  We construct an expansion of the solutions of the bi-confluent Heun equation
in terms of the Hermite functions. The series is governed by a three-term
recurrence relation between successive coefficients of the expansion. We
examine the restrictions that are imposed on the involved parameters in order
that the series terminates thus resulting in closed-form finite-sum solutions
of the bi-confluent Heun equation. A physical application of the closed-form
solutions is discussed. We present the five six-parametric potentials for which
the general solution of the one-dimensional Schr\"odinger equation is written
in terms of the bi-confluent Heun functions and further identify a particular
conditionally integrable potential for which the involved bi-confluent Heun
function admits a four-term finite-sum expansion in terms of the Hermite
functions. This is an infinite well defined on a half-axis. We present the
explicit solution of the one-dimensional Schr\"odinger equation for this
potential and discuss the bound states supported by the potential. We derive
the exact equation for the energy spectrum and construct an accurate
approximation for the bound-state energy levels.
