Optimal. Leaf size=24 \[ -\frac {1}{4 x}-\frac {3 \log (x)}{8}+\frac {3}{8} \log (3 x+2) \]
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Rubi [A] time = 0.01, antiderivative size = 24, 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 {1}{4 x}-\frac {3 \log (x)}{8}+\frac {3}{8} \log (3 x+2) \end {gather*}
Antiderivative was successfully verified.
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Rule 44
Rubi steps
\begin {align*} \int \frac {1}{x^2 (4+6 x)} \, dx &=\int \left (\frac {1}{4 x^2}-\frac {3}{8 x}+\frac {9}{8 (2+3 x)}\right ) \, dx\\ &=-\frac {1}{4 x}-\frac {3 \log (x)}{8}+\frac {3}{8} \log (2+3 x)\\ \end {align*}
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Mathematica [A] time = 0.00, size = 24, normalized size = 1.00 \begin {gather*} -\frac {1}{4 x}-\frac {3 \log (x)}{8}+\frac {3}{8} \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^2 (4+6 x)} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.59, size = 21, normalized size = 0.88 \begin {gather*} \frac {3 \, x \log \left (3 \, x + 2\right ) - 3 \, x \log \relax (x) - 2}{8 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.01, size = 20, normalized size = 0.83 \begin {gather*} -\frac {1}{4 \, x} + \frac {3}{8} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) - \frac {3}{8} \, \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 = 19, normalized size = 0.79 \begin {gather*} -\frac {3 \ln \relax (x )}{8}+\frac {3 \ln \left (3 x +2\right )}{8}-\frac {1}{4 x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.38, size = 18, normalized size = 0.75 \begin {gather*} -\frac {1}{4 \, x} + \frac {3}{8} \, \log \left (3 \, x + 2\right ) - \frac {3}{8} \, \log \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 18, normalized size = 0.75 \begin {gather*} -\frac {3\,\ln \left (\frac {x}{6\,x+4}\right )}{8}-\frac {1}{4\,x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.14, size = 20, normalized size = 0.83 \begin {gather*} - \frac {3 \log {\relax (x )}}{8} + \frac {3 \log {\left (x + \frac {2}{3} \right )}}{8} - \frac {1}{4 x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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