  We present the theoretical treatment of laser-assisted $(e,2e)$ ionizing
collisions in hydrogen for fast electrons, in the framework of the first-order
Born approximation at moderate laser intensities and photon energies beyond the
soft-photon approximation. The interaction of the laser field with the
incident, scattered, and ejected electrons is treated nonperturbatively by
using Gordon-Volkov wave functions, while the atomic dressing is treated by
using first-order perturbation theory. Within this semiperturbative formalism,
we obtain a closed-form formula for the nonlinear triple differential cross
section (TDCS), which is valid for linear as well circular polarizations.
Analytical simple expressions of TDCS are derived in the weak field domain and
low-photon energy limit. It was found that for non-resonant $(e,2e)$ reactions,
the analytical formulas obtained for the atomic matrix element in the
low-photon energy limit give a good agreement, qualitative and quantitative,
with the numerical semiperturbative model calculations. We study the influence
of the photon energy as well of the kinetic energy of the ejected electron on
the TDCS, in the asymmetric coplanar geometry, and show that the dressing of
the atomic target strongly influences the $(e, 2e)$ ionization process. We
discuss the exchange effects between the ejected and scattered electrons in the
TDCS.
