Optimal. Leaf size=31 \[ -\frac{1}{8 x^2}+\frac{3}{8 x}+\frac{9 \log (x)}{16}-\frac{9}{16} \log (3 x+2) \]
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Rubi [A] time = 0.0100174, antiderivative size = 31, 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} \[ -\frac{1}{8 x^2}+\frac{3}{8 x}+\frac{9 \log (x)}{16}-\frac{9}{16} \log (3 x+2) \]
Antiderivative was successfully verified.
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Rule 44
Rubi steps
\begin{align*} \int \frac{1}{x^3 (4+6 x)} \, dx &=\int \left (\frac{1}{4 x^3}-\frac{3}{8 x^2}+\frac{9}{16 x}-\frac{27}{16 (2+3 x)}\right ) \, dx\\ &=-\frac{1}{8 x^2}+\frac{3}{8 x}+\frac{9 \log (x)}{16}-\frac{9}{16} \log (2+3 x)\\ \end{align*}
Mathematica [A] time = 0.0028378, size = 31, normalized size = 1. \[ -\frac{1}{8 x^2}+\frac{3}{8 x}+\frac{9 \log (x)}{16}-\frac{9}{16} \log (3 x+2) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 24, normalized size = 0.8 \begin{align*} -{\frac{1}{8\,{x}^{2}}}+{\frac{3}{8\,x}}+{\frac{9\,\ln \left ( x \right ) }{16}}-{\frac{9\,\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 = 0.991965, size = 31, normalized size = 1. \begin{align*} \frac{3 \, x - 1}{8 \, x^{2}} - \frac{9}{16} \, \log \left (3 \, x + 2\right ) + \frac{9}{16} \, \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.74511, size = 77, normalized size = 2.48 \begin{align*} -\frac{9 \, x^{2} \log \left (3 \, x + 2\right ) - 9 \, x^{2} \log \left (x\right ) - 6 \, x + 2}{16 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.167408, size = 26, normalized size = 0.84 \begin{align*} \frac{9 \log{\left (x \right )}}{16} - \frac{9 \log{\left (x + \frac{2}{3} \right )}}{16} + \frac{3 x - 1}{8 x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16679, size = 34, normalized size = 1.1 \begin{align*} \frac{3 \, x - 1}{8 \, x^{2}} - \frac{9}{16} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) + \frac{9}{16} \, \log \left ({\left | x \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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