Sum of square roots of the difference between an arbitrary integer and a square

Question

I would like to know if there exists a closed form expression for the following series: $$\sum_{b=1}^{bmax} \sqrt{N-b^2}-b+1$$ where $ N>b^2$. I tried looking at the solutions to Sum of Square roots formula., but since in this case $N$ can be arbitrary, I am a bit stuck. Is it possible to find out and expression both the real and integer root case? Thanks in advance.