When we are faced with integrals like this, how would one know if the integral is solvable or unsolvable
$$f(x) = \int \sqrt{ P(t)} \,dt$$
By solvable I mean the integral can be broken down into an elementary function, and by unsolvable I mean the integral cannot be broken that to an elementary function “ Elliptic integral “ when the $P$ is a polynomial of degree $3$ or $4$
All elliptic integrals can be written in terms of three standard types: first, second and third $$F(x ; k) = \int_{0}^{x} \frac{dt}{\sqrt{\left(1 - t^2\right)\left(1 - k^2 t^2\right)}}$$ $$E(x;k) = \int_0^x \frac{\sqrt{1-k^2 t^2} }{\sqrt{1-t^2}}\,dt$$ $$\Pi(n ; \varphi \,|\,m) = \int_{0}^{\sin \varphi} \frac{1}{1-nt^2} \frac{dt}{\sqrt{\left(1-m t^2\right)\left(1-t^2\right) }}$$
Question: let $A$ and $B$ be polynomials of degree $2$,$3$ or $4$, without attempting to solve the integral, what is the condition that determines if the integral is elliptic or not $$f(x) = \int \sqrt{\frac{A(t)}{B(t)}} \,dt$$
