No computer algebra system implements a complete decision process for the integration of mixed transcendental and algebraic functions.
The integral from the excellent paper of Schultz may be solved by Maple if you convert the integrand to RootOf notation (Why this is not done internally in Maple is an interesting question?)
int(convert((29*x^2+18*x-3)/(x^6+4*x^5+6*x^4-12*x^3+33*x^2-16*x)^(1/2),RootOf),x);
My experiments suggest Maple has the best implementation of the Risch-Trager-Bronstein algorithm for the integration of purely algebraic integrals in terms of elementary functions (ref: table 1, section 3 of https://arxiv.org/pdf/2004.04910.pdf). However, Maple's implementation does not integrate expressions containing parameters or nested radicals (both of which has some support in AXIOM and FriCAS).
It would seem that some significant progress has been made in computing the logarithmic part of a mixed transcendental-algebraic integral by Miller [1]. Though, as far as I know, no computer algebra system has implemented his algorithm. It is also not clear if Miller's algorithm can deal with parameters, for example, the Risch-Trager-Bronstein algorithm has difficulties with the following pseudo-elliptic integral
$$\int\frac{\left(p x^2-q\right) \left(p x^2-x+q\right)dx}{x \left(p x^2+2 x+q\right) \sqrt{2 p^2x^4+2 p x^3+(4 p q+1) x^2+2 q x+2 q^2}} = - \frac{1}{\sqrt{2}}\log (x) + \frac{1}{\sqrt{2}}\log \left(\sqrt{2} y +2 p x^2+x+2q\right) - \frac{3}{\sqrt{5}}\tanh ^{-1}\left(\frac{\sqrt{5} y}{3 p x^2+3 q+x}\right),$$ where $y=\sqrt{2 p^2 x^4+2 p x^3+(4 pq+1)x^2+2 q x+2 q^2}$. My heuristic computes this integral quickly with the substitution $u=\frac{px^2+q}{p x}$.
[1] Miller, B. (2012). “On the Integration of Elementary Functions: Computing the Logarithmic Part”. Thesis (Ph.D.) Texas Tech University, Dept. of Mathematics and Statistics.