  The solution of the one-dimensional Schr\"odinger equation for a potential
involving an attractive $x^\frac{2}{3}$ and a repulsive centrifugal-barrier
$\sim x^{-2}$ terms is presented in terms of the non-integer-order Hermite
functions. The potential belongs to one of the five bi-confluent Heun families.
This is a conditionally integrable potential in that the strength of the
centrifugal-barrier term is fixed. The general solution of the problem is
composed using fundamental solutions each of which presents an irreducible
linear combination of two Hermite functions of a scaled and shifted argument.
The potential presents an infinitely extended confining well defined on the
positive semi-axis and sustains infinitely many bound states.
