Optimal. Leaf size=53 \[ -\frac {1}{128 x^2}+\frac {9}{128 x}+\frac {27}{128 (3 x+2)}+\frac {9}{128 (3 x+2)^2}+\frac {27 \log (x)}{128}-\frac {27}{128} \log (3 x+2) \]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 53, 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}{128 x^2}+\frac {9}{128 x}+\frac {27}{128 (3 x+2)}+\frac {9}{128 (3 x+2)^2}+\frac {27 \log (x)}{128}-\frac {27}{128} \log (3 x+2) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 44
Rubi steps
\begin {align*} \int \frac {1}{x^3 (4+6 x)^3} \, dx &=\int \left (\frac {1}{64 x^3}-\frac {9}{128 x^2}+\frac {27}{128 x}-\frac {27}{64 (2+3 x)^3}-\frac {81}{128 (2+3 x)^2}-\frac {81}{128 (2+3 x)}\right ) \, dx\\ &=-\frac {1}{128 x^2}+\frac {9}{128 x}+\frac {9}{128 (2+3 x)^2}+\frac {27}{128 (2+3 x)}+\frac {27 \log (x)}{128}-\frac {27}{128} \log (2+3 x)\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 44, normalized size = 0.83 \begin {gather*} \frac {1}{128} \left (\frac {2 \left (81 x^3+81 x^2+12 x-2\right )}{x^2 (3 x+2)^2}+27 \log (x)-27 \log (3 x+2)\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^3 (4+6 x)^3} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.95, size = 79, normalized size = 1.49 \begin {gather*} \frac {162 \, x^{3} + 162 \, x^{2} - 27 \, {\left (9 \, x^{4} + 12 \, x^{3} + 4 \, x^{2}\right )} \log \left (3 \, x + 2\right ) + 27 \, {\left (9 \, x^{4} + 12 \, x^{3} + 4 \, x^{2}\right )} \log \relax (x) + 24 \, x - 4}{128 \, {\left (9 \, x^{4} + 12 \, x^{3} + 4 \, x^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 1.07, size = 43, normalized size = 0.81 \begin {gather*} \frac {81 \, x^{3} + 81 \, x^{2} + 12 \, x - 2}{64 \, {\left (3 \, x^{2} + 2 \, x\right )}^{2}} - \frac {27}{128} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) + \frac {27}{128} \, \log \left ({\left | x \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 42, normalized size = 0.79 \begin {gather*} \frac {27 \ln \relax (x )}{128}-\frac {27 \ln \left (3 x +2\right )}{128}+\frac {9}{128 x}-\frac {1}{128 x^{2}}+\frac {9}{128 \left (3 x +2\right )^{2}}+\frac {27}{128 \left (3 x +2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.38, size = 48, normalized size = 0.91 \begin {gather*} \frac {81 \, x^{3} + 81 \, x^{2} + 12 \, x - 2}{64 \, {\left (9 \, x^{4} + 12 \, x^{3} + 4 \, x^{2}\right )}} - \frac {27}{128} \, \log \left (3 \, x + 2\right ) + \frac {27}{128} \, \log \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.09, size = 41, normalized size = 0.77 \begin {gather*} \frac {\frac {9\,x^3}{64}+\frac {9\,x^2}{64}+\frac {x}{48}-\frac {1}{288}}{x^4+\frac {4\,x^3}{3}+\frac {4\,x^2}{9}}-\frac {27\,\mathrm {atanh}\left (3\,x+1\right )}{64} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.18, size = 46, normalized size = 0.87 \begin {gather*} \frac {27 \log {\relax (x )}}{128} - \frac {27 \log {\left (x + \frac {2}{3} \right )}}{128} + \frac {81 x^{3} + 81 x^{2} + 12 x - 2}{576 x^{4} + 768 x^{3} + 256 x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________