  We present a bi-confluent Heun potential for the Schr\"odinger equation
involving inverse fractional powers and a repulsive centrifugal-barrier term
the strength of which is fixed to a constant. This is an infinite potential
well defined on the positive half-axis. Each of the fundamental solutions for
this conditionally integrable potential is written as an irreducible linear
combination of two Hermite functions of a shifted and scaled argument. We
present the general solution of the problem, derive the exact energy spectrum
equation and construct a highly accurate approximation for the bound-state
energy levels.
