Optimal. Leaf size=67 \[ -\frac{27}{256 x^2}+\frac{3}{128 x^3}-\frac{1}{256 x^4}+\frac{135}{256 x}+\frac{405}{512 (3 x+2)}+\frac{81}{512 (3 x+2)^2}+\frac{1215 \log (x)}{1024}-\frac{1215 \log (3 x+2)}{1024} \]
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Rubi [A] time = 0.0223792, antiderivative size = 67, 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} \[ -\frac{27}{256 x^2}+\frac{3}{128 x^3}-\frac{1}{256 x^4}+\frac{135}{256 x}+\frac{405}{512 (3 x+2)}+\frac{81}{512 (3 x+2)^2}+\frac{1215 \log (x)}{1024}-\frac{1215 \log (3 x+2)}{1024} \]
Antiderivative was successfully verified.
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Rule 44
Rubi steps
\begin{align*} \int \frac{1}{x^5 (4+6 x)^3} \, dx &=\int \left (\frac{1}{64 x^5}-\frac{9}{128 x^4}+\frac{27}{128 x^3}-\frac{135}{256 x^2}+\frac{1215}{1024 x}-\frac{243}{256 (2+3 x)^3}-\frac{1215}{512 (2+3 x)^2}-\frac{3645}{1024 (2+3 x)}\right ) \, dx\\ &=-\frac{1}{256 x^4}+\frac{3}{128 x^3}-\frac{27}{256 x^2}+\frac{135}{256 x}+\frac{81}{512 (2+3 x)^2}+\frac{405}{512 (2+3 x)}+\frac{1215 \log (x)}{1024}-\frac{1215 \log (2+3 x)}{1024}\\ \end{align*}
Mathematica [A] time = 0.032343, size = 54, normalized size = 0.81 \[ \frac{\frac{2 \left (3645 x^5+3645 x^4+540 x^3-90 x^2+24 x-8\right )}{x^4 (3 x+2)^2}+1215 \log (x)-1215 \log (3 x+2)}{1024} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 52, normalized size = 0.8 \begin{align*} -{\frac{1}{256\,{x}^{4}}}+{\frac{3}{128\,{x}^{3}}}-{\frac{27}{256\,{x}^{2}}}+{\frac{135}{256\,x}}+{\frac{81}{512\, \left ( 2+3\,x \right ) ^{2}}}+{\frac{405}{1024+1536\,x}}+{\frac{1215\,\ln \left ( x \right ) }{1024}}-{\frac{1215\,\ln \left ( 2+3\,x \right ) }{1024}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.07264, size = 78, normalized size = 1.16 \begin{align*} \frac{3645 \, x^{5} + 3645 \, x^{4} + 540 \, x^{3} - 90 \, x^{2} + 24 \, x - 8}{512 \,{\left (9 \, x^{6} + 12 \, x^{5} + 4 \, x^{4}\right )}} - \frac{1215}{1024} \, \log \left (3 \, x + 2\right ) + \frac{1215}{1024} \, \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.55242, size = 232, normalized size = 3.46 \begin{align*} \frac{7290 \, x^{5} + 7290 \, x^{4} + 1080 \, x^{3} - 180 \, x^{2} - 1215 \,{\left (9 \, x^{6} + 12 \, x^{5} + 4 \, x^{4}\right )} \log \left (3 \, x + 2\right ) + 1215 \,{\left (9 \, x^{6} + 12 \, x^{5} + 4 \, x^{4}\right )} \log \left (x\right ) + 48 \, x - 16}{1024 \,{\left (9 \, x^{6} + 12 \, x^{5} + 4 \, x^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.18851, size = 56, normalized size = 0.84 \begin{align*} \frac{1215 \log{\left (x \right )}}{1024} - \frac{1215 \log{\left (x + \frac{2}{3} \right )}}{1024} + \frac{3645 x^{5} + 3645 x^{4} + 540 x^{3} - 90 x^{2} + 24 x - 8}{4608 x^{6} + 6144 x^{5} + 2048 x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18215, size = 70, normalized size = 1.04 \begin{align*} \frac{3645 \, x^{5} + 3645 \, x^{4} + 540 \, x^{3} - 90 \, x^{2} + 24 \, x - 8}{512 \,{\left (3 \, x + 2\right )}^{2} x^{4}} - \frac{1215}{1024} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) + \frac{1215}{1024} \, \log \left ({\left | x \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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