Find all integer solutions of the Diophantine equation $$b=\sqrt{\sqrt{c}+\sqrt{ac+1}}$$
My work:
Squaring both sides twice $$b^4=c+2\sqrt{c\cdot(ac+1)}+ac+1$$
Since $b$ is an integer, $c$ and $ac+1$ both have to be squares as they have no common factor.
So we can say $$\begin{align} c=y^2\\ ac+1=x^2\\ x+y=b^2\\ \end{align}$$ where $x,y$ are some arbitrary integers. Eliminating $c$, we get $$x^2-ay^2=1$$ Thus we came to the Pell's equation. There are infinitely many solutions of this equation but, solutions with $x+y=$ square are rare. By trial and error one can find $1$ solution or maybe more. But I'm not able to do it by generalized method.
Any help is greatly appreciated.