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 = 0.01, antiderivative size = 23, normalized size of antiderivative = 1.00, 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*}
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Mathematica [A] time = 0.01, size = 23, normalized size = 1.00 \[ \frac {\log (x)}{3}-\log (x+1)+\frac {2}{3} \log (2 x+3) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 19, normalized size = 0.83 \[ \frac {2}{3} \, \log \left (2 \, x + 3\right ) - \log \left (x + 1\right ) + \frac {1}{3} \, \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.86, size = 22, normalized size = 0.96 \[ \frac {2}{3} \, \log \left ({\left | 2 \, x + 3 \right |}\right ) - \log \left ({\left | x + 1 \right |}\right ) + \frac {1}{3} \, \log \left ({\left | x \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 20, normalized size = 0.87 \[ \frac {\ln \relax (x )}{3}-\ln \left (x +1\right )+\frac {2 \ln \left (2 x +3\right )}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.51, size = 19, normalized size = 0.83 \[ \frac {2}{3} \, \log \left (2 \, x + 3\right ) - \log \left (x + 1\right ) + \frac {1}{3} \, \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 17, normalized size = 0.74 \[ \frac {2\,\ln \left (x+\frac {3}{2}\right )}{3}-\ln \left (x+1\right )+\frac {\ln \relax (x)}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.14, size = 19, normalized size = 0.83 \[ \frac {\log {\relax (x )}}{3} - \log {\left (x + 1 \right )} + \frac {2 \log {\left (x + \frac {3}{2} \right )}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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