  We present a conditionally integrable potential, belonging to the
bi-confluent Heun class, for which the Schr\"odinger equation is solved in
terms of the confluent hypergeometric functions. The potential involves an
attractive inverse square root term with arbitrary strength and a repulsive
centrifugal barrier core with the strength fixed to a constant. This is a
potential well defined on the half-axis. Each of the fundamental solutions
composing the general solution of the Schr\"odinger equation is written as an
irreducible linear combination, with non-constant coefficients, of two
confluent hypergeometric functions. We present the explicit solution in terms
of the non-integer order Hermite functions of scaled and shifted argument and
discuss the bound states supported by the potential. We derive the exact
equation for the energy spectrum and approximate that by a highly accurate
transcendental equation involving trigonometric functions. Finally, we
construct an accurate approximation for the bound-state energy levels.
