  Let $R$ be a (P.I.D) and let $T(V),\partial)$ be a free $R$-dga. The
quasi-isomorphism type of $(T(V),\partial)$ is the set, denoted
$\{(T(V),\partial)\}$, of all free dgas which are quasi-isomorphic to
$(T(V),\partial)$. In this paper we investigate to characterize and to compute
the set $\{(T(V),\partial)\}$ for a new class of free dgas called perfect (a
special kind of a perfect dga is the Adams-Hilton model of simply connected
CW-complex such that $H_{*}(X,R)$ is free). We show that if $(T(V),\partial)$
and $(T(W),\delta)$ are two perfect dgas, then $(T(W),\delta)\in
\{(T(V),\partial)\}$ if and only if their Whitehead exact sequences are
isomorphic. Moreover we show that every dga $(T(V),\partial)$ can be split to
give a pair $\big((T(V),\widetilde{\partial}),(\pi_{n})_{n\geq 2}\big)$
consisting with a perfect dga $(T(V),\widetilde{\partial})$ and a family of
extensions $(\pi_{n})_{n\geq 2}$ and we establish that if
$(T(W),\widetilde{\delta})\in \{(T(V),\widetilde{\partial})\}$ and if the
extensions $(\pi_{n})_{n\geq 2}$ and $(\pi'_{n})_{n\geq 2}$ are isomorphic (in
a certain sense), then $(T(W),\delta)\in \{(T(V),\partial)\}$.
