$x^3+Ax+B$ is not a square number but $x^3+Cx+D$ is a square

Question

Determine 4 distinct $A, B, C, D$ for any rational $x$ such that

1. either $x^3+Ax+B$ is not a square number but $x^3+Cx+D$ is a square.

or,

2. $x^3+Cx+D$ is not a square number but $x^3+Ax+B$ is a square.

What are the value of $A, B, C, D$? how to approach this problem?

this post is a consequence of a previous post