The approved answer has caused some risibility at mathoverflow, and I'll elaborate on Robin's
more reasonable comment (but I'm inclined to attribute the descent argument in this case to
Euler--at least he wrote it down). The version I give in an undergrad number theory class is this: First one develops the standard facts about Z[w] where w^2+w+1=0. (It has unique
factorization, 2 is prime, the units are 1,-1,w,-w,w^2 and -w^2, any element not 0 or a unit
has absolute value >1, and each congruence class mod 2 is represented by 0 or a unit). Then one notes that it's enough to prove:
Theorem--There are no a,b,c in R with a+b+c=0, abc a non-zero cube and a=b=1 mod 2.
The proof of the theorem is a reductio. Let H be max(/a/,/b/,/c/) and choose a solution
a,b,c with minimal H. (H^2 is an integer). a, b and c are evidently pairwise prime. Since
their product is a non-zero cube, each is (unit)(cube). Since a=b=1 mod 2, a=A^3 and
b=B^3, and we may assume that A=B=1 mod 2. Since abc is a cube, c=C^3 for some C in R. Since
2 divides c, 2 divides C and H is at least 8.
Now let S=Aw+Bw^2, T=Aw^2+Bw, and U=A+B. Then S+T+U=0 while STU is A^3+B^3=-C^3. Also S=T=1 mod 2, while max(/S/,/T/,/U/) is at most 2(H^(1/3)). This contradicts the minimality assumption.
As you know, your curve may have infinitely many rational points. Now suppose it has a rational point $(r/t,s/t)$, so $s^2/t^2=r^3/t^3+Ar/t+B$, so $(st^2)^2=(rt)^3+At^4(rt)+Bt^6$, so $(rt,st)$ is an integral point on the elliptic curve $y^2=x^3+A'x+B'$. It follows that there is no finite bound to the number of integral points on an elliptic curve. It suggests that the bigger the coefficients, the more integral points are possible.
A related question that may interest you is that of the rank of your curve, the number of independent generators of the group of rational points. It is believed, but, I think, not proved, that the rank is unbounded, but it's hard to find examples with large rank (where large might mean more than 20), and people do go to some effort to set new records. According to Wikipedia, curves with rank at least 28 are known.
Best Answer
Yes, one can detect if a point $P$ has infinite order, if one is allowed to apply the theorem of Mazur that classifies all the possible torsion group of elliptic curves defined over $\mathbb{Q}$, see Wikipedia link.
From the theorem one knows that if a point $P$ has order greater than $12$ then the order is infinite. So all you need to check is if $nP = \mathcal{O}$ for any $0 \leq n \leq 12$. Otherwise, the order of the point is infinite.