  Consider a third order linear differential equation $L(f)=0$, where
$L\in\mathbb{Q}(z)[\partial_z]$. We design an algorithm computing the
Liouvillian solutions of $L(f)=0$. The reducible cases devolve to the classical
case of second order operators, and in the irreducible cases, only finitely
many differential Galois groups are possible. The differential Galois group is
obtained through optimized computations of invariants and semi-invariants, and
if solvable, the solutions are returned as pullbacks and gauge transformations
of algebraic generalized hypergeometric function ${}_3F_2$. The computation
time is practical for reasonable size operators.
