We all know that there are algebraic numbers that can't be expressed by radical. For example the real root of the equation $x^5-x+1=0$ (which is near $-1.16$) is algebraic but can't be expressed by radicals.
We say that a function $f(x)$ is algebraic if there exists a polynomial $p(x,y)$ with integer coefficients such that $p\left(x,f(x)\right)=0$. Obviously, polynomial functions, rational functions and irrational functions are algebraic. Is there an algebraic function (defined by a series or by an integral) that isn't polynomial, rational or irrational?
Rational functions and root functions have a function term that is a radical expression. That are the explicit algebraic functions.
But just as there are algebraic numbers that cannot be represented by radicals, there are algebraic functions (the implicit algebraic functions) whose function term cannot be represented by a radical, see e. g. the algebraic function $f$ that is determined by the irreducible algebraic equation
$$f(x)^5-f(x)+1=0.$$
The function can be represented by series or integrals.
see e.g. Is a polynomial equation of degree $\ge 5$ not solvable by any way?