As long as the function $g(x)$ is well-behaved, we have the following very important result. Let
This result (and some related ones) is called the Fundamental Theorem of (Integral) Calculus.
Now let us apply that to your problem. We obtain
Use the above equation to solve for $f'(x)$ in terms of $L'(x)$. If you take $L(x)$ as known, you have found an explicit formula for $f'(x)$, and all you need to do is to integrate.
Now comes the unfortunate part. For most pleasant functions $L(x)$, the resulting integration problem will be either difficult or more often impossible (in terms of standard functions).
I hope that this gives you something to play with. You will find out why there is such a limited number of different arclength problems in calculus books!