I*y*(1/(x - I*y)^(5/2) - 1/(x + I*y)^(5/2))

I*Integrate[
  y*(1/(x - I*y)^(5/2) - 1/(x + I*y)^(5/2)), {x, -10, 10}, {y, 0, 10}]
% // N

(* -(8/3) Sqrt[5] (2 Sqrt[2] + Sqrt[-1 + 5 Sqrt[2]] - [Pi] - ArcCosh[3]) *)

(*  -2.31 *)

Quiet[Chop[
    I*NIntegrate[
      y *(1/(x - I y)^(5/2) - 1/(x + I y)^(5/2)), {x, -10, 10}, {y, 0,
        10}, Method -> #]]] & /@ {"LocalAdaptive",
{"EvenOddSubdivision", Method -> "LocalAdaptive"},
  "AdaptiveMonteCarlo", "QuasiMonteCarlo",
  "MonteCarlo", {"EvenOddSubdivision",
   Method -> "AdaptiveMonteCarlo"}, {"EvenOddSubdivision",
   Method -> "DuffyCoordinates"}}

expr = Simplify[
  TrigToExp[
   I*y*(1/(x - I*y)^(5/2) - 1/(x + I*y)^(5/2)) /. {x ->
      r*Cos[[CurlyPhi]], y -> r*Sin[[CurlyPhi]]}], {r > 0,
   0 < [CurlyPhi] < [Pi]}]

(* (E^(-((7 I [CurlyPhi])/
  2)) (-1 + E^(2 I [CurlyPhi])) (-1 + E^(5 I [CurlyPhi])))/(2 r^(
 3/2))  *)

Integrate[expr2*r, {r, 0, 10}, {[CurlyPhi], 0, [Pi]}] // N

(*   2.41   *)

Quiet[NIntegrate[expr2*r, {r, 0, 10}, {[CurlyPhi], 0, [Pi]},
    Method -> #]] & /@ {"LocalAdaptive", {"EvenOddSubdivision",
   Method -> "LocalAdaptive"}, "AdaptiveMonteCarlo",
  "QuasiMonteCarlo",
  "MonteCarlo", {"EvenOddSubdivision",
   Method -> "AdaptiveMonteCarlo"}, {"EvenOddSubdivision",
   Method -> "DuffyCoordinates"}}



 (*    {2.41, 2.41, 2.43, 2.42, 2.28, 2.38, 2.41}   *)

ceRe = FullSimplify[
   ComplexExpand[Re[I*y*(1/(x - I y)^(5/2) - 1/(x + I y)^(5/2))],
   TargetFunctions -> {Re, Im}], y >= 0 && x [Element] Reals]

NIntegrate[ceRe, {x, -10, 10}, {y, 0, 10}, WorkingPrecision -> 100,
  MaxRecursion -> 50, AccuracyGoal -> 6, PrecisionGoal -> 6,
  Exclusions -> {{0, 0}}]

(*   2.17329834703426526174107911927816608629584584526685923085449633834343
       9718373530841484083310764572423   *)

cetr = TrigToExp[ceRe]

(*   (I y)/(2 ((x - I y)/Sqrt[x^2 + y^2])^(5/2) (x^2 + y^2)^(5/4)) - (
     I y ((x - I y)/Sqrt[x^2 + y^2])^(5/2))/(2 (x^2 + y^2)^(5/4)) - (
     I y)/(2 ((x + I y)/Sqrt[x^2 + y^2])^(5/2) (x^2 + y^2)^(5/4)) + (
     I y ((x + I y)/Sqrt[x^2 + y^2])^(5/2))/(2 (x^2 + y^2)^(5/4))   *)


Integrate[cetr, {x, -10, 10}, {y, 0, 10}]
   (N[#1, 20] &)[%]

(*   (11/606 + (3 I)/
 202) ((112 + 92 I) Sqrt[-10 + 10 I] + (22 - 18 I) Sqrt[-5 -
 5 I] - (112 + 128 I) Sqrt[1 - 10 I] - (66 - 54 I) Sqrt[
 5 - 5 I] + (176 - 144 I) Sqrt[10] + (38 + 24 I) Sqrt[
 10 - 10 I] + (10 - 100 I) Sqrt[10 + 10 I] - (40 + 4 I) Sqrt[
 170 - 70 I] +
 16 Sqrt[1105 + 262 I] - (44 - 36 I) Sqrt[
 5] [Pi] - (40 + 4 I) Sqrt[10]
 ArcCoth[2 Sqrt[50/101 - (5 I)/101]] + (6 - 60 I) Sqrt[10]
 ArcCoth[Sqrt[1 + I]] - (88 - 72 I) Sqrt[5]
 ArcCoth[Sqrt[2]] - (6 - 60 I) Sqrt[10]
 ArcCoth[2/Sqrt[2 - I/5]] + (40 + 4 I) Sqrt[10]
 ArcCoth[(11/101 + (9 I)/101) Sqrt[10 - 100 I]] + (16 -
  160 I) Sqrt[5] ArcTan[(1 + I)/Sqrt[2]] + (16 - 160 I) Sqrt[5]
 ArcTanh[(1 + I)/Sqrt[2]] + (2 - 20 I) Sqrt[10]
 Log[1 - 1/2 (-1 + I)^(3/2)] + (18 + 22 I) (-1)^(1/4) Sqrt[5]
 Log[1 + 1/2 (-1 + I)^(3/2)] + (2 - 20 I) Sqrt[10]
 Log[1 - 1/Sqrt[1 + I]] - (2 - 20 I) Sqrt[10]
 Log[1 + 1/Sqrt[1 + I]] - (3 - 30 I) Sqrt[10]
 Log[((1 + 2 I) + 2 Sqrt[-1 + I]) (3 - 2 Sqrt[2])] + (3 -
  30 I) Sqrt[10] Log[2 - Sqrt[2]] - (3 - 30 I) Sqrt[10]
 Log[2 + Sqrt[2]] + (1 - 10 I) Sqrt[10]
 Log[-Sqrt[1 - I] + Sqrt[2]] - (1 - 10 I) Sqrt[10]
 Log[Sqrt[1 - I] + Sqrt[2]] - (10 + I) Sqrt[-5 - 5 I] Sqrt[1 - I]
 Log[-Sqrt[1 + I] + Sqrt[2]] + (10 + I) Sqrt[-5 - 5 I] Sqrt[1 - I]
 Log[Sqrt[1 + I] + Sqrt[2]] - (3 - 30 I) Sqrt[10]
 Log[(-(1/2) + I/2) Sqrt[1 + 10 I] + Sqrt[10]] + (3 - 30 I) Sqrt[
10] Log[(1/2 - I/2) Sqrt[1 + 10 I] + Sqrt[10]] - (1 - 10 I) Sqrt[
10] Log[2 Sqrt[5] - Sqrt[10 - I]] + (1 - I) Sqrt[100 - 495 I]
 Log[2 Sqrt[5] - Sqrt[10 - I]] + (1 - 10 I) Sqrt[10]
 Log[2 Sqrt[5] + Sqrt[10 - I]] - (1 - I) Sqrt[100 - 495 I]
 Log[2 Sqrt[5] + Sqrt[10 - I]])   *)

(*    8/3 (2 Sqrt[10] - Sqrt[-5 + 25 Sqrt[2]])    *)

(*  2.1732996388253877021 + 0.*10^-20 I  *)

I*Integrate[y*(1/(x - I*y)^(5/2) - 1/(x + I*y)^(5/2)), {y, 0, 10}, {x, -10, 10}]
N[%]
(*
8/3 Sqrt[5 (7 + 5 Sqrt[2] - 4 Sqrt[-2 + 10 Sqrt[2]])]
2.1733
*)