@sand15 

Thank you. There must be more than one path to deal with the unavoidable case analysis depending on the relations between a, b, c, d, t.  I have updated my response for a case where b abd c cannot be freely chosen.

Since the location of the pole at x=-t/d with respect to m and n is important I would (for a new attempt) rewrite the problem this way

(f*x^2 + g*x + h)/(x + p);
                            2          
                         f x  + g x + h
                         --------------
                             x + p     

to have one parameter less to deal with. Maybe helpfull if the OP is interesed to reduce the set of possible solutions.

@acer
Now its clear. There is use for functional operators and unapply, which both can produce the same output.
Thank you for the detailed answer.

@acer 

Beautiful! Your answer anticipates a follow-up question.

However, I still haven't figured out why my "poor man's attempts" didn't work.
In the meantime I found that the arrow operator and unapply are treated differently.

Using unapply, my attempts can be fixed. 

for i from 1 to 5 do 
   x-> `^`(x,i);
   unapply(`^`(x,i),x);
end do;

But this is somehow not satisfying because up until now I thought unapply and -> are equivalent.  Aren't they?

x -> `^`(x, i);
unapply(`^`(x, i), x);
         proc (x) options operator, arrow; x^i end proc

         proc (x) options operator, arrow; x^i end proc

@acer Thank you. That makes sense.

An example of "the" equations would be helpful to better understand what you are trying to achieve

there are no determinants in your post

@Ronan 

The start page is another hard to reproduce trouble maker. Have you tried to disable it.

@acer Thanks for the 'Simple way' to check

What you are describing sounds very familiar to me (and I think I have even posted something here, that I cannot find straight away). It is most likely not a bug but related to Windows (7 and 10) and your personal setup.

A few questions:

What happens if you wait (for me Maple sometimes started after a few minutes)?

Any recent changes on your computer or network configuration?

Does Maple start always from the start menu (i.e. without double click on a file)

It is super annoying and not reproducible. For me it disappeared as it appeared several times over the past 5 years. I remember that this effect was more pronounced after a fresh system start-up. Also changing/ connecting/disconnecting networks seemed toplay a role.

PS.:My biggest suspicion at the moment is Java in combination with Windows.

@ That's what I get

An example would be helpfull.

Maple uses the method exact to solve the ode. Slighty better results can be obtained this way

factor(ode);
%*denom(lhs(%));
dsolve(%)

This way the first 5 solutions can be identified as the roots of the first term encricled in red. The sixth solution (in yellow) is now what we would expect since a different method (separable) was applied

Your original sixth solution is implicit which can be made explicit this way

simplify(isolate(diff(sol[6],x),diff(y(x),x)));dsolve(%)

I can't tell why Maple returns an implicit solution

@nm 

Interesting design decision that seems to save memory space, as no power statements are required in arithmetic expressions.

dismantle(a^b);

POWER(3)
   NAME(4): a
   NAME(4): b

@jganding 

This makes it a bit more generic

expr :=1* Unit('h');
evalf(convert(expr, 'unit_free'))*indets(expr, specfunc(Units:-Unit))[];

Do you have an example?