Euler's substitution:

Integrals of the form $\int R\left(x,\sqrt{\left.a{x}^2+bx+c\right)}\right.dx$ are calculated with the aid of one of the following three Euler substitutions:

i. $\sqrt{a{x}^2+bx+c}=t\pm x\sqrt{a}$ if $a>0$

ii. $\sqrt{a{x}^2+bx+c}=tx\pm \sqrt{c}$ if $c>0$

iii.$\sqrt{a{x}^2+bx+c}=(x-a)t$ if $a{x}^2+bx+c=a(x-a)(x-b)$

i.e., if $\alpha$ is a real root of $a{x}^2+bx+c=0$