P:=proc(p,x,y)
  local pp:=expand(p), ld:=ldegree(pp,{x,y}), a:=floor(ld/2), b:=ld-a;
  coeff(coeff(pp,x,a),y,b)*x^a*y^b
end proc;

P(13*x^2*y^2 + x*y^2 + 2*y*x^2,x,y) # x*y^2
P(100*x^2*y^2 + 35*y*x + 45*x,x,y)   # 0

The double integral diverges.
I have supposed that Dgamma = D(gamma), Dphi = D(phi).
(Use gamma1 instead of gamma = used by Maple.) 
 

g(a) = piecewise(a < 0, 1/4*2^(1/2)*(a^2)^(1/4)*exp(-1/2*a^2)*BesselK(1/4,1/2*a^2),
-1/2*Pi^(1/2)*2^(1/4)*exp(-1/2*a^2)*(2^(1/2)*CylinderD(3/2,-2^(1/2)*a)+
2*a*CylinderD(1/2,-2^(1/2)*a)));

Note that actually the second branch is valid for a<0 too!

Windows:

src:=cat(kernelopts(mapledir), "\\samples"):
dest:="D:\\tmp\\samples\\":
system( cat("xcopy \"", src, "\" ", dest, " /E") );

No, if   f := y z x + exp(x) cos(y) + g(y, z)
your ff is not correct.

IntWithConst:=proc(f::algebraic, x::name, C::name:=_F)
  local u:=indets(f, And(name, Not(constant))) minus {x};
  int(f, x) + C(u[])
end:

A := exp(x)*cos(y) + y*z:

f:=IntWithConst(A, x);
        f := y*z*x + exp(x)*cos(y) + _F(y, z)

diff(f,y);
       z*x - exp(x)*sin(y) + diff(_F(y, z), y)

The principle is simple: if diff(f(x,y), x) = 0 then f(x,y) is constant with respect to x, which means that f(x,y) depends only on y, i.e. f(x,y) = g(y).

If you want to see this in Maple:

pdsolve( diff(f(x,y), x) = 0, f(x,y) );

         f(x, y) = _F1(y)

Actually, the fact is true only locally: in a non-convex domain, f(x,y) could depend on x (!). Maths is difficult...

So, you want a partial fraction form. Unfortunately Maple does it only for rational expressions and a substitution is needed.

S:=sqrt(N__s):
subs(x=S, convert(subs(S=x,S^2=x^2, ex1), parfrac));

A simpler method is to use the view option

plot3d(4*x^2+9*y^2, x=-5..5, y=-5..5, view=0..100);

The loci are two circles centered at B and C.

(The geometry package cannot be used generally for such problems; they must be solved directly.)

Download geom-loci-ellipse-circles.mw

You have some inaccurate formulations, starting with the definition of a defective eigenvalue (see the cited wiki article).
Anyway, the defective eigenvectors can be obtained directly from the Jordan form of the matrix (they are the columns of Q):

A :=Matrix([[1,-2],[2,5]]):
J,Q:=LinearAlgebra:-JordanForm(A, output=['J','Q']);

v1:=Q[..,1];
v2:=Q[..,2];

(A-3).v1,  (A-3).v2,  (A-3)^2 . v2;

restart;
HB1 := HeunB(5/2, -5^(1/4)*(2*M - R)/sqrt((2*M - R)*(4*M - R)), (17*sqrt(5))/10, (3*5^(3/4)*(2*M - R))/(2*sqrt((2*M - R)*(4*M - R))), 5^(1/4)*sqrt((2*M - R)*(4*M - R))*r^2/(2*(2*M - R)*R^2)):
A:=op(5, HB1):
answer := series(eval(series(eval(HB1, A=z), z), z=A), r);

Your difficulties are related to maths rather than Maple.

C is a curve represented geometrically as a segment with endpoints (1,0), (2,2).
So, you have to find a parametric representation for C [for a computation by hand].

You are asked for the path integral (not line integral -- Maple terminology).  The computation is straightforward using VectorCalculus.

VectorCalculus:-PathInt( 2*x + y^2, [x,y] = Line(<1,0>, <2,2>) );

        (13*sqrt(5))/3

You do not need Maple for this. Denote by {0, 1, a} the set, where 0 and 1 are the min and max elements.
Then there is a unique lattice (0 < a < 1), whose graph is

    0
    |
    a
    |
    1

Of course, if the max and min elements are not distinguished, there are 3! = 6 (isomorphic) lattices. 

radsimp does a symbolic simplification, which means (among other things) that Maple is allowed to choose for sqrt(x^2) any of the values x or -x. Actually this command is deprecated, see the help page.

radsimp( b-a - sqrt((b-a)^2) ); 
    2 b - 2 a

Note that it is not correct to speak about a right or wrong result here beacuse the sign of b-a is not known.
Thr preferred approach is to use assuming: