Optimal. Leaf size=40 \[ \frac {1}{3} \sqrt {2+4 x+3 x^2}-\frac {2 \sinh ^{-1}\left (\frac {2+3 x}{\sqrt {2}}\right )}{3 \sqrt {3}} \]
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Rubi [A]
time = 0.01, antiderivative size = 40, 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 = {654, 633, 221}
\begin {gather*} \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}} \end {gather*}
Antiderivative was successfully verified.
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Rule 221
Rule 633
Rule 654
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 {\text {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*}
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Mathematica [A]
time = 0.08, size = 48, normalized size = 1.20 \begin {gather*} \frac {1}{9} \left (3 \sqrt {2+4 x+3 x^2}+2 \sqrt {3} \log \left (-2-3 x+\sqrt {6+12 x+9 x^2}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.00, size = 30, normalized size = 0.75
method | result | size |
default | \(\frac {\sqrt {3 x^{2}+4 x +2}}{3}-\frac {2 \sqrt {3}\, \arcsinh \left (\frac {3 \sqrt {2}\, \left (x +\frac {2}{3}\right )}{2}\right )}{9}\) | \(30\) |
risch | \(\frac {\sqrt {3 x^{2}+4 x +2}}{3}-\frac {2 \sqrt {3}\, \arcsinh \left (\frac {3 \sqrt {2}\, \left (x +\frac {2}{3}\right )}{2}\right )}{9}\) | \(30\) |
trager | \(\frac {\sqrt {3 x^{2}+4 x +2}}{3}+\frac {2 \RootOf \left (\textit {\_Z}^{2}-3\right ) \ln \left (-3 \RootOf \left (\textit {\_Z}^{2}-3\right ) x -2 \RootOf \left (\textit {\_Z}^{2}-3\right )+3 \sqrt {3 x^{2}+4 x +2}\right )}{9}\) | \(57\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 31, normalized size = 0.78 \begin {gather*} -\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.37, size = 52, normalized size = 1.30 \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 - 5\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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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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Giac [A]
time = 1.41, size = 48, normalized size = 1.20 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.11, size = 44, normalized size = 1.10 \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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