Optimal. Leaf size=35 \[ \frac{\tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{\sqrt{3}} \]
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Rubi [A] time = 0.0081287, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {621, 206} \[ \frac{\tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{\sqrt{3}} \]
Antiderivative was successfully verified.
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Rule 621
Rule 206
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{2+5 x+3 x^2}} \, dx &=2 \operatorname{Subst}\left (\int \frac{1}{12-x^2} \, dx,x,\frac{5+6 x}{\sqrt{2+5 x+3 x^2}}\right )\\ &=\frac{\tanh ^{-1}\left (\frac{5+6 x}{2 \sqrt{3} \sqrt{2+5 x+3 x^2}}\right )}{\sqrt{3}}\\ \end{align*}
Mathematica [A] time = 0.0055392, size = 28, normalized size = 0.8 \[ \frac{\log \left (2 \sqrt{9 x^2+15 x+6}+6 x+5\right )}{\sqrt{3}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.047, size = 30, normalized size = 0.9 \begin{align*}{\frac{\sqrt{3}}{3}\ln \left ({\frac{\sqrt{3}}{3} \left ({\frac{5}{2}}+3\,x \right ) }+\sqrt{3\,{x}^{2}+5\,x+2} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.57506, size = 38, normalized size = 1.09 \begin{align*} \frac{1}{3} \, \sqrt{3} \log \left (2 \, \sqrt{3} \sqrt{3 \, x^{2} + 5 \, x + 2} + 6 \, x + 5\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.07287, size = 111, normalized size = 3.17 \begin{align*} \frac{1}{6} \, \sqrt{3} \log \left (4 \, \sqrt{3} \sqrt{3 \, x^{2} + 5 \, x + 2}{\left (6 \, x + 5\right )} + 72 \, x^{2} + 120 \, x + 49\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{3 x^{2} + 5 x + 2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.28827, size = 46, normalized size = 1.31 \begin{align*} -\frac{1}{3} \, \sqrt{3} \log \left ({\left | -2 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )} - 5 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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