  We give an effective upper bound of |Bir(X)| for the birational automorphism
group of an irregular n-fold (with n = 3) of general type in terms of the
volume V = V(X) under an ''albanese smoothness and simplicity'' condition. To
be precise, |Bir(X)| < d_3 V^{10}. An optimum linear bound |Bir(X)|-1 <
(1/3)(42)^3 V is obtained for those 3-folds with non-maximal albanese
dimension. For all n > 2, a bound |Bir(X)| < d_n V^{10} is obtained when alb_X
is generically finite, alb(X) is smooth and Alb(X) is simple.
