How to Collect $x$ powers to group $\sqrt{x}$ and $\sqrt{1+x}$ together?

Question

I have been failing to apply Collect[#, x]& to render a like-power series of $x^k$. Here $k = \frac52, \frac32, \frac12,\frac{-1}2$, etc.

The major issue is:

terms like $\sqrt{x}$ and $\sqrt{x+y}$ are not grouped together even though I thought in Collect[#, x]& things like $y$ would be treated as constants. (this can't be fixed by With, right?)

For my purpose, $\sqrt{x^5 +y^5}$ is considered the same power as $x^{\frac52}$, and I believe there are no terms like $\frac{\sqrt{x+y}}{\sqrt{x+2y}}$ or $\sqrt{a x^2 + bx}$.

Question:

How do I arrange the expression (named myFun in the code block) into a series of like-powers of $x$? I'd like to group all terms of the form $\sqrt{x+y}$ together, including $y = 1$ and $y=0$.

(I also would like to see myFun in series of like-powers of $L$, where $L$ is another variable shown in the codes. Attempting at $L$-series poses basically the issues.)

My ultimate goal is to identify the expression as "linear $x$ plus perturbation" for asymptotic $L$ (actually at just $L \gtrapprox 3$ or 4). Therefore, it is essential that I'll be able to demonstrate the final result as something like "$c_0 + c_1 x +{}$other powers (positive or negative)". Consequently, Together or Collect[#, {Sqrt[x], Sqrt[1+x-y]} }]& based methods might need to recruit additional tricks to achieve this.

(Substantial Edit: Originally the post contained paragraphs about failure of Simplify inside Collect, which was distracting.)

Mathematically, $x$ and $L$ are the only two variables, and they are independent. All others are considered parameters with ranges specified in the block.

(* WARNING !!! HUGE OUTPUT !!! *)

ClearAll[myFun, tmp, ps (* power sum *), L, x, y, Y, tmp];
ps[j_, M_] := 2/5 ((j + M)^(5/2) - (1 + j)^(5/2)) + ((j + M)^(3/2) + (1 + j)^(3/2))/2 + 1/8 (Sqrt[j + M] - Sqrt[1 + j]) + 1/1920 (1/(j + M)^(3/2) - 1/ ((1 + j)^(3/2)));
tmp = (y (Y + L) - x Y)/(y + Y);
tmp = ps[x, tmp] Sqrt[1/(L + x + Y)] - ps[x - y, tmp] Sqrt[1/(L + x - y)];
myFun = Normal@Series[tmp, {L, \[Infinity], 4}]

(* Collect[ExpandAll@myFun,x,
      Simplify[#, 
        0<x<1 && 0<y<1 && y+Y >0 && L >= 3
    ]&  ] gives many terms that obviously can be combined *)

I'm also open to any methods other than Collect that produce results similar to what I described above. (This might be a mathematically ill-posed question ... but there seems to be some wiggle room)

My failed attempts thus far include:

(a) Using Pattern to try to capture "any sqrt-like function involving $x$", with BlankNullSequence which I don't really know how ... Collect[ myFun, Power[ {___, x, ___ }, _ ] ] is obviously wrong.

(b) Doing the arrangement on SeriesData, that is, without Normal@. This gave strange results that I'm not sure how to make use of.

There are some posts about Collect but I didn't see any addressing non-integer powers. Others I don't think apply to my situation, like this (esp. the FAQ therein) or that.

Any input would be appreciated. Thanks.

P.S.

If the expression seems too large to read, feel free to reduce the order of Series from 4 to 3. One can also remove the terms in ps[j_, M_] starting at the $+\frac18$ and onward.

Answer 1

This is just addressing your ultimate goal. You can do

List @@ ExpandAll @ myFun;
Cases[%, #] & /@ {c0_ /; FreeQ[c0, x], x * c1_ /; FreeQ[c1, x]};
split = Plus@@@Append[%, Complement[%%, Flatten@%]];

It might be more readable to do

{constant, linear, perturbation} = split

Check that the split pieces sum to the original expression:

Plus @@ split == ExpandAll @ myFun

True