Optimal. Leaf size=23 \[ \frac{\log (x)}{3}-\log (x+1)+\frac{2}{3} \log (2 x+3) \]
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Rubi [A] time = 0.0080618, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062, Rules used = {72} \[ \frac{\log (x)}{3}-\log (x+1)+\frac{2}{3} \log (2 x+3) \]
Antiderivative was successfully verified.
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Rule 72
Rubi steps
\begin{align*} \int \frac{1}{x (1+x) (3+2 x)} \, dx &=\int \left (\frac{1}{-1-x}+\frac{1}{3 x}+\frac{4}{3 (3+2 x)}\right ) \, dx\\ &=\frac{\log (x)}{3}-\log (1+x)+\frac{2}{3} \log (3+2 x)\\ \end{align*}
Mathematica [A] time = 0.0044875, size = 23, normalized size = 1. \[ \frac{\log (x)}{3}-\log (x+1)+\frac{2}{3} \log (2 x+3) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 20, normalized size = 0.9 \begin{align*}{\frac{\ln \left ( x \right ) }{3}}-\ln \left ( 1+x \right ) +{\frac{2\,\ln \left ( 3+2\,x \right ) }{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.934752, size = 26, normalized size = 1.13 \begin{align*} \frac{2}{3} \, \log \left (2 \, x + 3\right ) - \log \left (x + 1\right ) + \frac{1}{3} \, \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.90344, size = 59, normalized size = 2.57 \begin{align*} \frac{2}{3} \, \log \left (2 \, x + 3\right ) - \log \left (x + 1\right ) + \frac{1}{3} \, \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.117938, size = 19, normalized size = 0.83 \begin{align*} \frac{\log{\left (x \right )}}{3} - \log{\left (x + 1 \right )} + \frac{2 \log{\left (x + \frac{3}{2} \right )}}{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.05534, size = 30, normalized size = 1.3 \begin{align*} \frac{2}{3} \, \log \left ({\left | 2 \, x + 3 \right |}\right ) - \log \left ({\left | x + 1 \right |}\right ) + \frac{1}{3} \, \log \left ({\left | x \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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