Integer solutions of (x+1)(xy+1)=z^3 Consider the equation
$$(x+1)(xy+1)=z^3,$$
where $x,y$ and $z$ are positive integers with $x$ and $y$ both at least $2$ (and so $z$ is necessarily at least $3$). For every $z\geq 3$, there exists the solution
$$x=z-1 \quad \text{and} \quad y=z+1.$$
My question is, if one imposes the constraint that
$$x^{1/2} \leq y \leq x^2 \leq y^4,$$
can there be any other integer solutions (with $x,y \geq 2$)? 
Moreover, $z$ may be assumed for my purposes to be even with at least three prime divisors.
Thank you in advance for any advice.
Edit: there was a typo in my bounds relating $x$ and $y$.

Here is another family of solutions.
For natural $q$:
$x=3,y=144 q^{3} + 360 q^{2} + 300 q + 83,z=12q+10$.
Yours is linear in $y$, so solutions are positive integers
of the form $y=\frac{z^{3} -  x - 1}{x^{2} + x}$.
Plugging small values of $x$, more solutions may come from
integers:
x= 2   y= (1/6) * (z^3 - 3)
x= 3   y= (1/12) * (z^3 - 4)
x= 4   y= (1/20) * (z^3 - 5)
x= 5   y= (1/30) * (z^3 - 6)
x= 6   y= (1/42) * (z^3 - 7)
x= 7   y= (1/56) * (z - 2) * (z^2 + 2*z + 4)
x= 8   y= (1/72) * (z^3 - 9)
x= 9   y= (1/90) * (z^3 - 10)
x= 10   y= (1/110) * (z^3 - 11)
x= 11   y= (1/132) * (z^3 - 12)
x= 12   y= (1/156) * (z^3 - 13)
x= 13   y= (1/182) * (z^3 - 14)
x= 14   y= (1/210) * (z^3 - 15)

