I am reading some stuff on Risch's algorithm here (Wikipedia in French), about finding explicitly some primitive of functions in terms of "elementary functions" (composition of polynomials, log, exp, trigonometric functions, ...).
They mention that $$f(x) = \frac{x}{\sqrt{x^{4}+10x^{2}-96x-71}}$$ has an explicit primitive, of the form $-1/8 \cdot \log(P(x) \sqrt{x^{4}+10x^{2}-96x-71} - Q(x))$ for some $P, Q \in \mathbb Z[x]$. On the other hand, they say that the integral of $$f(x) = \frac{x}{\sqrt{x^{4}+10x^{2}-96x-72}}$$ cannot be expressed using elementary functions.
They claim that this is because the Galois group of (the splitting field of) the polynomial $x^{4}+10x^{2}-96x-72$ is $S_4$ (24 elements) while the one of $x^{4}+10x^{2}-96x-71$ is $D_4$ (8 elements). I don't understand how this can influence Risch's algorithm. Both groups are solvable, so I'm not sure what the problem really is.
Can anyone explain the relation between the Galois groups of $P$ and the expression of $\int \frac{x}{\sqrt{P(x)}} dx$ via elementary functions? I did not find anything in the book "Bronstein, Symbolic Integration I" which does cover Risch's algorithm and a bit of differential Galois theory.