Optimal. Leaf size=39 \[ \frac {1}{32 (2+3 x)^2}+\frac {1}{32 (2+3 x)}+\frac {\log (x)}{64}-\frac {1}{64} \log (2+3 x) \]
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Rubi [A]
time = 0.01, antiderivative size = 39, 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 = {46}
\begin {gather*} \frac {1}{32 (3 x+2)}+\frac {1}{32 (3 x+2)^2}+\frac {\log (x)}{64}-\frac {1}{64} \log (3 x+2) \end {gather*}
Antiderivative was successfully verified.
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Rule 46
Rubi steps
\begin {align*} \int \frac {1}{x (4+6 x)^3} \, dx &=\int \left (\frac {1}{64 x}-\frac {3}{16 (2+3 x)^3}-\frac {3}{32 (2+3 x)^2}-\frac {3}{64 (2+3 x)}\right ) \, dx\\ &=\frac {1}{32 (2+3 x)^2}+\frac {1}{32 (2+3 x)}+\frac {\log (x)}{64}-\frac {1}{64} \log (2+3 x)\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 29, normalized size = 0.74 \begin {gather*} \frac {1}{64} \left (\frac {6 (1+x)}{(2+3 x)^2}+\log (-6 x)-\log (4+6 x)\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 32, normalized size = 0.82
method | result | size |
risch | \(\frac {\frac {3 x}{32}+\frac {3}{32}}{\left (2+3 x \right )^{2}}+\frac {\ln \left (x \right )}{64}-\frac {\ln \left (2+3 x \right )}{64}\) | \(28\) |
norman | \(\frac {-\frac {3}{16} x -\frac {27}{128} x^{2}}{\left (2+3 x \right )^{2}}+\frac {\ln \left (x \right )}{64}-\frac {\ln \left (2+3 x \right )}{64}\) | \(31\) |
default | \(\frac {1}{32 \left (2+3 x \right )^{2}}+\frac {1}{64+96 x}+\frac {\ln \left (x \right )}{64}-\frac {\ln \left (2+3 x \right )}{64}\) | \(32\) |
meijerg | \(\frac {3}{128}+\frac {\ln \left (x \right )}{64}-\frac {\ln \left (2\right )}{64}+\frac {\ln \left (3\right )}{64}-\frac {3 x \left (\frac {9 x}{2}+4\right )}{256 \left (1+\frac {3 x}{2}\right )^{2}}-\frac {\ln \left (1+\frac {3 x}{2}\right )}{64}\) | \(38\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 30, normalized size = 0.77 \begin {gather*} \frac {3 \, {\left (x + 1\right )}}{32 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} - \frac {1}{64} \, \log \left (3 \, x + 2\right ) + \frac {1}{64} \, \log \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.50, size = 50, normalized size = 1.28 \begin {gather*} -\frac {{\left (9 \, x^{2} + 12 \, x + 4\right )} \log \left (3 \, x + 2\right ) - {\left (9 \, x^{2} + 12 \, x + 4\right )} \log \left (x\right ) - 6 \, x - 6}{64 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.04, size = 27, normalized size = 0.69 \begin {gather*} \frac {3 x + 3}{288 x^{2} + 384 x + 128} + \frac {\log {\left (x \right )}}{64} - \frac {\log {\left (x + \frac {2}{3} \right )}}{64} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.97, size = 27, normalized size = 0.69 \begin {gather*} \frac {3 \, {\left (x + 1\right )}}{32 \, {\left (3 \, x + 2\right )}^{2}} - \frac {1}{64} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) + \frac {1}{64} \, \log \left ({\left | x \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.13, size = 29, normalized size = 0.74 \begin {gather*} \frac {1}{32\,\left (3\,x+2\right )}-\frac {\ln \left (\frac {6\,x+4}{x}\right )}{64}+\frac {1}{8\,{\left (6\,x+4\right )}^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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