Optimal. Leaf size=38 \[ -\frac {1}{12 x^3}+\frac {3}{16 x^2}-\frac {9}{16 x}-\frac {27 \log (x)}{32}+\frac {27}{32} \log (3 x+2) \]
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Rubi [A] time = 0.01, antiderivative size = 38, 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} \begin {gather*} \frac {3}{16 x^2}-\frac {1}{12 x^3}-\frac {9}{16 x}-\frac {27 \log (x)}{32}+\frac {27}{32} \log (3 x+2) \end {gather*}
Antiderivative was successfully verified.
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Rule 44
Rubi steps
\begin {align*} \int \frac {1}{x^4 (4+6 x)} \, dx &=\int \left (\frac {1}{4 x^4}-\frac {3}{8 x^3}+\frac {9}{16 x^2}-\frac {27}{32 x}+\frac {81}{32 (2+3 x)}\right ) \, dx\\ &=-\frac {1}{12 x^3}+\frac {3}{16 x^2}-\frac {9}{16 x}-\frac {27 \log (x)}{32}+\frac {27}{32} \log (2+3 x)\\ \end {align*}
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Mathematica [A] time = 0.00, size = 38, normalized size = 1.00 \begin {gather*} -\frac {1}{12 x^3}+\frac {3}{16 x^2}-\frac {9}{16 x}-\frac {27 \log (x)}{32}+\frac {27}{32} \log (3 x+2) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^4 (4+6 x)} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.57, size = 33, normalized size = 0.87 \begin {gather*} \frac {81 \, x^{3} \log \left (3 \, x + 2\right ) - 81 \, x^{3} \log \relax (x) - 54 \, x^{2} + 18 \, x - 8}{96 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.03, size = 30, normalized size = 0.79 \begin {gather*} -\frac {27 \, x^{2} - 9 \, x + 4}{48 \, x^{3}} + \frac {27}{32} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) - \frac {27}{32} \, \log \left ({\left | x \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 29, normalized size = 0.76 \begin {gather*} -\frac {27 \ln \relax (x )}{32}+\frac {27 \ln \left (3 x +2\right )}{32}-\frac {9}{16 x}+\frac {3}{16 x^{2}}-\frac {1}{12 x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.35, size = 28, normalized size = 0.74 \begin {gather*} -\frac {27 \, x^{2} - 9 \, x + 4}{48 \, x^{3}} + \frac {27}{32} \, \log \left (3 \, x + 2\right ) - \frac {27}{32} \, \log \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.09, size = 24, normalized size = 0.63 \begin {gather*} \frac {27\,\mathrm {atanh}\left (3\,x+1\right )}{16}-\frac {\frac {9\,x^2}{16}-\frac {3\,x}{16}+\frac {1}{12}}{x^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.16, size = 31, normalized size = 0.82 \begin {gather*} - \frac {27 \log {\relax (x )}}{32} + \frac {27 \log {\left (x + \frac {2}{3} \right )}}{32} + \frac {- 27 x^{2} + 9 x - 4}{48 x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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