Optimal. Leaf size=59 \[ \frac {1}{6} (2+3 x) \sqrt {-2+4 x+3 x^2}-\frac {5 \tanh ^{-1}\left (\frac {2+3 x}{\sqrt {3} \sqrt {-2+4 x+3 x^2}}\right )}{3 \sqrt {3}} \]
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Rubi [A]
time = 0.01, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {626, 635, 212}
\begin {gather*} \frac {1}{6} (3 x+2) \sqrt {3 x^2+4 x-2}-\frac {5 \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 212
Rule 626
Rule 635
Rubi steps
\begin {align*} \int \sqrt {-2+4 x+3 x^2} \, dx &=\frac {1}{6} (2+3 x) \sqrt {-2+4 x+3 x^2}-\frac {5}{3} \int \frac {1}{\sqrt {-2+4 x+3 x^2}} \, dx\\ &=\frac {1}{6} (2+3 x) \sqrt {-2+4 x+3 x^2}-\frac {10}{3} \text {Subst}\left (\int \frac {1}{12-x^2} \, dx,x,\frac {4+6 x}{\sqrt {-2+4 x+3 x^2}}\right )\\ &=\frac {1}{6} (2+3 x) \sqrt {-2+4 x+3 x^2}-\frac {5 \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 = 0.17, size = 61, normalized size = 1.03 \begin {gather*} \frac {1}{6} (2+3 x) \sqrt {-2+4 x+3 x^2}-\frac {10 \tanh ^{-1}\left (\frac {\sqrt {-6+12 x+9 x^2}}{2+\sqrt {10}+3 x}\right )}{3 \sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.55, size = 50, normalized size = 0.85
method | result | size |
default | \(\frac {\left (6 x +4\right ) \sqrt {3 x^{2}+4 x -2}}{12}-\frac {5 \ln \left (\frac {\left (2+3 x \right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+4 x -2}\right ) \sqrt {3}}{9}\) | \(50\) |
risch | \(\frac {\left (2+3 x \right ) \sqrt {3 x^{2}+4 x -2}}{6}-\frac {5 \ln \left (\frac {\left (2+3 x \right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+4 x -2}\right ) \sqrt {3}}{9}\) | \(50\) |
trager | \(\left (\frac {1}{3}+\frac {x}{2}\right ) \sqrt {3 x^{2}+4 x -2}-\frac {5 \RootOf \left (\textit {\_Z}^{2}-3\right ) \ln \left (3 \RootOf \left (\textit {\_Z}^{2}-3\right ) x +3 \sqrt {3 x^{2}+4 x -2}+2 \RootOf \left (\textit {\_Z}^{2}-3\right )\right )}{9}\) | \(61\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 58, normalized size = 0.98 \begin {gather*} \frac {1}{2} \, \sqrt {3 \, x^{2} + 4 \, x - 2} x - \frac {5}{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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Fricas [A]
time = 2.58, size = 58, normalized size = 0.98 \begin {gather*} \frac {1}{6} \, \sqrt {3 \, x^{2} + 4 \, x - 2} {\left (3 \, x + 2\right )} + \frac {5}{18} \, \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 ) \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 \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 = 0.54, size = 54, normalized size = 0.92 \begin {gather*} \frac {1}{6} \, \sqrt {3 \, x^{2} + 4 \, x - 2} {\left (3 \, x + 2\right )} + \frac {5}{9} \, \sqrt {3} \log \left ({\left | -\sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 4 \, x - 2}\right )} - 2 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.19, size = 48, normalized size = 0.81 \begin {gather*} \left (\frac {x}{2}+\frac {1}{3}\right )\,\sqrt {3\,x^2+4\,x-2}-\frac {5\,\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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