The answer may depend on what limitations you place on f, g and h in terms of continuity and differentiability. If f is continuous I can almost trivially construct functions g and h that would satisfy your criteria but be only piecewise continuous.
Let me struggle through this in my own way.
I find it useful to give familiar-looking names to things wherever possible.
So let's put
z-1(c) = y
and ∫z(x)dx = Z,
and I'll call the integral-derivative we have to evaluate F.
So we have to evaluate F = (d/dc)( Z(1) -Z(y) ),
How does this sound?
Your denominator is a polynomial in k; so it can be written a product of terms like (k - zi) where the zi are zeros of the polynomial. If you break this product into partial fractions, you'll automatically get a log on integration.