The specific space that you mention---smooth functions with compact support---is not a $DF$ space.  Under certain situations, an $LF$ space can be a $DFN$ space, e.g., when it is a strict inductive limit of finite dimensional spaces.  In general, a (strict) $LF$ space as introduced in the seminal article by Dieudonné and Schwartz is NOT a $DF$ space but it CAN be in special circumstances, e.g., if it is the strict inductive limit of a sequence of Banach spaces.  Good references are "Espaces vectorielles topologiques"  by Grothendieck (available also in english translation) and the first volume of " Topological vector spaces " by Köthe.