  We construct two new exactly solvable potentials giving rise to bound-state
solutions to the Schr\"odinger equation, which can be written in terms of the
recently introduced Laguerre- or Jacobi-type $X_1$ exceptional orthogonal
polynomials. These potentials, extending either the radial oscillator or the
Scarf I potential by the addition of some rational terms, turn out to be
translationally shape invariant as their standard counterparts and isospectral
to them.
