The magnetic moment operator being
${\mu} = -\frac{\mu_B}{\hbar}(L+2\,S)$,
a $$\mu = -\frac{\mu_\text{B}}{\hbar}(L+2S),$$ a direct evaluation of its absolute value would be
$|\mu| = (\mu^2)^{1/2} = \frac{\mu_B}{\hbar}(L^2+4\,S^2+4\,L\cdot S)^{1/2}$
whose $$|\mu| = (\mu^2)^{1/2} = \frac{\mu_\text{B}}{\hbar}(L^2 + 4S^2 + 4L\cdot S)^{1/2}$$ whose expectation value is
$\langle|\mu|\rangle = \mu_B \left[ 2j(j+1) - l(l+1) +2\,s(s+1) \right]^{1/2}$.
in $$\langle|\mu|\rangle = \mu_\text{B} \left[ 2j(j+1) - l(l+1) +2s(s+1)\right]^{1/2}$$ in which we have used the equality $~~2\,L\cdot S = J^2 - L^2 - S^2$$2L \cdot S = J^2 - L^2 - S^2$.
Now, a different approach could be to use the proportionality of $\mu$ and $J$ through the Landè g-factor $\mu = -g_J \frac{\mu_\text{B}}{\hbar}J$,
$\mu = -g_J \frac{\mu_B}{\hbar} J$
which which would give
$\langle|\mu|\rangle = (\mu^2)^{1/2} = \frac{\mu_B}{\hbar}\,g_J(J^2)^{1/2} = \mu_B\,g_J[j(j+1)]^{1/2} = \mu_B[(\frac{3}{2}+\frac{s(s+1)-l(l+1)}{2j(j+1)})][j(j+1)]^{1/2}$
which $$\langle|\mu|\rangle = (\mu^2)^{1/2} = \frac{\mu_\text{B}}{\hbar}g_J(J^2)^{1/2} = \mu_\text{B} g_J[j(j+1)]^{1/2} \\ = \mu_\text{B} \left[\frac{3}{2}+\frac{s(s+1)-l(l+1)}{2j(j+1)}\right][j(j+1)]^{1/2}$$ which is clearly different from the equation above. Putting some values for $l$, $s$, and $j$, one indeed obtains different results.
I tend to think that the first result is correct, but I cannot think about valid reasons to confute the second derivation.
