Optimal. Leaf size=49 \[ -\frac {1}{48 x^3}+\frac {3}{32 x^2}-\frac {27}{64 x}-\frac {27}{64 (3 x+2)}-\frac {27 \log (x)}{32}+\frac {27}{32} \log (3 x+2) \]
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Rubi [A] time = 0.02, antiderivative size = 49, normalized size of antiderivative = 1.00, 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} \[ \frac {3}{32 x^2}-\frac {1}{48 x^3}-\frac {27}{64 x}-\frac {27}{64 (3 x+2)}-\frac {27 \log (x)}{32}+\frac {27}{32} \log (3 x+2) \]
Antiderivative was successfully verified.
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Rule 44
Rubi steps
\begin {align*} \int \frac {1}{x^4 (4+6 x)^2} \, dx &=\int \left (\frac {1}{16 x^4}-\frac {3}{16 x^3}+\frac {27}{64 x^2}-\frac {27}{32 x}+\frac {81}{64 (2+3 x)^2}+\frac {81}{32 (2+3 x)}\right ) \, dx\\ &=-\frac {1}{48 x^3}+\frac {3}{32 x^2}-\frac {27}{64 x}-\frac {27}{64 (2+3 x)}-\frac {27 \log (x)}{32}+\frac {27}{32} \log (2+3 x)\\ \end {align*}
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Mathematica [A] time = 0.03, size = 44, normalized size = 0.90 \[ \frac {1}{192} \left (-\frac {4 \left (81 x^3+27 x^2-6 x+2\right )}{x^3 (3 x+2)}-162 \log (x)+162 \log (3 x+2)\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 64, normalized size = 1.31 \[ -\frac {162 \, x^{3} + 54 \, x^{2} - 81 \, {\left (3 \, x^{4} + 2 \, x^{3}\right )} \log \left (3 \, x + 2\right ) + 81 \, {\left (3 \, x^{4} + 2 \, x^{3}\right )} \log \relax (x) - 12 \, x + 4}{96 \, {\left (3 \, x^{4} + 2 \, x^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.23, size = 60, normalized size = 1.22 \[ -\frac {27}{64 \, {\left (3 \, x + 2\right )}} - \frac {9 \, {\left (\frac {60}{3 \, x + 2} - \frac {72}{{\left (3 \, x + 2\right )}^{2}} - 13\right )}}{128 \, {\left (\frac {2}{3 \, x + 2} - 1\right )}^{3}} - \frac {27}{32} \, \log \left ({\left | -\frac {2}{3 \, x + 2} + 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 38, normalized size = 0.78 \[ -\frac {27 \ln \relax (x )}{32}+\frac {27 \ln \left (3 x +2\right )}{32}-\frac {27}{64 x}+\frac {3}{32 x^{2}}-\frac {1}{48 x^{3}}-\frac {27}{64 \left (3 x +2\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.31, size = 43, normalized size = 0.88 \[ -\frac {81 \, x^{3} + 27 \, x^{2} - 6 \, x + 2}{48 \, {\left (3 \, x^{4} + 2 \, x^{3}\right )}} + \frac {27}{32} \, \log \left (3 \, x + 2\right ) - \frac {27}{32} \, \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.09, size = 37, normalized size = 0.76 \[ \frac {27\,\mathrm {atanh}\left (3\,x+1\right )}{16}-\frac {\frac {9\,x^3}{16}+\frac {3\,x^2}{16}-\frac {x}{24}+\frac {1}{72}}{x^4+\frac {2\,x^3}{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.17, size = 41, normalized size = 0.84 \[ - \frac {27 \log {\relax (x )}}{32} + \frac {27 \log {\left (x + \frac {2}{3} \right )}}{32} + \frac {- 81 x^{3} - 27 x^{2} + 6 x - 2}{144 x^{4} + 96 x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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