Optimal. Leaf size=42 \[ -\frac{1}{32 x^2}+\frac{3}{16 x}+\frac{9}{32 (3 x+2)}+\frac{27 \log (x)}{64}-\frac{27}{64} \log (3 x+2) \]
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Rubi [A] time = 0.0135602, antiderivative size = 42, 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} \[ -\frac{1}{32 x^2}+\frac{3}{16 x}+\frac{9}{32 (3 x+2)}+\frac{27 \log (x)}{64}-\frac{27}{64} \log (3 x+2) \]
Antiderivative was successfully verified.
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Rule 44
Rubi steps
\begin{align*} \int \frac{1}{x^3 (4+6 x)^2} \, dx &=\int \left (\frac{1}{16 x^3}-\frac{3}{16 x^2}+\frac{27}{64 x}-\frac{27}{32 (2+3 x)^2}-\frac{81}{64 (2+3 x)}\right ) \, dx\\ &=-\frac{1}{32 x^2}+\frac{3}{16 x}+\frac{9}{32 (2+3 x)}+\frac{27 \log (x)}{64}-\frac{27}{64} \log (2+3 x)\\ \end{align*}
Mathematica [A] time = 0.0192639, size = 36, normalized size = 0.86 \[ \frac{1}{64} \left (-\frac{2}{x^2}+\frac{12}{x}+\frac{18}{3 x+2}+27 \log (x)-27 \log (3 x+2)\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 33, normalized size = 0.8 \begin{align*} -{\frac{1}{32\,{x}^{2}}}+{\frac{3}{16\,x}}+{\frac{9}{64+96\,x}}+{\frac{27\,\ln \left ( x \right ) }{64}}-{\frac{27\,\ln \left ( 2+3\,x \right ) }{64}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.08064, size = 51, normalized size = 1.21 \begin{align*} \frac{27 \, x^{2} + 9 \, x - 2}{32 \,{\left (3 \, x^{3} + 2 \, x^{2}\right )}} - \frac{27}{64} \, \log \left (3 \, x + 2\right ) + \frac{27}{64} \, \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.73493, size = 140, normalized size = 3.33 \begin{align*} \frac{54 \, x^{2} - 27 \,{\left (3 \, x^{3} + 2 \, x^{2}\right )} \log \left (3 \, x + 2\right ) + 27 \,{\left (3 \, x^{3} + 2 \, x^{2}\right )} \log \left (x\right ) + 18 \, x - 4}{64 \,{\left (3 \, x^{3} + 2 \, x^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.268462, size = 36, normalized size = 0.86 \begin{align*} \frac{27 \log{\left (x \right )}}{64} - \frac{27 \log{\left (x + \frac{2}{3} \right )}}{64} + \frac{27 x^{2} + 9 x - 2}{96 x^{3} + 64 x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.22409, size = 69, normalized size = 1.64 \begin{align*} \frac{9}{32 \,{\left (3 \, x + 2\right )}} - \frac{9 \,{\left (\frac{12}{3 \, x + 2} - 5\right )}}{128 \,{\left (\frac{2}{3 \, x + 2} - 1\right )}^{2}} + \frac{27}{64} \, \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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