Optimal. Leaf size=40 \[ \frac{1}{3} \sqrt{3 x^2+4 x+2}-\frac{2 \sinh ^{-1}\left (\frac{3 x+2}{\sqrt{2}}\right )}{3 \sqrt{3}} \]
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Rubi [A] time = 0.0177499, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {640, 619, 215} \[ \frac{1}{3} \sqrt{3 x^2+4 x+2}-\frac{2 \sinh ^{-1}\left (\frac{3 x+2}{\sqrt{2}}\right )}{3 \sqrt{3}} \]
Antiderivative was successfully verified.
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Rule 640
Rule 619
Rule 215
Rubi steps
\begin{align*} \int \frac{x}{\sqrt{2+4 x+3 x^2}} \, dx &=\frac{1}{3} \sqrt{2+4 x+3 x^2}-\frac{2}{3} \int \frac{1}{\sqrt{2+4 x+3 x^2}} \, dx\\ &=\frac{1}{3} \sqrt{2+4 x+3 x^2}-\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{8}}} \, dx,x,4+6 x\right )}{3 \sqrt{6}}\\ &=\frac{1}{3} \sqrt{2+4 x+3 x^2}-\frac{2 \sinh ^{-1}\left (\frac{2+3 x}{\sqrt{2}}\right )}{3 \sqrt{3}}\\ \end{align*}
Mathematica [A] time = 0.0123403, size = 40, normalized size = 1. \[ \frac{1}{9} \left (3 \sqrt{3 x^2+4 x+2}-2 \sqrt{3} \sinh ^{-1}\left (\frac{3 x+2}{\sqrt{2}}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 30, normalized size = 0.8 \begin{align*}{\frac{1}{3}\sqrt{3\,{x}^{2}+4\,x+2}}-{\frac{2\,\sqrt{3}}{9}{\it Arcsinh} \left ({\frac{3\,\sqrt{2}}{2} \left ( x+{\frac{2}{3}} \right ) } \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.47642, size = 42, normalized size = 1.05 \begin{align*} -\frac{2}{9} \, \sqrt{3} \operatorname{arsinh}\left (\frac{1}{2} \, \sqrt{2}{\left (3 \, x + 2\right )}\right ) + \frac{1}{3} \, \sqrt{3 \, x^{2} + 4 \, x + 2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.02219, size = 142, normalized size = 3.55 \begin{align*} \frac{1}{9} \, \sqrt{3} \log \left (\sqrt{3} \sqrt{3 \, x^{2} + 4 \, x + 2}{\left (3 \, x + 2\right )} - 9 \, x^{2} - 12 \, x - 5\right ) + \frac{1}{3} \, \sqrt{3 \, x^{2} + 4 \, x + 2} \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{x}{\sqrt{3 x^{2} + 4 x + 2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17975, size = 65, normalized size = 1.62 \begin{align*} \frac{2}{9} \, \sqrt{3} \log \left (-\sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 4 \, x + 2}\right )} - 2\right ) + \frac{1}{3} \, \sqrt{3 \, x^{2} + 4 \, x + 2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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