Integrand size = 16, antiderivative size = 41 \[ \int \frac {1}{x^4 \left (13+\frac {2}{x}+15 x\right )} \, dx=-\frac {1}{4 x^2}+\frac {13}{4 x}+\frac {139 \log (x)}{8}+\frac {27}{56} \log (2+3 x)-\frac {125}{7} \log (1+5 x) \]
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Time = 0.02 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {1400, 723, 814} \[ \int \frac {1}{x^4 \left (13+\frac {2}{x}+15 x\right )} \, dx=-\frac {1}{4 x^2}+\frac {13}{4 x}+\frac {139 \log (x)}{8}+\frac {27}{56} \log (3 x+2)-\frac {125}{7} \log (5 x+1) \]
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Rule 723
Rule 814
Rule 1400
Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{x^3 \left (2+13 x+15 x^2\right )} \, dx \\ & = -\frac {1}{4 x^2}+\frac {1}{2} \int \frac {-13-15 x}{x^2 \left (2+13 x+15 x^2\right )} \, dx \\ & = -\frac {1}{4 x^2}+\frac {1}{2} \int \left (-\frac {13}{2 x^2}+\frac {139}{4 x}+\frac {81}{28 (2+3 x)}-\frac {1250}{7 (1+5 x)}\right ) \, dx \\ & = -\frac {1}{4 x^2}+\frac {13}{4 x}+\frac {139 \log (x)}{8}+\frac {27}{56} \log (2+3 x)-\frac {125}{7} \log (1+5 x) \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^4 \left (13+\frac {2}{x}+15 x\right )} \, dx=-\frac {1}{4 x^2}+\frac {13}{4 x}+\frac {139 \log (x)}{8}+\frac {27}{56} \log (2+3 x)-\frac {125}{7} \log (1+5 x) \]
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Time = 0.05 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.76
method | result | size |
risch | \(\frac {\frac {13 x}{4}-\frac {1}{4}}{x^{2}}+\frac {139 \ln \left (x \right )}{8}+\frac {27 \ln \left (3 x +2\right )}{56}-\frac {125 \ln \left (1+5 x \right )}{7}\) | \(31\) |
default | \(-\frac {1}{4 x^{2}}+\frac {13}{4 x}+\frac {139 \ln \left (x \right )}{8}+\frac {27 \ln \left (3 x +2\right )}{56}-\frac {125 \ln \left (1+5 x \right )}{7}\) | \(32\) |
norman | \(\frac {-\frac {1}{4} x +\frac {13}{4} x^{2}}{x^{3}}+\frac {139 \ln \left (x \right )}{8}+\frac {27 \ln \left (3 x +2\right )}{56}-\frac {125 \ln \left (1+5 x \right )}{7}\) | \(35\) |
parallelrisch | \(\frac {973 \ln \left (x \right ) x^{2}-1000 \ln \left (x +\frac {1}{5}\right ) x^{2}+27 \ln \left (x +\frac {2}{3}\right ) x^{2}-14+182 x}{56 x^{2}}\) | \(36\) |
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Time = 0.29 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.95 \[ \int \frac {1}{x^4 \left (13+\frac {2}{x}+15 x\right )} \, dx=-\frac {1000 \, x^{2} \log \left (5 \, x + 1\right ) - 27 \, x^{2} \log \left (3 \, x + 2\right ) - 973 \, x^{2} \log \left (x\right ) - 182 \, x + 14}{56 \, x^{2}} \]
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Time = 0.09 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.88 \[ \int \frac {1}{x^4 \left (13+\frac {2}{x}+15 x\right )} \, dx=\frac {139 \log {\left (x \right )}}{8} - \frac {125 \log {\left (x + \frac {1}{5} \right )}}{7} + \frac {27 \log {\left (x + \frac {2}{3} \right )}}{56} + \frac {13 x - 1}{4 x^{2}} \]
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Time = 0.18 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.76 \[ \int \frac {1}{x^4 \left (13+\frac {2}{x}+15 x\right )} \, dx=\frac {13 \, x - 1}{4 \, x^{2}} - \frac {125}{7} \, \log \left (5 \, x + 1\right ) + \frac {27}{56} \, \log \left (3 \, x + 2\right ) + \frac {139}{8} \, \log \left (x\right ) \]
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Time = 0.28 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.83 \[ \int \frac {1}{x^4 \left (13+\frac {2}{x}+15 x\right )} \, dx=\frac {13 \, x - 1}{4 \, x^{2}} - \frac {125}{7} \, \log \left ({\left | 5 \, x + 1 \right |}\right ) + \frac {27}{56} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) + \frac {139}{8} \, \log \left ({\left | x \right |}\right ) \]
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Time = 9.33 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.63 \[ \int \frac {1}{x^4 \left (13+\frac {2}{x}+15 x\right )} \, dx=\frac {27\,\ln \left (x+\frac {2}{3}\right )}{56}-\frac {125\,\ln \left (x+\frac {1}{5}\right )}{7}+\frac {139\,\ln \left (x\right )}{8}+\frac {\frac {13\,x}{4}-\frac {1}{4}}{x^2} \]
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