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[tex]\int R(x,\sqrt{ax^2+bx+c})[/tex]

You can use Euler's substitutions:

[tex]1. \sqrt{ax^2+bx+c} = t \pm \sqrt{a}x, a>0 [/tex]

[tex]2. \sqrt{ax^2+bx+c} = tx \pm \sqrt{c}, c>0 [/tex]

[tex]3. \sqrt{ax^2+bx+c} = \sqrt{a(x-{x_{1}})(x-x_{2})}=t(x-x_{1})=t(x-x_{2}) [/tex]

if [tex] x_{1}, x_{2} [/tex] are real.

I understand how to mechanically evaluate the integrals but cannot find any justification or derivation of the substitution formulas, especially when to use, for example, t+x as opposed to t-x in the first instance.Can anyone help please? Thanks!