Nice idea!
As long as the function $g(x)$ is well-behaved, we have the following very important result. Let $$G(x)=\int_c^x g(t)dt$$. Then $G'(x)=g(x)$.
This result (and some related ones) is called the Fundamental Theorem of (Integral) Calculus.
Now let us apply that to your problem. We obtain $$L'(x)=\sqrt{1+(f'(x))^2}$$
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!