Optimal. Leaf size=35 \[ -\frac{1}{16 x}-\frac{3}{16 (3 x+2)}-\frac{3 \log (x)}{16}+\frac{3}{16} \log (3 x+2) \]
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Rubi [A] time = 0.0099562, antiderivative size = 35, normalized size of antiderivative = 1., 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} \[ -\frac{1}{16 x}-\frac{3}{16 (3 x+2)}-\frac{3 \log (x)}{16}+\frac{3}{16} \log (3 x+2) \]
Antiderivative was successfully verified.
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Rule 44
Rubi steps
\begin{align*} \int \frac{1}{x^2 (4+6 x)^2} \, dx &=\int \left (\frac{1}{16 x^2}-\frac{3}{16 x}+\frac{9}{16 (2+3 x)^2}+\frac{9}{16 (2+3 x)}\right ) \, dx\\ &=-\frac{1}{16 x}-\frac{3}{16 (2+3 x)}-\frac{3 \log (x)}{16}+\frac{3}{16} \log (2+3 x)\\ \end{align*}
Mathematica [A] time = 0.0163434, size = 31, normalized size = 0.89 \[ \frac{1}{16} \left (-\frac{1}{x}-\frac{3}{3 x+2}-3 \log (x)+3 \log (3 x+2)\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 28, normalized size = 0.8 \begin{align*} -{\frac{1}{16\,x}}-{\frac{3}{32+48\,x}}-{\frac{3\,\ln \left ( x \right ) }{16}}+{\frac{3\,\ln \left ( 2+3\,x \right ) }{16}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01809, size = 42, normalized size = 1.2 \begin{align*} -\frac{3 \, x + 1}{8 \,{\left (3 \, x^{2} + 2 \, x\right )}} + \frac{3}{16} \, \log \left (3 \, x + 2\right ) - \frac{3}{16} \, \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.73492, size = 116, normalized size = 3.31 \begin{align*} \frac{3 \,{\left (3 \, x^{2} + 2 \, x\right )} \log \left (3 \, x + 2\right ) - 3 \,{\left (3 \, x^{2} + 2 \, x\right )} \log \left (x\right ) - 6 \, x - 2}{16 \,{\left (3 \, x^{2} + 2 \, x\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.164169, size = 29, normalized size = 0.83 \begin{align*} - \frac{3 x + 1}{24 x^{2} + 16 x} - \frac{3 \log{\left (x \right )}}{16} + \frac{3 \log{\left (x + \frac{2}{3} \right )}}{16} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.2096, size = 54, normalized size = 1.54 \begin{align*} -\frac{3}{16 \,{\left (3 \, x + 2\right )}} + \frac{3}{32 \,{\left (\frac{2}{3 \, x + 2} - 1\right )}} - \frac{3}{16} \, \log \left ({\left | -\frac{2}{3 \, x + 2} + 1 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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