Optimal. Leaf size=84 \[ \frac{1}{3} \log \left (x^2+x+1\right )+\frac{1}{24} \log \left (2 x^2-x+2\right )-\frac{5}{2 x}-\frac{3 \log (x)}{4}+\frac{5}{12} \sqrt{\frac{5}{3}} \tan ^{-1}\left (\frac{1-4 x}{\sqrt{15}}\right )-\frac{10 \tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{3 \sqrt{3}} \]
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Rubi [A] time = 0.151957, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 6, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.171, Rules used = {2087, 800, 634, 618, 204, 628} \[ \frac{1}{3} \log \left (x^2+x+1\right )+\frac{1}{24} \log \left (2 x^2-x+2\right )-\frac{5}{2 x}-\frac{3 \log (x)}{4}+\frac{5}{12} \sqrt{\frac{5}{3}} \tan ^{-1}\left (\frac{1-4 x}{\sqrt{15}}\right )-\frac{10 \tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{3 \sqrt{3}} \]
Antiderivative was successfully verified.
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Rule 2087
Rule 800
Rule 634
Rule 618
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{5+x+3 x^2+2 x^3}{x^2 \left (2+x+3 x^2+x^3+2 x^4\right )} \, dx &=-\left (\frac{1}{3} \int \frac{-6+4 x}{x^2 \left (4-2 x+4 x^2\right )} \, dx\right )+\frac{1}{3} \int \frac{24+16 x}{x^2 \left (4+4 x+4 x^2\right )} \, dx\\ &=\frac{1}{3} \int \left (\frac{6}{x^2}-\frac{2}{x}+\frac{2 (-2+x)}{1+x+x^2}\right ) \, dx-\frac{1}{3} \int \left (-\frac{3}{2 x^2}+\frac{1}{4 x}+\frac{13-2 x}{4 \left (2-x+2 x^2\right )}\right ) \, dx\\ &=-\frac{5}{2 x}-\frac{3 \log (x)}{4}-\frac{1}{12} \int \frac{13-2 x}{2-x+2 x^2} \, dx+\frac{2}{3} \int \frac{-2+x}{1+x+x^2} \, dx\\ &=-\frac{5}{2 x}-\frac{3 \log (x)}{4}+\frac{1}{24} \int \frac{-1+4 x}{2-x+2 x^2} \, dx+\frac{1}{3} \int \frac{1+2 x}{1+x+x^2} \, dx-\frac{25}{24} \int \frac{1}{2-x+2 x^2} \, dx-\frac{5}{3} \int \frac{1}{1+x+x^2} \, dx\\ &=-\frac{5}{2 x}-\frac{3 \log (x)}{4}+\frac{1}{3} \log \left (1+x+x^2\right )+\frac{1}{24} \log \left (2-x+2 x^2\right )+\frac{25}{12} \operatorname{Subst}\left (\int \frac{1}{-15-x^2} \, dx,x,-1+4 x\right )+\frac{10}{3} \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1+2 x\right )\\ &=-\frac{5}{2 x}+\frac{5}{12} \sqrt{\frac{5}{3}} \tan ^{-1}\left (\frac{1-4 x}{\sqrt{15}}\right )-\frac{10 \tan ^{-1}\left (\frac{1+2 x}{\sqrt{3}}\right )}{3 \sqrt{3}}-\frac{3 \log (x)}{4}+\frac{1}{3} \log \left (1+x+x^2\right )+\frac{1}{24} \log \left (2-x+2 x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0332324, size = 78, normalized size = 0.93 \[ -\frac{-24 x \log \left (x^2+x+1\right )-3 x \log \left (2 x^2-x+2\right )+54 x \log (x)+80 \sqrt{3} x \tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )+10 \sqrt{15} x \tan ^{-1}\left (\frac{4 x-1}{\sqrt{15}}\right )+180}{72 x} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 65, normalized size = 0.8 \begin{align*} -{\frac{5}{2\,x}}-{\frac{3\,\ln \left ( x \right ) }{4}}+{\frac{\ln \left ( 2\,{x}^{2}-x+2 \right ) }{24}}-{\frac{5\,\sqrt{15}}{36}\arctan \left ({\frac{ \left ( -1+4\,x \right ) \sqrt{15}}{15}} \right ) }+{\frac{\ln \left ({x}^{2}+x+1 \right ) }{3}}-{\frac{10\,\sqrt{3}}{9}\arctan \left ({\frac{ \left ( 1+2\,x \right ) \sqrt{3}}{3}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.51129, size = 86, normalized size = 1.02 \begin{align*} -\frac{5}{36} \, \sqrt{15} \arctan \left (\frac{1}{15} \, \sqrt{15}{\left (4 \, x - 1\right )}\right ) - \frac{10}{9} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) - \frac{5}{2 \, x} + \frac{1}{24} \, \log \left (2 \, x^{2} - x + 2\right ) + \frac{1}{3} \, \log \left (x^{2} + x + 1\right ) - \frac{3}{4} \, \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.40958, size = 250, normalized size = 2.98 \begin{align*} -\frac{10 \, \sqrt{5} \sqrt{3} x \arctan \left (\frac{1}{15} \, \sqrt{5} \sqrt{3}{\left (4 \, x - 1\right )}\right ) + 80 \, \sqrt{3} x \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) - 3 \, x \log \left (2 \, x^{2} - x + 2\right ) - 24 \, x \log \left (x^{2} + x + 1\right ) + 54 \, x \log \left (x\right ) + 180}{72 \, x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.275504, size = 87, normalized size = 1.04 \begin{align*} - \frac{3 \log{\left (x \right )}}{4} + \frac{\log{\left (x^{2} - \frac{x}{2} + 1 \right )}}{24} + \frac{\log{\left (x^{2} + x + 1 \right )}}{3} - \frac{5 \sqrt{15} \operatorname{atan}{\left (\frac{4 \sqrt{15} x}{15} - \frac{\sqrt{15}}{15} \right )}}{36} - \frac{10 \sqrt{3} \operatorname{atan}{\left (\frac{2 \sqrt{3} x}{3} + \frac{\sqrt{3}}{3} \right )}}{9} - \frac{5}{2 x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19833, size = 88, normalized size = 1.05 \begin{align*} -\frac{5}{36} \, \sqrt{15} \arctan \left (\frac{1}{15} \, \sqrt{15}{\left (4 \, x - 1\right )}\right ) - \frac{10}{9} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) - \frac{5}{2 \, x} + \frac{1}{24} \, \log \left (2 \, x^{2} - x + 2\right ) + \frac{1}{3} \, \log \left (x^{2} + x + 1\right ) - \frac{3}{4} \, \log \left ({\left | x \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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