In the following integral:
$\displaystyle \int\frac{x+1}{(x-1)\sqrt{x^3+x^2+x}}\text{d}x$
the substitution $t=\displaystyle\frac{x+1}{x-1}$ gives a straightforward integrable form $\displaystyle \int\frac{\text{d}p}{\sqrt{p^2+a}} $. My question is, how does one come up with this substitution? Its not obvious, at least, not to me. I tried $t=1/x$ and $t=\dfrac{1}{x-1}$, which apparently returned a more complex looking expression. So how do I find the proper substitution?
Again, if I try $t=\dfrac{x+1}{x-1}$ for the slightly altered integral,
$\displaystyle \int\frac{x+1}{(x-1)\sqrt{x^3+x^2+2x}}\text{d}x$
it doesn't seem to reduce to a form that I can integrate. How should I approach this one?