Optimal. Leaf size=28 \[ \frac{(3 x+2) \log (3 x+2)}{3 \sqrt{-(3 x+2)^2}} \]
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Rubi [A] time = 0.008147, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {247, 15, 29} \[ \frac{(3 x+2) \log (3 x+2)}{3 \sqrt{-(3 x+2)^2}} \]
Antiderivative was successfully verified.
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Rule 247
Rule 15
Rule 29
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{-(2+3 x)^2}} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{\sqrt{-x^2}} \, dx,x,2+3 x\right )\\ &=\frac{(2+3 x) \operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,2+3 x\right )}{3 \sqrt{-(2+3 x)^2}}\\ &=\frac{(2+3 x) \log (2+3 x)}{3 \sqrt{-(2+3 x)^2}}\\ \end{align*}
Mathematica [A] time = 0.0057903, size = 28, normalized size = 1. \[ \frac{(3 x+2) \log (3 x+2)}{3 \sqrt{-(3 x+2)^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 25, normalized size = 0.9 \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 [C] time = 1.46836, size = 8, normalized size = 0.29 \begin{align*} \frac{1}{3} i \, \log \left (x + \frac{2}{3}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] time = 1.18553, size = 28, normalized size = 1. \begin{align*} -\frac{1}{3} i \, \log \left (x + \frac{2}{3}\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{- \left (3 x + 2\right )^{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] time = 1.16224, size = 31, normalized size = 1.11 \begin{align*} \frac{i \, \log \left ({\left (-3 i \, x - 2 i\right )} \mathrm{sgn}\left (-3 \, x - 2\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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