Optimal. Leaf size=53 \[ -\frac{1}{128 x^2}+\frac{9}{128 x}+\frac{27}{128 (3 x+2)}+\frac{9}{128 (3 x+2)^2}+\frac{27 \log (x)}{128}-\frac{27}{128} \log (3 x+2) \]
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Rubi [A] time = 0.015826, antiderivative size = 53, 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} \[ -\frac{1}{128 x^2}+\frac{9}{128 x}+\frac{27}{128 (3 x+2)}+\frac{9}{128 (3 x+2)^2}+\frac{27 \log (x)}{128}-\frac{27}{128} \log (3 x+2) \]
Antiderivative was successfully verified.
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Rule 44
Rubi steps
\begin{align*} \int \frac{1}{x^3 (4+6 x)^3} \, dx &=\int \left (\frac{1}{64 x^3}-\frac{9}{128 x^2}+\frac{27}{128 x}-\frac{27}{64 (2+3 x)^3}-\frac{81}{128 (2+3 x)^2}-\frac{81}{128 (2+3 x)}\right ) \, dx\\ &=-\frac{1}{128 x^2}+\frac{9}{128 x}+\frac{9}{128 (2+3 x)^2}+\frac{27}{128 (2+3 x)}+\frac{27 \log (x)}{128}-\frac{27}{128} \log (2+3 x)\\ \end{align*}
Mathematica [A] time = 0.0369721, size = 44, normalized size = 0.83 \[ \frac{1}{128} \left (\frac{2 \left (81 x^3+81 x^2+12 x-2\right )}{x^2 (3 x+2)^2}+27 \log (x)-27 \log (3 x+2)\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.011, size = 42, normalized size = 0.8 \begin{align*} -{\frac{1}{128\,{x}^{2}}}+{\frac{9}{128\,x}}+{\frac{9}{128\, \left ( 2+3\,x \right ) ^{2}}}+{\frac{27}{256+384\,x}}+{\frac{27\,\ln \left ( x \right ) }{128}}-{\frac{27\,\ln \left ( 2+3\,x \right ) }{128}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.0443, size = 65, normalized size = 1.23 \begin{align*} \frac{81 \, x^{3} + 81 \, x^{2} + 12 \, x - 2}{64 \,{\left (9 \, x^{4} + 12 \, x^{3} + 4 \, x^{2}\right )}} - \frac{27}{128} \, \log \left (3 \, x + 2\right ) + \frac{27}{128} \, \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.80194, size = 193, normalized size = 3.64 \begin{align*} \frac{162 \, x^{3} + 162 \, x^{2} - 27 \,{\left (9 \, x^{4} + 12 \, x^{3} + 4 \, x^{2}\right )} \log \left (3 \, x + 2\right ) + 27 \,{\left (9 \, x^{4} + 12 \, x^{3} + 4 \, x^{2}\right )} \log \left (x\right ) + 24 \, x - 4}{128 \,{\left (9 \, x^{4} + 12 \, x^{3} + 4 \, x^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.239392, size = 46, normalized size = 0.87 \begin{align*} \frac{27 \log{\left (x \right )}}{128} - \frac{27 \log{\left (x + \frac{2}{3} \right )}}{128} + \frac{81 x^{3} + 81 x^{2} + 12 x - 2}{576 x^{4} + 768 x^{3} + 256 x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.23302, size = 58, normalized size = 1.09 \begin{align*} \frac{81 \, x^{3} + 81 \, x^{2} + 12 \, x - 2}{64 \,{\left (3 \, x^{2} + 2 \, x\right )}^{2}} - \frac{27}{128} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) + \frac{27}{128} \, \log \left ({\left | x \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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