Optimal. Leaf size=37 \[ \frac{5}{6} \log \left (3 x^2-4 x+11\right )-\frac{103 \tan ^{-1}\left (\frac{2-3 x}{\sqrt{29}}\right )}{3 \sqrt{29}} \]
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Rubi [A] time = 0.0223993, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {634, 618, 204, 628} \[ \frac{5}{6} \log \left (3 x^2-4 x+11\right )-\frac{103 \tan ^{-1}\left (\frac{2-3 x}{\sqrt{29}}\right )}{3 \sqrt{29}} \]
Antiderivative was successfully verified.
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Rule 634
Rule 618
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{31+5 x}{11-4 x+3 x^2} \, dx &=\frac{5}{6} \int \frac{-4+6 x}{11-4 x+3 x^2} \, dx+\frac{103}{3} \int \frac{1}{11-4 x+3 x^2} \, dx\\ &=\frac{5}{6} \log \left (11-4 x+3 x^2\right )-\frac{206}{3} \operatorname{Subst}\left (\int \frac{1}{-116-x^2} \, dx,x,-4+6 x\right )\\ &=-\frac{103 \tan ^{-1}\left (\frac{2-3 x}{\sqrt{29}}\right )}{3 \sqrt{29}}+\frac{5}{6} \log \left (11-4 x+3 x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0136187, size = 37, normalized size = 1. \[ \frac{5}{6} \log \left (3 x^2-4 x+11\right )+\frac{103 \tan ^{-1}\left (\frac{3 x-2}{\sqrt{29}}\right )}{3 \sqrt{29}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 31, normalized size = 0.8 \begin{align*}{\frac{5\,\ln \left ( 3\,{x}^{2}-4\,x+11 \right ) }{6}}+{\frac{103\,\sqrt{29}}{87}\arctan \left ({\frac{ \left ( 6\,x-4 \right ) \sqrt{29}}{58}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.65553, size = 41, normalized size = 1.11 \begin{align*} \frac{103}{87} \, \sqrt{29} \arctan \left (\frac{1}{29} \, \sqrt{29}{\left (3 \, x - 2\right )}\right ) + \frac{5}{6} \, \log \left (3 \, x^{2} - 4 \, x + 11\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.16902, size = 104, normalized size = 2.81 \begin{align*} \frac{103}{87} \, \sqrt{29} \arctan \left (\frac{1}{29} \, \sqrt{29}{\left (3 \, x - 2\right )}\right ) + \frac{5}{6} \, \log \left (3 \, x^{2} - 4 \, x + 11\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.111909, size = 44, normalized size = 1.19 \begin{align*} \frac{5 \log{\left (x^{2} - \frac{4 x}{3} + \frac{11}{3} \right )}}{6} + \frac{103 \sqrt{29} \operatorname{atan}{\left (\frac{3 \sqrt{29} x}{29} - \frac{2 \sqrt{29}}{29} \right )}}{87} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13874, size = 41, normalized size = 1.11 \begin{align*} \frac{103}{87} \, \sqrt{29} \arctan \left (\frac{1}{29} \, \sqrt{29}{\left (3 \, x - 2\right )}\right ) + \frac{5}{6} \, \log \left (3 \, x^{2} - 4 \, x + 11\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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