Optimal. Leaf size=57 \[ \frac {1}{3} \sqrt {3 x^2+5 x+2}-\frac {5 \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{6 \sqrt {3}} \]
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Rubi [A] time = 0.01, antiderivative size = 57, 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+5 x+2}-\frac {5 \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{6 \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+5 x+3 x^2}} \, dx &=\frac {1}{3} \sqrt {2+5 x+3 x^2}-\frac {5}{6} \int \frac {1}{\sqrt {2+5 x+3 x^2}} \, dx\\ &=\frac {1}{3} \sqrt {2+5 x+3 x^2}-\frac {5}{3} \operatorname {Subst}\left (\int \frac {1}{12-x^2} \, dx,x,\frac {5+6 x}{\sqrt {2+5 x+3 x^2}}\right )\\ &=\frac {1}{3} \sqrt {2+5 x+3 x^2}-\frac {5 \tanh ^{-1}\left (\frac {5+6 x}{2 \sqrt {3} \sqrt {2+5 x+3 x^2}}\right )}{6 \sqrt {3}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 52, normalized size = 0.91 \begin {gather*} \frac {1}{18} \left (6 \sqrt {3 x^2+5 x+2}-5 \sqrt {3} \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {9 x^2+15 x+6}}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.13, size = 54, normalized size = 0.95 \begin {gather*} \frac {1}{3} \sqrt {3 x^2+5 x+2}-\frac {5 \tanh ^{-1}\left (\frac {\sqrt {3 x^2+5 x+2}}{\sqrt {3} (x+1)}\right )}{3 \sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 53, normalized size = 0.93 \begin {gather*} \frac {5}{36} \, \sqrt {3} \log \left (-4 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (6 \, x + 5\right )} + 72 \, x^{2} + 120 \, x + 49\right ) + \frac {1}{3} \, \sqrt {3 \, x^{2} + 5 \, x + 2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 49, normalized size = 0.86 \begin {gather*} \frac {5}{18} \, \sqrt {3} \log \left ({\left | -2 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )} - 5 \right |}\right ) + \frac {1}{3} \, \sqrt {3 \, x^{2} + 5 \, x + 2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 45, normalized size = 0.79 \begin {gather*} -\frac {5 \sqrt {3}\, \ln \left (\frac {\left (3 x +\frac {5}{2}\right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x +2}\right )}{18}+\frac {\sqrt {3 x^{2}+5 x +2}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.97, size = 43, normalized size = 0.75 \begin {gather*} -\frac {5}{18} \, \sqrt {3} \log \left (2 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} + 6 \, x + 5\right ) + \frac {1}{3} \, \sqrt {3 \, x^{2} + 5 \, x + 2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.09, size = 44, normalized size = 0.77 \begin {gather*} \frac {\sqrt {3\,x^2+5\,x+2}}{3}-\frac {5\,\sqrt {3}\,\ln \left (\sqrt {3\,x^2+5\,x+2}+\frac {\sqrt {3}\,\left (3\,x+\frac {5}{2}\right )}{3}\right )}{18} \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 {\left (x + 1\right ) \left (3 x + 2\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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