# 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.

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 $$\int P_0(x)^s\sqrt[n]{(p_0+p_1x+...+p_mx^m)^k}\ dx$$.
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$$\int\sqrt[n]{(p_0+p_1x)^kP(x)^{sn}}\ dx$$ is elementary.

$$\int\sqrt[n]{(p_0+p_1x+...+p_mx^m)^{kn}P(x)^{sn}}\ dx$$ is elementary.
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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.
$$\int\sqrt{(p_0+p_1x+...+p_mx^m)^{2k}P(x)^{2s}}\ dx$$ is elementary.
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