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 = 0.02, antiderivative size = 54, normalized size of antiderivative = 1.00, 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} \begin {gather*} \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}} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 621
Rule 640
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*}
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Mathematica [A] time = 0.02, size = 49, normalized size = 0.91 \begin {gather*} \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 ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.10, size = 54, normalized size = 1.00 \begin {gather*} \frac {1}{3} \sqrt {3 x^2+4 x-2}+\frac {2 \log \left (\sqrt {3} \sqrt {3 x^2+4 x-2}-3 x-2\right )}{3 \sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 53, normalized size = 0.98 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 49, normalized size = 0.91 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 45, normalized size = 0.83 \begin {gather*} -\frac {2 \sqrt {3}\, \ln \left (\frac {\left (3 x +2\right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+4 x -2}\right )}{9}+\frac {\sqrt {3 x^{2}+4 x -2}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.98, size = 43, normalized size = 0.80 \begin {gather*} -\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.11, size = 44, normalized size = 0.81 \begin {gather*} \frac {\sqrt {3\,x^2+4\,x-2}}{3}-\frac {2\,\sqrt {3}\,\ln \left (\sqrt {3\,x^2+4\,x-2}+\frac {\sqrt {3}\,\left (3\,x+2\right )}{3}\right )}{9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x}{\sqrt {3 x^{2} + 4 x - 2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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