  In this paper, we introduce a family of sextic potentials that are exactly
solvable, and for the first time, a family of triple-well potentials with their
whole energy spectrum and wavefunctions using supersymmetry method. It was
suggested since three decades ago that all "additive" or "translational" shape
invariant superpotentials formed by two combination of functions have been
found and their list was already exhausted by the well-known exactly solvable
potentials that are available in most textbooks and furthermore, there are no
others. We have devised a new family of superpotentials formed by a linear
combination of three functions (two monomials and one rational) and where the
change of parameter function is linear in four parameters. This new family of
potentials with superpotential $W(x,A,B,D,G) = Ax^3 + Bx -\frac{Dx}{1+Gx^2}$
will extend the list of exactly solvable Schr\"odinger equations. We have shown
that the energy of the bound states is rational in the quantum number.
Furthermore, approximating the potential around the central well by a harmonic
oscillator, as a usual practice, is not valid. The two outer wells affect
noticeably the probability density distribution of the excited states. We have
noticed that the populations of the triple-well potentials are localized in the
two outer wells. These results have potential applications to explore more
physical phenomena such as tunneling effect, and instantons dynamics.
