Integrand size = 43, antiderivative size = 63 \[ \int \frac {-32+5 x-27 x^2+4 x^3}{-70-299 x-286 x^2+50 x^3-13 x^4+30 x^5} \, dx=\frac {3988 \arctan \left (\frac {1+2 x}{\sqrt {19}}\right )}{13685 \sqrt {19}}-\frac {3146 \log (7-3 x)}{80155}-\frac {334}{323} \log (1+2 x)+\frac {4822 \log (2+5 x)}{4879}+\frac {11049 \log \left (5+x+x^2\right )}{260015} \]
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Time = 0.06 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.116, Rules used = {2099, 648, 632, 210, 642} \[ \int \frac {-32+5 x-27 x^2+4 x^3}{-70-299 x-286 x^2+50 x^3-13 x^4+30 x^5} \, dx=\frac {3988 \arctan \left (\frac {2 x+1}{\sqrt {19}}\right )}{13685 \sqrt {19}}+\frac {11049 \log \left (x^2+x+5\right )}{260015}-\frac {3146 \log (7-3 x)}{80155}-\frac {334}{323} \log (2 x+1)+\frac {4822 \log (5 x+2)}{4879} \]
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Rule 210
Rule 632
Rule 642
Rule 648
Rule 2099
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {668}{323 (1+2 x)}-\frac {9438}{80155 (-7+3 x)}+\frac {24110}{4879 (2+5 x)}+\frac {48935+22098 x}{260015 \left (5+x+x^2\right )}\right ) \, dx \\ & = -\frac {3146 \log (7-3 x)}{80155}-\frac {334}{323} \log (1+2 x)+\frac {4822 \log (2+5 x)}{4879}+\frac {\int \frac {48935+22098 x}{5+x+x^2} \, dx}{260015} \\ & = -\frac {3146 \log (7-3 x)}{80155}-\frac {334}{323} \log (1+2 x)+\frac {4822 \log (2+5 x)}{4879}+\frac {11049 \int \frac {1+2 x}{5+x+x^2} \, dx}{260015}+\frac {1994 \int \frac {1}{5+x+x^2} \, dx}{13685} \\ & = -\frac {3146 \log (7-3 x)}{80155}-\frac {334}{323} \log (1+2 x)+\frac {4822 \log (2+5 x)}{4879}+\frac {11049 \log \left (5+x+x^2\right )}{260015}-\frac {3988 \text {Subst}\left (\int \frac {1}{-19-x^2} \, dx,x,1+2 x\right )}{13685} \\ & = \frac {3988 \arctan \left (\frac {1+2 x}{\sqrt {19}}\right )}{13685 \sqrt {19}}-\frac {3146 \log (7-3 x)}{80155}-\frac {334}{323} \log (1+2 x)+\frac {4822 \log (2+5 x)}{4879}+\frac {11049 \log \left (5+x+x^2\right )}{260015} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.90 \[ \int \frac {-32+5 x-27 x^2+4 x^3}{-70-299 x-286 x^2+50 x^3-13 x^4+30 x^5} \, dx=\frac {163508 \sqrt {19} \arctan \left (\frac {1+2 x}{\sqrt {19}}\right )-418418 \log (7-3 x)-11023670 \log (1+2 x)+10536070 \log (2+5 x)+453009 \log \left (5+x+x^2\right )}{10660615} \]
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Time = 0.07 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.81
method | result | size |
default | \(\frac {4822 \ln \left (5 x +2\right )}{4879}-\frac {3146 \ln \left (3 x -7\right )}{80155}+\frac {11049 \ln \left (x^{2}+x +5\right )}{260015}+\frac {3988 \arctan \left (\frac {\left (1+2 x \right ) \sqrt {19}}{19}\right ) \sqrt {19}}{260015}-\frac {334 \ln \left (1+2 x \right )}{323}\) | \(51\) |
risch | \(-\frac {3146 \ln \left (3 x -7\right )}{80155}+\frac {4822 \ln \left (5 x +2\right )}{4879}-\frac {334 \ln \left (1+2 x \right )}{323}+\frac {11049 \ln \left (15904144 x^{2}+15904144 x +79520720\right )}{260015}+\frac {3988 \sqrt {19}\, \arctan \left (\frac {\left (3988 x +1994\right ) \sqrt {19}}{37886}\right )}{260015}\) | \(55\) |
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Time = 0.26 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.79 \[ \int \frac {-32+5 x-27 x^2+4 x^3}{-70-299 x-286 x^2+50 x^3-13 x^4+30 x^5} \, dx=\frac {3988}{260015} \, \sqrt {19} \arctan \left (\frac {1}{19} \, \sqrt {19} {\left (2 \, x + 1\right )}\right ) + \frac {11049}{260015} \, \log \left (x^{2} + x + 5\right ) + \frac {4822}{4879} \, \log \left (5 \, x + 2\right ) - \frac {3146}{80155} \, \log \left (3 \, x - 7\right ) - \frac {334}{323} \, \log \left (2 \, x + 1\right ) \]
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Time = 0.20 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.08 \[ \int \frac {-32+5 x-27 x^2+4 x^3}{-70-299 x-286 x^2+50 x^3-13 x^4+30 x^5} \, dx=- \frac {3146 \log {\left (x - \frac {7}{3} \right )}}{80155} + \frac {4822 \log {\left (x + \frac {2}{5} \right )}}{4879} - \frac {334 \log {\left (x + \frac {1}{2} \right )}}{323} + \frac {11049 \log {\left (x^{2} + x + 5 \right )}}{260015} + \frac {3988 \sqrt {19} \operatorname {atan}{\left (\frac {2 \sqrt {19} x}{19} + \frac {\sqrt {19}}{19} \right )}}{260015} \]
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Time = 0.29 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.79 \[ \int \frac {-32+5 x-27 x^2+4 x^3}{-70-299 x-286 x^2+50 x^3-13 x^4+30 x^5} \, dx=\frac {3988}{260015} \, \sqrt {19} \arctan \left (\frac {1}{19} \, \sqrt {19} {\left (2 \, x + 1\right )}\right ) + \frac {11049}{260015} \, \log \left (x^{2} + x + 5\right ) + \frac {4822}{4879} \, \log \left (5 \, x + 2\right ) - \frac {3146}{80155} \, \log \left (3 \, x - 7\right ) - \frac {334}{323} \, \log \left (2 \, x + 1\right ) \]
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Time = 0.28 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.84 \[ \int \frac {-32+5 x-27 x^2+4 x^3}{-70-299 x-286 x^2+50 x^3-13 x^4+30 x^5} \, dx=\frac {3988}{260015} \, \sqrt {19} \arctan \left (\frac {1}{19} \, \sqrt {19} {\left (2 \, x + 1\right )}\right ) + \frac {11049}{260015} \, \log \left (x^{2} + x + 5\right ) + \frac {4822}{4879} \, \log \left ({\left | 5 \, x + 2 \right |}\right ) - \frac {3146}{80155} \, \log \left ({\left | 3 \, x - 7 \right |}\right ) - \frac {334}{323} \, \log \left ({\left | 2 \, x + 1 \right |}\right ) \]
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Time = 0.34 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.92 \[ \int \frac {-32+5 x-27 x^2+4 x^3}{-70-299 x-286 x^2+50 x^3-13 x^4+30 x^5} \, dx=\frac {4822\,\ln \left (x+\frac {2}{5}\right )}{4879}-\frac {334\,\ln \left (x+\frac {1}{2}\right )}{323}-\frac {3146\,\ln \left (x-\frac {7}{3}\right )}{80155}-\ln \left (x+\frac {1}{2}-\frac {\sqrt {19}\,1{}\mathrm {i}}{2}\right )\,\left (-\frac {11049}{260015}+\frac {\sqrt {19}\,1994{}\mathrm {i}}{260015}\right )+\ln \left (x+\frac {1}{2}+\frac {\sqrt {19}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {11049}{260015}+\frac {\sqrt {19}\,1994{}\mathrm {i}}{260015}\right ) \]
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