The postmodern story centered around supersymmetry (SUSY) tells us
that although it started as a failure in experimental high-energy
physics, it quickly found plan B as a successful and very productive
solvable-model-building method in 1D quantum mechanics. For the
present purposes the initial reference A. Sinha and P. Roy,
Czechoslovak J. Phys. 54 (2004) 129 - 138 used the SUSY technique
and produced the large family of solvable potentials including also
the one mentioned in the title. The rediscovery (not using SUSY) of
one particular member of the family came with Cari\v{n}ena et al [1]
in 2008. In a very detailed comment on reference [1] a return to
SUSY origins has been added, by Fellows and Smith [23], in 2009.
They were not aware of the Sinha's and Roy's paper -- this I already
mentioned in my recent comment MR 2525883 on the latter comment.
Unfortunately, they did not pay attention to the 1D variable-mass
Schr\"{o}dinger equations to which SUSY is also known to apply
extremely well. The paper under review partially fills the gap by
taking the Cari\v{n}ena's ``newly discovered solvable nonlinear
oscillator" and by allowing the variability of the ``mass" $m=m(x)$.
Making three concrete choices of the latter function, the authors
construct three new models. Perhaps, it's time for some fourth team
to rederive these results using SUSY.

MR2545636 Kraenkel, R. A.; Senthilvelan, M. On the solutions of the
position-dependent effective mass Schrödinger equation of a
nonlinear oscillator related with the isotonic oscillator. J. Phys.
A 42 (2009), no. 41, 415303, 10 pp. 81Qxx