Optimal. Leaf size=72 \[ \frac{1}{3} \log \left (x^2+x+1\right )+\frac{1}{6} \log \left (2 x^2-x+2\right )+x-\frac{1}{3} \sqrt{\frac{5}{3}} \tan ^{-1}\left (\frac{1-4 x}{\sqrt{15}}\right )-\frac{10 \tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{3 \sqrt{3}} \]
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Rubi [A] time = 0.0926255, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 5, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.152, Rules used = {2075, 634, 618, 204, 628} \[ \frac{1}{3} \log \left (x^2+x+1\right )+\frac{1}{6} \log \left (2 x^2-x+2\right )+x-\frac{1}{3} \sqrt{\frac{5}{3}} \tan ^{-1}\left (\frac{1-4 x}{\sqrt{15}}\right )-\frac{10 \tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{3 \sqrt{3}} \]
Antiderivative was successfully verified.
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Rule 2075
Rule 634
Rule 618
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{x \left (5+x+3 x^2+2 x^3\right )}{2+x+3 x^2+x^3+2 x^4} \, dx &=\int \left (1+\frac{2 (-2+x)}{3 \left (1+x+x^2\right )}+\frac{2 (1+x)}{3 \left (2-x+2 x^2\right )}\right ) \, dx\\ &=x+\frac{2}{3} \int \frac{-2+x}{1+x+x^2} \, dx+\frac{2}{3} \int \frac{1+x}{2-x+2 x^2} \, dx\\ &=x+\frac{1}{6} \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{5}{6} \int \frac{1}{2-x+2 x^2} \, dx-\frac{5}{3} \int \frac{1}{1+x+x^2} \, dx\\ &=x+\frac{1}{3} \log \left (1+x+x^2\right )+\frac{1}{6} \log \left (2-x+2 x^2\right )-\frac{5}{3} \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 )\\ &=x-\frac{1}{3} \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{1}{3} \log \left (1+x+x^2\right )+\frac{1}{6} \log \left (2-x+2 x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0217062, size = 69, normalized size = 0.96 \[ \frac{1}{18} \left (3 \left (2 \log \left (x^2+x+1\right )+\log \left (2 x^2-x+2\right )+6 x\right )-20 \sqrt{3} \tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )+2 \sqrt{15} \tan ^{-1}\left (\frac{4 x-1}{\sqrt{15}}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 57, normalized size = 0.8 \begin{align*} x+{\frac{\ln \left ( 2\,{x}^{2}-x+2 \right ) }{6}}+{\frac{\sqrt{15}}{9}\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 = 1.62814, size = 76, normalized size = 1.06 \begin{align*} \frac{1}{9} \, \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 ) + x + \frac{1}{6} \, \log \left (2 \, x^{2} - x + 2\right ) + \frac{1}{3} \, \log \left (x^{2} + x + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.38692, size = 212, normalized size = 2.94 \begin{align*} \frac{1}{9} \, \sqrt{5} \sqrt{3} \arctan \left (\frac{1}{15} \, \sqrt{5} \sqrt{3}{\left (4 \, x - 1\right )}\right ) - \frac{10}{9} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) + x + \frac{1}{6} \, \log \left (2 \, x^{2} - x + 2\right ) + \frac{1}{3} \, \log \left (x^{2} + x + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.19997, size = 75, normalized size = 1.04 \begin{align*} x + \frac{\log{\left (x^{2} - \frac{x}{2} + 1 \right )}}{6} + \frac{\log{\left (x^{2} + x + 1 \right )}}{3} + \frac{\sqrt{15} \operatorname{atan}{\left (\frac{4 \sqrt{15} x}{15} - \frac{\sqrt{15}}{15} \right )}}{9} - \frac{10 \sqrt{3} \operatorname{atan}{\left (\frac{2 \sqrt{3} x}{3} + \frac{\sqrt{3}}{3} \right )}}{9} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.24324, size = 76, normalized size = 1.06 \begin{align*} \frac{1}{9} \, \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 ) + x + \frac{1}{6} \, \log \left (2 \, x^{2} - x + 2\right ) + \frac{1}{3} \, \log \left (x^{2} + x + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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