Computing an integral : Mathematica seems to say it’s diverging or not defined and I think it is

I would like to compute :

Element[a, Integers] Element[b, Integers] Integrate[Exp[-a^2*(x (1 - x/b^2))^2], {x, 0, Infinity}] 

which is obviously not diverging. But for some reason Mathematica can’t solve it. So I tried to change the variable : $ x \rightarrow x (1 – x/b^2)$ . Now :

Integrate[Exp[-a^2*x^2]*1/(1 - (2 x)/b^2), {x, 0, Infinity}] 

And then I get the answer :

ConditionalExpression[1/8 b^2 E^(-(1/4) a^2 b^4) (2 \[Pi] Erfi[1/2 Sqrt[a^2] b^2] + 2 ExpIntegralEi[(a^2 b^4)/4] + 2 Log[a^2] + Log[1/(a^2 b^4)]  - 4 Log[-(1/b^2)] - Log[a^2 b^4]),  (Re[b^2] < 0 || b^2 \[NotElement] Reals) && Re[a^2] > 0] 

which implies that $ b^2<0$ , because of the $ – 4 Log[-(1/b^2)]$ I assume. But I don’t get it. Even if $ b^2>0$ it’s converging ! Could it mean it exists but it can’t be solved analytically ?

Or what’s the mistake ? And is there a way to go around that issue ?