Optimal. Leaf size=35 \[ \frac {\tanh ^{-1}\left (\frac {5+6 x}{2 \sqrt {3} \sqrt {2+5 x+3 x^2}}\right )}{\sqrt {3}} \]
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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 = {635, 212}
\begin {gather*} \frac {\tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{\sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 635
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {2+5 x+3 x^2}} \, dx &=2 \text {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 = 0.07, size = 30, normalized size = 0.86 \begin {gather*} \frac {2 \tanh ^{-1}\left (\frac {\sqrt {\frac {2}{3}+\frac {5 x}{3}+x^2}}{1+x}\right )}{\sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.46, size = 30, normalized size = 0.86
method | result | size |
default | \(\frac {\ln \left (\frac {\left (\frac {5}{2}+3 x \right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x +2}\right ) \sqrt {3}}{3}\) | \(30\) |
trager | \(-\frac {\RootOf \left (\textit {\_Z}^{2}-3\right ) \ln \left (-6 \RootOf \left (\textit {\_Z}^{2}-3\right ) x +6 \sqrt {3 x^{2}+5 x +2}-5 \RootOf \left (\textit {\_Z}^{2}-3\right )\right )}{3}\) | \(42\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.52, size = 28, normalized size = 0.80 \begin {gather*} \frac {1}{3} \, \sqrt {3} \log \left (2 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} + 6 \, x + 5\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.74, size = 38, normalized size = 1.09 \begin {gather*} \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 + 49\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 \frac {1}{\sqrt {3 x^{2} + 5 x + 2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.16, size = 54, normalized size = 1.54 \begin {gather*} \frac {1}{12} \, \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (6 \, x + 5\right )} + \frac {1}{72} \, \sqrt {3} \log \left ({\left | -2 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )} - 5 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.24, size = 26, normalized size = 0.74 \begin {gather*} \frac {\sqrt {3}\,\ln \left (\sqrt {3}\,\left (x+\frac {5}{6}\right )+\sqrt {3\,x^2+5\,x+2}\right )}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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