Reference Request: Quadratic Bivariate Diophantine Equations

I'm interested in Diophantine equations in two variables of degree up to 2; so

$$ax^2+bxy+cy^2+dx+ey+f=0$$

where $$a,b,c,d,e,f$$ are integers and we are looking for all integer solutions. All over the internet it is claimed that there is a general algorithm for finding all integer solutions of this, that was first described by Lagrange (this is claimed for example here and here, and here about a more restricted equation). However, I wasn't able to find such an algorithm in any book on elementary number theory or on diophantine equations. Absurdly, even the book "Quadratic Diophantine Equations" by Titu doesn't have an algorithm for the full problem above! (obviously, it has algorithms for special cases.) The only source I found that claims it describes an algorithm is this webpage, but it is rather informal and I'd like to see a full proof in a serious book. Is there a book or (less preferably) a paper that describes the full algorithm?