Arc length of quadratic curve

Question

I would like to find the arc length of a curve from $a\le t\le b$, the curve is $t^2A+tB+C$ $$arcLength=\int_{a}^{b}\sqrt{(2At+B)^2+1}\,dt$$ I am having trouble getting rid of $t$ (the variable) (variables $A$, $B$, $a$, $b$, and $C$ are constants)

I need help solving the above integral to get the arc length

Thanks in Advanced

Answer 1

It's already in a good form. The next step, with the standard techniques? A trig substitution $2At+B=\tan\theta$.

Answer 2

More a comment than answer, but easier to enter as an answer.

Wolfy gives four forms for the indefinite integral, one of which is

$\dfrac{\sqrt{(2 A t + B)^2 + 1} (2 A t + B) + \sinh^{-1}(2 A t + B)}{4 A} + constant $