Motivation behind Euler substitution in Integrals

Question

As the title suggests, I want to know the motivation behind the Euler substitutions.How someone arrived at these substitutions.

To remove any doubts, by euler substitutions i mean -

If we have a rational function of $x$ and $\sqrt{ax^2+bx+c}$ ,

Case I. If $a>0$ , put $\sqrt{ax^2+bx+c}=t \pm x\sqrt{a}$

Case II. If $c>0$ , put $\sqrt{ax^2+bx+c}=tx \pm \sqrt{c}$

Case III. If the trinomial has real roots m,n, then put $\sqrt{ax^2+bx+c}= t(x-m)$

Answer 1

When considering $t=\sqrt{ax^2+bx+c}$ , you will find that $x$ in terms of $t$ is tedious.

But when considering e.g. $\sqrt{ax^2+bx+c}=t\pm x\sqrt{a}$ , $\sqrt{ax^2+bx+c}=tx\pm\sqrt{c}$ instead, you will find that $x$ in terms of $t$ are easier and give the rational functions.