WLOG let a = 1, b = B and c = C. You get . The substitution I used (the standard Euler substitution, just to set the record straight) is . Arguably simpler .....?? Arguably easier ....?? But ...... intuitively way less obvious .. ..??
Personally, I prefer ..... easy to remember for this type of integrand viz. and the algebra for getting the stuff that's needed (dx and x in terms of u) is easy.
Eg. To solve for x in terms of u: square both sides of , subtract from both sides, group x-terms, solve for x.
Note: In my original post I used z (not u) as my new variable, but u's OK I guess.