Integrability of a polynomial under root Under what conditions can $$\int\sqrt[n]{a_0\cdot x^m+a_1 \cdot x^{m-1}+...+a_m}\ dx$$ be expressed in terms of elementary functions?
$\sqrt{x^2+3x+5}$ can be expressed, while $\sqrt{x^3+3x+5}$ not. 
 A: I don't have a general, comprehensive theory. I have only some examples of somewhat more general cases that I have derived from integral tables.
Let $k,m,s\in\mathbb{N_0}$,
let $n\in\mathbb{N_{>0}}$,
let $p_0,p_1,...,p_m\in\mathbb{C}$,
let $P,P_0$ denote polynomial functions,
let $R$ denote a rational function.
I found that $\int\sqrt[n]{(p_0+p_1x+...+p_mx^m)^kP(x)^{sn}}\ dx$ is included in $\int R(x,\sqrt[n]{(p_0+p_1x+...+p_mx^m)^k})\ dx$ because it can be attributed to integrals of the form $\int P_0(x)^s\sqrt[n]{(p_0+p_1x+...+p_mx^m)^k}\ dx$.
$\ $
$\int\sqrt[n]{(p_0+p_1x)^kP(x)^{sn}}\ dx$ is elementary.
$\int\sqrt[n]{(p_0+p_1x+...+p_mx^m)^{kn}P(x)^{sn}}\ dx$ is elementary.
$\ $
for $n=2$:
Let's consider the  integrals $\int R(x,\sqrt{p_0+p_1x+...+p_mx^m})\ dx$.  
Under certain conditions, some of these integrals are elementary. The conditions are written e.g. in:
Goursat, É: A Course in Mathematical Analysis
and in
Gradshteyn, I. S.; Ryzhik, I. M.: Table of Integrals, Series, and Products.  
The cases $m=0$ and $m=1$ are given above.
case $m=2$:
$\int\sqrt{(p_0+p_1x+p_2x^2)^kP(x)^{2s}}\ dx$ is elementary.
The cases $m=3$ and $m=4$ with radicands without multiple roots are called the elliptic integrals. Those elliptic integrals that have an elementary antiderivative are called pseudo-elliptic integrals.
The cases $m\ge 5$ with radicands without multiple roots are called the hyperelliptic integrals.
For real-valued radicands, the remarks about multiple roots doesn't matter.
$\int\sqrt{(p_0+p_1x+...+p_mx^m)^{2k}P(x)^{2s}}\ dx$ is elementary.
$\ $
Maybe an algorithm derived from Risch algorithm can decide if a whole class of integrands has elementary antiderivatives. Maybe the following can help.
Ng, E. W.: Symbolic integration of a class of algebraic functions. [by an algorithmic approach]. 1974
Bertrand, L.: On the implementation of a new algorithm for the computation of hyperelliptic integrals. In: Proceedings of the international symposium on Symbolic and algebraic computation - ISSAC '94. 1994
Bertrand, L.: Symbolic Computation of Hyperelliptic Integrals and Arithmetic in the Jacobian of the Curve. 1995
Labahn, G.; Mutrie, M.: Reduction of Elliptic Integrals to Legendre Normal Form. 1997
Carlson, B. C.: Toward Symbolic Integration of Elliptic Integrals. 1999
Krason, P.; Milewski, J.: New approach to certain real hyper-elliptic integrals. 2019
