Bott, Cattaneo and Rossi defined invariants of long knots $\mathbb R^n \hookrightarrow \mathbb R^{n+2}$ as combinations of configuration space integrals for $n$ odd $\geq 3$. Here, we give a more flexible definition of these invariants. Our definition allows us to interpret these invariants as counts of diagrams. It extends to long knots inside more general $(n+2)$-manifolds, called asymptotic homology $\mathbb R^{n+2}$, and provides invariants of these knots.