Optimal. Leaf size=28 \[ \frac {1}{8 (3 x+2)}+\frac {\log (x)}{16}-\frac {1}{16} \log (3 x+2) \]
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Rubi [A] time = 0.01, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {44} \[ \frac {1}{8 (3 x+2)}+\frac {\log (x)}{16}-\frac {1}{16} \log (3 x+2) \]
Antiderivative was successfully verified.
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Rule 44
Rubi steps
\begin {align*} \int \frac {1}{x (4+6 x)^2} \, dx &=\int \left (\frac {1}{16 x}-\frac {3}{8 (2+3 x)^2}-\frac {3}{16 (2+3 x)}\right ) \, dx\\ &=\frac {1}{8 (2+3 x)}+\frac {\log (x)}{16}-\frac {1}{16} \log (2+3 x)\\ \end {align*}
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Mathematica [A] time = 0.01, size = 26, normalized size = 0.93 \[ \frac {1}{16} \left (\frac {2}{3 x+2}+\log (-6 x)-\log (6 x+4)\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 32, normalized size = 1.14 \[ -\frac {{\left (3 \, x + 2\right )} \log \left (3 \, x + 2\right ) - {\left (3 \, x + 2\right )} \log \relax (x) - 2}{16 \, {\left (3 \, x + 2\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.11, size = 25, normalized size = 0.89 \[ \frac {1}{8 \, {\left (3 \, x + 2\right )}} + \frac {1}{16} \, \log \left ({\left | -\frac {2}{3 \, x + 2} + 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 23, normalized size = 0.82 \[ \frac {\ln \relax (x )}{16}-\frac {\ln \left (3 x +2\right )}{16}+\frac {1}{24 x +16} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.37, size = 22, normalized size = 0.79 \[ \frac {1}{8 \, {\left (3 \, x + 2\right )}} - \frac {1}{16} \, \log \left (3 \, x + 2\right ) + \frac {1}{16} \, \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 20, normalized size = 0.71 \[ \frac {1}{8\,\left (3\,x+2\right )}-\frac {\ln \left (\frac {6\,x+4}{x}\right )}{16} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.14, size = 19, normalized size = 0.68 \[ \frac {\log {\relax (x )}}{16} - \frac {\log {\left (x + \frac {2}{3} \right )}}{16} + \frac {1}{24 x + 16} \]
Verification of antiderivative is not currently implemented for this CAS.
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