I asked [a question](https://math.stackexchange.com/questions/500589/integrals-of-sqrtx-sqrt-phantom-dots-sqrtx1-in-elementary-functions) at Math.SE last year and later offered a bounty for it, but it remains unsolved even in the simplest case. So I finally decided to repost this case here:


Is it possible to express the following indefinite integral in elementary functions?
$${\large\int}\sqrt{x+\sqrt{x+\sqrt{x+1}}}\ dx$$



Is it possible to express $\int\sqrt{x+\sqrt{x+\sqrt{x+1}}}dx$ in elementary functions?
The answer is 'no'. Making the substitution
$$
x = \frac{(t-1)(t-5)(t^2+2t+5)}{16t^2},
$$
one finds
$$
{\textstyle\sqrt{x+\sqrt{x+\sqrt{x+1}}}\,\mathrm{d}x}
 = \frac{(t^2-2t+5)(t^2-5)\sqrt{t^4{-}2t^2{-}40t+25}\ \mathrm{d}t}{32t^4}.
$$
Denote the right hand side of the above equation by $\beta$. Now, setting 
$$
Q(t) = \frac{(25-25t-14t^2-3t^3+t^4)}{96t^3},
$$
one finds that 
$$
\beta = \mathrm{d}\left(Q(t)\sqrt{t^4{-}2t^2{-}40t+25}\right)
+ \frac{(10+15t-2t^2+t^3)\,\mathrm{d}t}{8t \sqrt{t^4-2t^2-40t+25}}\,
$$
The second term on the right hand side is in normal form for differentials on the elliptic curve $s^2 = t^4-2t^2-40t+25$ that are odd with respect to the involution $(t,s)\mapsto(t,-s)$. Since this term represents a differential that has two poles of order $2$ (over the two points where $t=\infty$) and two poles of order $1$ (over the two points where $t=0$), an application of Liouville's Theorem (on integration in elementary terms, with the differential field taken to be the field of meromorphic functions on the elliptic curve) shows that this term is not integrable in elementary terms. Hence $\beta$ is not, and hence the original integrand is not.


