Write it as follows:

  eq2:=(Y+sqrt(Y^2-X^2*c^2))/2 - c;
  L2:=[Y=(b^2+a^2), X=2*a];

Evaluating with L2 gives the original equation.

Now solve it with 'paper and pencil' using Maple:

  isolate(eq2, (Y^2-X^2*c^2)^(1/2));
  lhs(%)^2=rhs(%)^2;
  lhs(%)-rhs(%)=0;
  collect(%,c);

You achieve at (-X^2-4)*c^2+4*Y*c = 0.

It has c=0 as solution. Dividing for c != 0 will
give the other solution 4*Y/(X^2+4), which Maple
gives as well and which has to be understood 
(evaluating in L2 gives the initial problem).

  sol2:=[solve(%,c)];

                                      4 Y
                         sol2 := [0, ------]
                                      2
                                     X  + 4


First that is formal, i.e. as far as it exists.

Second it means: it is necessary for eq2 = 0.

You want it to be sufficient as well. It is not:

  tst2:=subs(a=2,b=2,L2); # plot original problem

                        tst2 := [Y = 8, X = 4]

  eval(sol2,tst2);
                                [0, 8/5]

  eval(eq2,tst2); eval(%, c=8/5): simplify(%);

                                     2 1/2
                           (64 - 16 c )
                       4 + --------------- - c
                                  2


                                 24/5

I hope I made not too many errors ...

rationalize(A(x,1)): 
a:=numer(%): b:=denom(%%):

a*convert(1/b,fullparfrac) 
int(%,x=0..infinity);
                                   1/2
                               -2 2

Here are some variants:

time(add(isprime(n), n = 1 .. 11112)); # yours, my PC is slower

                                0.061

time(`+`(op(select(isprime,[$1..11112]))));

                                0.062

numtheory:-pi(11112):
time(`+`(ithprime(i) $ i=1..%));

                                0.014

The last version is a bit unfair: to find that number Maple needs
about the same time, so it is actually ~ 0.03 sec. Up to 10^5 it
is faster.

Note however that this usage of 'time' does not measure the time
needed to create necessary data structures, see the help.

If you want it faster search the board for 'alec' and 'isprime'
and http://www.mapleprimes.com/blog/alec/the-eratosthenes-sieve
will show you related compiled versions (of course numbers must
fit into a compiled integer), ES2 is from the link.

Then after compiling (which is very quick and only once): 

time(`+`(op(convert(ES2(10^5),listlist)))); 

                                0.016

time(`+`(op(convert(ES2(10^6),listlist)))); 

                                0.125

while the non-compiled versions from the beginning would need
7 - 8 seconds for the last result.

Of course you could do adding in a compiled version as well and
that should be even faster.

Here are some variants:

time(add(isprime(n), n = 1 .. 11112)); # yours, my PC is slower

                                0.061

time(`+`(op(select(isprime,[$1..11112]))));

                                0.062

numtheory:-pi(11112):
time(`+`(ithprime(i) $ i=1..%));

                                0.014

The last version is a bit unfair: to find that number Maple needs
about the same time, so it is actually ~ 0.03 sec. Up to 10^5 it
is faster.

Note however that this usage of 'time' does not measure the time
needed to create necessary data structures, see the help.

If you want it faster search the board for 'alec' and 'isprime'
and http://www.mapleprimes.com/blog/alec/the-eratosthenes-sieve
will show you related compiled versions (of course numbers must
fit into a compiled integer), ES2 is from the link.

Then after compiling (which is very quick and only once): 

time(`+`(op(convert(ES2(10^5),listlist)))); 

                                0.016

time(`+`(op(convert(ES2(10^6),listlist)))); 

                                0.125

while the non-compiled versions from the beginning would need
7 - 8 seconds for the last result.

Of course you could do adding in a compiled version as well and
that should be even faster.