Optimal. Leaf size=63 \[ \frac{11049 \log \left (x^2+x+5\right )}{260015}-\frac{3146 \log (7-3 x)}{80155}-\frac{334}{323} \log (2 x+1)+\frac{4822 \log (5 x+2)}{4879}+\frac{3988 \tan ^{-1}\left (\frac{2 x+1}{\sqrt{19}}\right )}{13685 \sqrt{19}} \]
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Rubi [A] time = 0.0867445, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 43, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.116, Rules used = {2074, 634, 618, 204, 628} \[ \frac{11049 \log \left (x^2+x+5\right )}{260015}-\frac{3146 \log (7-3 x)}{80155}-\frac{334}{323} \log (2 x+1)+\frac{4822 \log (5 x+2)}{4879}+\frac{3988 \tan ^{-1}\left (\frac{2 x+1}{\sqrt{19}}\right )}{13685 \sqrt{19}} \]
Antiderivative was successfully verified.
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Rule 2074
Rule 634
Rule 618
Rule 204
Rule 628
Rubi steps
\begin{align*} \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 &=\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 \operatorname{Subst}\left (\int \frac{1}{-19-x^2} \, dx,x,1+2 x\right )}{13685}\\ &=\frac{3988 \tan ^{-1}\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*}
Mathematica [A] time = 0.0265904, size = 57, normalized size = 0.9 \[ \frac{453009 \log \left (x^2+x+5\right )-418418 \log (7-3 x)-11023670 \log (2 x+1)+10536070 \log (5 x+2)+163508 \sqrt{19} \tan ^{-1}\left (\frac{2 x+1}{\sqrt{19}}\right )}{10660615} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.012, size = 51, normalized size = 0.8 \begin{align*}{\frac{4822\,\ln \left ( 2+5\,x \right ) }{4879}}-{\frac{3146\,\ln \left ( 3\,x-7 \right ) }{80155}}-{\frac{334\,\ln \left ( 1+2\,x \right ) }{323}}+{\frac{11049\,\ln \left ({x}^{2}+x+5 \right ) }{260015}}+{\frac{3988\,\sqrt{19}}{260015}\arctan \left ({\frac{ \left ( 1+2\,x \right ) \sqrt{19}}{19}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.41807, size = 68, normalized size = 1.08 \begin{align*} \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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.90173, size = 216, normalized size = 3.43 \begin{align*} \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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.317007, size = 68, normalized size = 1.08 \begin{align*} - \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.06043, size = 72, normalized size = 1.14 \begin{align*} \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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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