Take
$$f(x)=2x\arccos\left(\frac{x^2+d^2-1}{2xd}\right)$$
and try and find
$$ I(x)=\int_{d-1}^{3}dx f(x) \sqrt{\left(\frac{x-1}{x}\right)}\left(3-x\right)^3 $$
You'll find the result is messy (see Simple Integral Involving Radicals: Why Does Mathematica Fail?)
This is caused by the $\arccos$, which in the referenced question has been approximated by a Taylor series about $x=d-1$ truncated to the first term: even in truncated form the integral is all over the place.
Can anyone think of a good approximation to the $\arccos$ function valid when $x\approx d-1$ that doesn't involve radicals, allowing easier integration of $I(x)$?