Optimal. Leaf size=54 \[ \frac{1}{3} \sqrt{3 x^2+4 x-2}-\frac{2 \tanh ^{-1}\left (\frac{3 x+2}{\sqrt{3} \sqrt{3 x^2+4 x-2}}\right )}{3 \sqrt{3}} \]
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Rubi [A] time = 0.015042, antiderivative size = 54, 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, 621, 206} \[ \frac{1}{3} \sqrt{3 x^2+4 x-2}-\frac{2 \tanh ^{-1}\left (\frac{3 x+2}{\sqrt{3} \sqrt{3 x^2+4 x-2}}\right )}{3 \sqrt{3}} \]
Antiderivative was successfully verified.
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Rule 640
Rule 621
Rule 206
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{4}{3} \operatorname{Subst}\left (\int \frac{1}{12-x^2} \, dx,x,\frac{4+6 x}{\sqrt{-2+4 x+3 x^2}}\right )\\ &=\frac{1}{3} \sqrt{-2+4 x+3 x^2}-\frac{2 \tanh ^{-1}\left (\frac{2+3 x}{\sqrt{3} \sqrt{-2+4 x+3 x^2}}\right )}{3 \sqrt{3}}\\ \end{align*}
Mathematica [A] time = 0.0167977, size = 49, normalized size = 0.91 \[ \frac{1}{9} \left (3 \sqrt{3 x^2+4 x-2}-2 \sqrt{3} \tanh ^{-1}\left (\frac{3 x+2}{\sqrt{9 x^2+12 x-6}}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.041, size = 45, normalized size = 0.8 \begin{align*}{\frac{1}{3}\sqrt{3\,{x}^{2}+4\,x-2}}-{\frac{2\,\sqrt{3}}{9}\ln \left ({\frac{ \left ( 2+3\,x \right ) \sqrt{3}}{3}}+\sqrt{3\,{x}^{2}+4\,x-2} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.48325, size = 58, normalized size = 1.07 \begin{align*} -\frac{2}{9} \, \sqrt{3} \log \left (2 \, \sqrt{3} \sqrt{3 \, x^{2} + 4 \, x - 2} + 6 \, x + 4\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 = 2.45922, size = 143, normalized size = 2.65 \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 - 1\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 = 1.17741, size = 66, normalized size = 1.22 \begin{align*} \frac{2}{9} \, \sqrt{3} \log \left ({\left | -\sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 4 \, x - 2}\right )} - 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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