Optimal. Leaf size=46 \[ -\frac {1}{64 x}-\frac {3}{32 (3 x+2)}-\frac {3}{64 (3 x+2)^2}-\frac {9 \log (x)}{128}+\frac {9}{128} \log (3 x+2) \]
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Rubi [A] time = 0.02, antiderivative size = 46, 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}{64 x}-\frac {3}{32 (3 x+2)}-\frac {3}{64 (3 x+2)^2}-\frac {9 \log (x)}{128}+\frac {9}{128} \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)^3} \, dx &=\int \left (\frac {1}{64 x^2}-\frac {9}{128 x}+\frac {9}{32 (2+3 x)^3}+\frac {9}{32 (2+3 x)^2}+\frac {27}{128 (2+3 x)}\right ) \, dx\\ &=-\frac {1}{64 x}-\frac {3}{64 (2+3 x)^2}-\frac {3}{32 (2+3 x)}-\frac {9 \log (x)}{128}+\frac {9}{128} \log (2+3 x)\\ \end {align*}
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Mathematica [A] time = 0.02, size = 39, normalized size = 0.85 \begin {gather*} \frac {1}{128} \left (-\frac {2 \left (27 x^2+27 x+4\right )}{x (3 x+2)^2}-9 \log (x)+9 \log (3 x+2)\right ) \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)^3} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.92, size = 68, normalized size = 1.48 \begin {gather*} -\frac {54 \, x^{2} - 9 \, {\left (9 \, x^{3} + 12 \, x^{2} + 4 \, x\right )} \log \left (3 \, x + 2\right ) + 9 \, {\left (9 \, x^{3} + 12 \, x^{2} + 4 \, x\right )} \log \relax (x) + 54 \, x + 8}{128 \, {\left (9 \, x^{3} + 12 \, x^{2} + 4 \, x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.89, size = 37, normalized size = 0.80 \begin {gather*} -\frac {27 \, x^{2} + 27 \, x + 4}{64 \, {\left (3 \, x + 2\right )}^{2} x} + \frac {9}{128} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) - \frac {9}{128} \, \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 = 37, normalized size = 0.80 \begin {gather*} -\frac {9 \ln \relax (x )}{128}+\frac {9 \ln \left (3 x +2\right )}{128}-\frac {1}{64 x}-\frac {3}{64 \left (3 x +2\right )^{2}}-\frac {3}{32 \left (3 x +2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.39, size = 41, normalized size = 0.89 \begin {gather*} -\frac {27 \, x^{2} + 27 \, x + 4}{64 \, {\left (9 \, x^{3} + 12 \, x^{2} + 4 \, x\right )}} + \frac {9}{128} \, \log \left (3 \, x + 2\right ) - \frac {9}{128} \, \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 = 35, normalized size = 0.76 \begin {gather*} \frac {9\,\mathrm {atanh}\left (3\,x+1\right )}{64}-\frac {\frac {3\,x^2}{64}+\frac {3\,x}{64}+\frac {1}{144}}{x^3+\frac {4\,x^2}{3}+\frac {4\,x}{9}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.18, size = 41, normalized size = 0.89 \begin {gather*} \frac {- 27 x^{2} - 27 x - 4}{576 x^{3} + 768 x^{2} + 256 x} - \frac {9 \log {\relax (x )}}{128} + \frac {9 \log {\left (x + \frac {2}{3} \right )}}{128} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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