  When performing cosmological inference, standard analyses of the
Lyman-$\alpha$ (Ly$\alpha$) three-dimensional correlation functions only
consider the information carried by the distinct peak produced by baryon
acoustic oscillations (BAO). In this work, we address whether this compression
is sufficient to capture all the relevant cosmological information carried by
these functions. We do this by performing a direct fit to the full shape,
including all physical scales without compression, of synthetic Ly$\alpha$
auto-correlation functions and cross-correlations with quasars at effective
redshift $z_{\rm{eff}}=2.3$, assuming a DESI-like survey, and providing a
comparison to the classic method applied to the same dataset. Our approach
leads to a $3.5\%$ constraint on the matter density $\Omega_{\rm{M}}$, which is
about three to four times better than what BAO alone can probe. The growth term
$f \sigma_{8} (z_{\rm{eff}})$ is constrained to the $10\%$ level, and the
spectral index $n_{\rm{s}}$ to $\sim 3-4\%$. We demonstrate that the extra
information resulting from our `direct fit' approach, except for the
$n_{\rm{s}}$ constraint, can be traced back to the Alcock-Paczy\'nski effect
and redshift space distortion information.
