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Store pointers or objects in classes?

Just a design / optimization issue. When do you store pointers or objects and why? For example, I believe that both of these work (excluding compilation errors):

class A{
  std::unique_ptr<Object> object_ptr;
};

A::A():object_ptr(new Object()){}

class B{
  Object object;
};

B::B():object(Object()){}

I believe that one difference occurs when creating an instance on the stack or heap?

For instance:

   int main(){
      std::unique_ptr<A> a_ptr;
      std::unique_ptr<B> b_ptr;
      a_ptr = new A(); //(*object_ptr) on heap, (*a_ptr) on heap?
      b_ptr = new B(); //(*object_ptr) on heap, object on heap?

      A a;   //a on stack, (*object_ptr) on heap?
      B b;   //b on stack, object on stack?
}

In addition, there sizeof(A)should be < sizeof(B)? Are there any other problems that I am missing? (Daniel reminded me of the issue of inheritance in his related article in the comments)

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