What is the class/set which integral the operator map algebraic function to

Question

Integral is an operator, and it map a function to other one, $\int f(x_1,x_2,\dots,x_n)dx_1dx_2\dots dx_n=F(x_1,x_2,\dots,x_n)$.

Now if $f(x_1,x_2,\dots,x_n)dx_1dx_2\dots dx_n$ is defined over $\mathbb{C}$ and is algebraic function,then,what class is $F(x_1,x_2,\dots,x_n)$?