  In this paper, we present a method of deriving extended Prelle-Singer
method's quantifiers from Darboux Polynomials for third-order nonlinear
ordinary differential equations. By knowing the Darboux polynomials and its
cofactors, we extract the extended Prelle-Singer method's quantities without
evaluating the Prelle-Singer method's determining equations. We consider three
different cases of known Darboux polynomials. In the first case, we prove the
integrability of the given third-order nonlinear equation by utilizing the
Prelle-Singer method's quantifiers from the two known Darboux polynomials. If
we know only one Darboux polynomial, then the integrability of the given
equation will be dealt as case $2$. Likewise, case $3$ discuss the
integrability of the given system where we have two Darboux polynomials and one
set of Prelle-Singer method quantity. The established interconnection not only
helps in deriving the integrable quantifiers without solving the underlying
determining equations. It also provides a way to prove the complete
integrability and helps us in deriving the general solution of the given
equation. We demonstrate the utility of this procedure with three different
examples.
