  We use techniques in the shuffle algebra to present a formula for the
partition function of a one-dimensional log-gas comprised of particles of
(possibly) different integer charges at certain inverse temperature $\beta$ in
terms of the Berezin integral of an associated non-homogeneous alternating
tensor. This generalizes previously known results by removing the restriction
on the number of species of odd charge. Our methods provide a unified framework
extending the de Bruijn integral identities from classical $\beta$-ensembles
($\beta$ = 1, 2, 4) to multicomponent ensembles, as well as to iterated
integrals of more general determinantal integrands.
