Getting lower/upper bounds for $a$ and $b$, when solving an equation in positive integers

Question

I have the following equation:

$$\frac{a(a + 3)(a(r - 5) + (12 - r))}{9}=\frac{b (9 + b (-14 + r) - r)}{3}\tag1$$

I want to solve that equation in positive integers for $a$, $b$, and $r$.

Is there a way to find a lower/upper bound for one ($a$) or two ($a$ and $b$) variables when I set $r$ to be kown? So that I can use software to look between or in these bounds?

Answer 1

Some solutions $(r,a,b)$ with fixed $r$ (equivalent elliptic equation):

(15, 117, 2340)
(16, 47, 450)
(19, 5, 14)
(24, 3, 6)
(30, 1545, 43860)
(31, 5, 10)
(45, 12, 30)
(46, 9, 20)
(129, 48, 204)
(139, 32, 112)
(177, 9, 18)
(186, 3792, 138336)
(220, 35, 126)
(244, 5, 8)
(553, 45, 180)
(573, 33, 114)
(576, 243, 2214)

Some solutions $(a,b,r)$ with fixed $a$ (equivalent Pell equation):

(5, 8, 244)
(5, 10, 31)
(5, 14, 19)
(9, 18, 177)
(9, 20, 46)
(12, 30, 45)
(32, 112, 139)
(33, 114, 573)
(35, 126, 220)
(45, 180, 553)
(47, 450, 16)
(48, 204, 129)
(63, 294, 3750)
(77, 396, 3889)
(116, 728, 46750)
(117, 2340, 15)
(159, 1166, 6826)
(240, 2156, 2098129)
(243, 2214, 576)
(357, 3906, 72807)
(372, 4154, 2509849)

I think no simple way to find bounds for $a,b$.