Kolchin (1973) and van der Put & Singer (2003) give detailed accounts of Picard–Vessiot theory.
Unfortunately it is hard to tell exactly what they proved as the concept of being "solvable by quadratures" is not defined precisely or used consistently in their papers.
Kolchin (1946, 1948) gave precise definitions of the necessary concepts and proved a rigorous version of this theorem.
A Picard–Vessiot extension is Liouvillian if and only if the identity component of its differential Galois group is solvable (Kolchin 1948, p. 38, van der Put & Singer 2003, Theorem 1.39).
More precisely, extensions by algebraic functions correspond to finite differential Galois groups, extensions by integrals correspond to subquotients of the differential Galois group that are 1-dimensional and unipotent, and extensions by exponentials of integrals correspond to subquotients of the differential Galois group that are 1-dimensional and reductive (tori).