  Integration by parts plays a crucial role in mathematical analysis, e.g.,
during the proof of necessary optimality conditions in the calculus of
variations and optimal control. Motivated by this fact, we construct a new,
right-weighted generalized fractional derivative in the Riemann-Liouville sense
with its associated integral for the recently introduced weighted generalized
fractional derivative with Mittag-Leffler kernel. We rewrite these operators
equivalently in effective series, proving some interesting properties relating
to the left and the right fractional operators. These results permit us to
obtain the corresponding integration by parts formula. With the new general
formula, we obtain an appropriate weighted Euler-Lagrange equation for dynamic
optimization, extending those existing in the literature. We end with the
application of an optimization variational problem to the quantum mechanics
framework.
