"that this integral is uncomputable; "
Yes, that is correct. This problem is from the book Handbook of ﬁrst order partial diﬀerential equations, Volume 1, by Polyanin, Zaitsev, Moussiaux (2002).
This is problem 22.214.171.124 at page 38
The book gives only the principal integral to each pde. The the principal integral for this PDE is
So the general solution is an arbitrary function of the above principal integral. The integral can be left unevaluated since it has no antiderivative. This is how the book gives solutions to all the PDE's in this chapter. Here is from the introduction