Optimal. Leaf size=60 \[ \frac{9}{256 x^2}-\frac{1}{192 x^3}-\frac{27}{128 x}-\frac{27}{64 (3 x+2)}-\frac{27}{256 (3 x+2)^2}-\frac{135 \log (x)}{256}+\frac{135}{256} \log (3 x+2) \]
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Rubi [A] time = 0.0207767, antiderivative size = 60, 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} \[ \frac{9}{256 x^2}-\frac{1}{192 x^3}-\frac{27}{128 x}-\frac{27}{64 (3 x+2)}-\frac{27}{256 (3 x+2)^2}-\frac{135 \log (x)}{256}+\frac{135}{256} \log (3 x+2) \]
Antiderivative was successfully verified.
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Rule 44
Rubi steps
\begin{align*} \int \frac{1}{x^4 (4+6 x)^3} \, dx &=\int \left (\frac{1}{64 x^4}-\frac{9}{128 x^3}+\frac{27}{128 x^2}-\frac{135}{256 x}+\frac{81}{128 (2+3 x)^3}+\frac{81}{64 (2+3 x)^2}+\frac{405}{256 (2+3 x)}\right ) \, dx\\ &=-\frac{1}{192 x^3}+\frac{9}{256 x^2}-\frac{27}{128 x}-\frac{27}{256 (2+3 x)^2}-\frac{27}{64 (2+3 x)}-\frac{135 \log (x)}{256}+\frac{135}{256} \log (2+3 x)\\ \end{align*}
Mathematica [A] time = 0.0320831, size = 49, normalized size = 0.82 \[ \frac{1}{768} \left (-\frac{2 \left (1215 x^4+1215 x^3+180 x^2-30 x+8\right )}{x^3 (3 x+2)^2}-405 \log (x)+405 \log (3 x+2)\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 47, normalized size = 0.8 \begin{align*} -{\frac{1}{192\,{x}^{3}}}+{\frac{9}{256\,{x}^{2}}}-{\frac{27}{128\,x}}-{\frac{27}{256\, \left ( 2+3\,x \right ) ^{2}}}-{\frac{27}{128+192\,x}}-{\frac{135\,\ln \left ( x \right ) }{256}}+{\frac{135\,\ln \left ( 2+3\,x \right ) }{256}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00904, size = 72, normalized size = 1.2 \begin{align*} -\frac{1215 \, x^{4} + 1215 \, x^{3} + 180 \, x^{2} - 30 \, x + 8}{384 \,{\left (9 \, x^{5} + 12 \, x^{4} + 4 \, x^{3}\right )}} + \frac{135}{256} \, \log \left (3 \, x + 2\right ) - \frac{135}{256} \, \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.79541, size = 215, normalized size = 3.58 \begin{align*} -\frac{2430 \, x^{4} + 2430 \, x^{3} + 360 \, x^{2} - 405 \,{\left (9 \, x^{5} + 12 \, x^{4} + 4 \, x^{3}\right )} \log \left (3 \, x + 2\right ) + 405 \,{\left (9 \, x^{5} + 12 \, x^{4} + 4 \, x^{3}\right )} \log \left (x\right ) - 60 \, x + 16}{768 \,{\left (9 \, x^{5} + 12 \, x^{4} + 4 \, x^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.204437, size = 51, normalized size = 0.85 \begin{align*} - \frac{135 \log{\left (x \right )}}{256} + \frac{135 \log{\left (x + \frac{2}{3} \right )}}{256} - \frac{1215 x^{4} + 1215 x^{3} + 180 x^{2} - 30 x + 8}{3456 x^{5} + 4608 x^{4} + 1536 x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19775, size = 63, normalized size = 1.05 \begin{align*} -\frac{1215 \, x^{4} + 1215 \, x^{3} + 180 \, x^{2} - 30 \, x + 8}{384 \,{\left (3 \, x + 2\right )}^{2} x^{3}} + \frac{135}{256} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) - \frac{135}{256} \, \log \left ({\left | x \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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