{16} \]
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