Suppose I had a function
$$f(x) = \frac{2x}{\sqrt{x^3 + 2x + 5}},$$
that I wanted to integrate on the interval $[\pi, 2\pi]$. Can Gauss quadrature of order $2$ (ie. with two points $\pm\frac{1}{\sqrt{3}}$) integrate this exactly? It seems so as I believe it can integrate all polynomials up to degree $3$? But what about the square root, does this prevent it from integrating the function exactly? If so, what would be a good way to improve the approximation for numerical integration of this function on this domain?
This is not a polynomial, not just because of the square root but also because of the division. So the answer is no. Since you're far away from any singularities, it nevertheless seems reasonable to use quadrature (e.g. Gauss of order $2$) directly without any transformations. I don't see any transformation that would obviously improve the quadrature.