Optimal. Leaf size=29 \[ \frac{(3 x+2) \log (3 x+2)}{3 \sqrt{9 x^2+12 x+4}} \]
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Rubi [A] time = 0.0048338, antiderivative size = 29, normalized size of antiderivative = 1., 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 = {608, 31} \[ \frac{(3 x+2) \log (3 x+2)}{3 \sqrt{9 x^2+12 x+4}} \]
Antiderivative was successfully verified.
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Rule 608
Rule 31
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{4+12 x+9 x^2}} \, dx &=\frac{(6+9 x) \int \frac{1}{6+9 x} \, dx}{\sqrt{4+12 x+9 x^2}}\\ &=\frac{(2+3 x) \log (2+3 x)}{3 \sqrt{4+12 x+9 x^2}}\\ \end{align*}
Mathematica [A] time = 0.0063044, size = 26, normalized size = 0.9 \[ \frac{(3 x+2) \log (3 x+2)}{3 \sqrt{(3 x+2)^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.126, size = 23, normalized size = 0.8 \begin{align*}{\frac{ \left ( 2+3\,x \right ) \ln \left ( 2+3\,x \right ) }{3}{\frac{1}{\sqrt{ \left ( 2+3\,x \right ) ^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.66952, size = 8, normalized size = 0.28 \begin{align*} \frac{1}{3} \, \log \left (x + \frac{2}{3}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.18526, size = 24, normalized size = 0.83 \begin{align*} \frac{1}{3} \, \log \left (3 \, x + 2\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{9 x^{2} + 12 x + 4}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.22704, size = 34, normalized size = 1.17 \begin{align*} \frac{\log \left ({\left | 3 \, x + 2 \right |}{\left | \mathrm{sgn}\left (3 \, x + 2\right ) \right |}\right )}{3 \, \mathrm{sgn}\left (3 \, x + 2\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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