Optimal. Leaf size=36 \[ -\frac{\tanh ^{-1}\left (\frac{5 x+4}{2 \sqrt{2} \sqrt{3 x^2+5 x+2}}\right )}{\sqrt{2}} \]
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Rubi [A] time = 0.0110107, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {724, 206} \[ -\frac{\tanh ^{-1}\left (\frac{5 x+4}{2 \sqrt{2} \sqrt{3 x^2+5 x+2}}\right )}{\sqrt{2}} \]
Antiderivative was successfully verified.
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Rule 724
Rule 206
Rubi steps
\begin{align*} \int \frac{1}{x \sqrt{2+5 x+3 x^2}} \, dx &=-\left (2 \operatorname{Subst}\left (\int \frac{1}{8-x^2} \, dx,x,\frac{4+5 x}{\sqrt{2+5 x+3 x^2}}\right )\right )\\ &=-\frac{\tanh ^{-1}\left (\frac{4+5 x}{2 \sqrt{2} \sqrt{2+5 x+3 x^2}}\right )}{\sqrt{2}}\\ \end{align*}
Mathematica [A] time = 0.0102785, size = 31, normalized size = 0.86 \[ -\frac{\tanh ^{-1}\left (\frac{5 x+4}{2 \sqrt{6 x^2+10 x+4}}\right )}{\sqrt{2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.041, size = 29, normalized size = 0.8 \begin{align*} -{\frac{\sqrt{2}}{2}{\it Artanh} \left ({\frac{ \left ( 4+5\,x \right ) \sqrt{2}}{4}{\frac{1}{\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 = 1.52742, size = 47, normalized size = 1.31 \begin{align*} -\frac{1}{2} \, \sqrt{2} \log \left (\frac{2 \, \sqrt{2} \sqrt{3 \, x^{2} + 5 \, x + 2}}{{\left | x \right |}} + \frac{4}{{\left | x \right |}} + 5\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.16917, size = 119, normalized size = 3.31 \begin{align*} \frac{1}{4} \, \sqrt{2} \log \left (-\frac{4 \, \sqrt{2} \sqrt{3 \, x^{2} + 5 \, x + 2}{\left (5 \, x + 4\right )} - 49 \, x^{2} - 80 \, x - 32}{x^{2}}\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}{x \sqrt{\left (x + 1\right ) \left (3 x + 2\right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.13422, size = 82, normalized size = 2.28 \begin{align*} -\frac{1}{2} \, \sqrt{2} \log \left ({\left | -\sqrt{3} x + \sqrt{2} + \sqrt{3 \, x^{2} + 5 \, x + 2} \right |}\right ) + \frac{1}{2} \, \sqrt{2} \log \left ({\left | -\sqrt{3} x - \sqrt{2} + \sqrt{3 \, x^{2} + 5 \, x + 2} \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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