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

A Schr\"odinger potential involving $x^\frac{2}{3}$ and  centrifugal-barrier terms conditionally integrable in terms of the confluent  hypergeometric functions