Optimal. Leaf size=35 \[ \frac {\tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x-2}}\right )}{\sqrt {3}} \]
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Rubi [A] time = 0.01, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {621, 206} \[ \frac {\tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x-2}}\right )}{\sqrt {3}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 621
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {-2+5 x+3 x^2}} \, dx &=2 \operatorname {Subst}\left (\int \frac {1}{12-x^2} \, dx,x,\frac {5+6 x}{\sqrt {-2+5 x+3 x^2}}\right )\\ &=\frac {\tanh ^{-1}\left (\frac {5+6 x}{2 \sqrt {3} \sqrt {-2+5 x+3 x^2}}\right )}{\sqrt {3}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 28, normalized size = 0.80 \[ \frac {\log \left (2 \sqrt {9 x^2+15 x-6}+6 x+5\right )}{\sqrt {3}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.03, size = 38, normalized size = 1.09 \[ \frac {1}{6} \, \sqrt {3} \log \left (4 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x - 2} {\left (6 \, x + 5\right )} + 72 \, x^{2} + 120 \, x + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.73, size = 34, normalized size = 0.97 \[ -\frac {1}{3} \, \sqrt {3} \log \left ({\left | -2 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x - 2}\right )} - 5 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 30, normalized size = 0.86 \[ \frac {\sqrt {3}\, \ln \left (\frac {\left (3 x +\frac {5}{2}\right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x -2}\right )}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.93, size = 28, normalized size = 0.80 \[ \frac {1}{3} \, \sqrt {3} \log \left (2 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x - 2} + 6 \, x + 5\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.23, size = 26, normalized size = 0.74 \[ \frac {\sqrt {3}\,\ln \left (\sqrt {3}\,\left (x+\frac {5}{6}\right )+\sqrt {3\,x^2+5\,x-2}\right )}{3} \]
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 \[ \int \frac {1}{\sqrt {3 x^{2} + 5 x - 2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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