But the result is real.

Here is your integral (the one only with respect to x that you show above and where I fix the aA to a A -- i.e., add a space):

temp = Integrate[((-((
        8 a A x y)/((a^2 - 2 a x + x^2 + y^2) (a^2 + 2 a x + x^2 + 
           y^2))))*
      z0)/((a A) ((x0 - x)^2 + (y0 - y)^2 + (z0 - z)^2)^(3/2)), x];

now test it with random values for the unspecified parameters:

In[95]:= Table[
  Im[temp /. {x0 -> Random[], y0 -> Random[], z0 -> Random[], 
      A -> Random[], a -> Random[], y -> Random[], z -> Random[], 
      x -> Random[]} // N], {1000}] // Union

Out[95]= {0.}

The imaginary part is always 0.

It doesn't change anything. :D

If the integration over x is wrong than everything is wrong. That's one reason, and the other reason is that integrating this result above also over y is (I guess) to complex for mathematica and I can wait for 5 minutes but still nothing happens so I am forced to abort the process.

AND I am highly suspecting that those complex numbers are the main reason, why Mathematica can't do the second integration over y.

To Frank: I didn't like your idea, because in the end I would like to have the result: B(x0,y0,z0) if my x0,y0,z0 are the constants I am talking about, but I tried it anyway. (see below).

To David: Yours doesn't work either.

I still get a complex result. Here is the code:

In[2]:= Elpolje = Simplify[-Grad[Koncna[x, y], {x, y}]]

Out[2]= {(
 4 a A (a^2 - x^2 + y^2))/((a^2 - 2 a x + x^2 + y^2) (a^2 + 2 a x + 
    x^2 + y^2)), -((
  8 a A x y)/((a^2 - 2 a x + x^2 + y^2) (a^2 + 2 a x + x^2 + y^2)))}

The integral

In[8]:= a = 20
y0 = 1
x0 = 1
z0 = 1

Out[8]= 20

Out[9]= 1

Out[10]= 1

Out[11]= 1

In[12]:= Integrate[(Elpolje[[2]]*
    z0)/((aA) ((x0 - x)^2 + (y0 - y)^2 + (z0 - z)^2)^(3/2)), x]

Out[12]= -(1/aA)
 160 A y ((-482379 + 157585 x + 326374 y + 1628 x y - 168785 y^2 - 
       20 x y^2 + 4796 y^3 + 8 x y^3 - 1600 y^4 + 326374 z + 
       1628 x z - 6408 y z - 24 x y z + 4812 y^2 z + 8 x y^2 z - 
       8 y^3 z - 166391 z^2 - 826 x z^2 + 3228 y z^2 + 12 x y z^2 - 
       2410 y^2 z^2 - 4 x y^2 z^2 + 4 y^3 z^2 + 3212 z^3 + 12 x z^3 - 
       24 y z^3 + 4 y^2 z^3 - 813 z^4 - 3 x z^4 + 6 y z^4 - y^2 z^4 + 
       6 z^5 - z^6)/(Sqrt[
       3 - 2 x + x^2 - 2 y + y^2 - 2 z + 
        z^2] (51719068962 - 52755965394 y + 27935849089 y^2 - 
         1568132376 y^3 + 535095200 y^4 - 10253120 y^5 + 
         2560064 y^6 - 52755965394 z + 2089332928 y z - 
         1583724024 y^2 z + 28405040 y^3 z - 10330368 y^4 z + 
         25728 y^5 z + 27422649161 z^2 - 1068053936 y z^2 + 
         806167828 y^2 z^2 - 14318520 y^3 z^2 + 5178112 y^4 z^2 - 
         12864 y^5 z^2 - 1052462288 z^3 + 23490768 y z^3 - 
         14344696 y^2 z^3 + 116064 y^3 z^3 - 12928 y^4 z^3 + 
         272886176 z^4 - 6001972 y z^4 + 3634790 y^2 z^4 - 
         29096 y^3 z^4 + 3232 y^4 z^4 - 5898548 z^5 + 77792 y z^5 - 
         29192 y^2 z^5 + 48 y^3 z^5 + 1013294 z^6 - 13152 y z^6 + 
         4884 y^2 z^6 - 8 y^3 z^6 - 12992 z^7 + 80 y z^7 - 
         8 y^2 z^7 + 1654 z^8 - 10 y z^8 + y^2 z^8 - 10 z^9 + z^10)) +
     Log[-((320 y (-363 I + (38 + 2 I) y + 2 I z - I z^2) Sqrt[
        3 - 2 x + x^2 - 2 y + y^2 - 2 z + z^2])/(-20 + x - I y)) + (
      320 (6171 y - 
         6897 x y + (692 + 1009 I) y^2 + (38 - 1085 I) x y^2 - (405 - 
            74 I) y^3 + (38 + 2 I) x y^3 + (2 - 38 I) y^4 + 692 y z + 
         38 x y z - (8 - 74 I) y^2 z + 2 I x y^2 z + 2 y^3 z - 
         350 y z^2 - 19 x y z^2 + (4 - 37 I) y^2 z^2 - I x y^2 z^2 - 
         y^3 z^2 + 4 y z^3 - y z^4))/((-20 I + I x + y) Sqrt[
       363 - (2 - 38 I) y - 2 z + z^2])]/(
    160 y (-363 I + (38 + 2 I) y + 2 I z - I z^2) Sqrt[
     363 - (2 - 38 I) y - 2 z + z^2]) + (
    I Log[(
       320 I y (-443 + (2 + 42 I) y + 2 z - z^2) Sqrt[
        3 - 2 x + x^2 - 2 y + y^2 - 2 z + z^2])/(20 + x - I y) - (
       320 (-10189 y + 
          9303 x y + (932 + 1409 I) y^2 - (42 + 
             1325 I) x y^2 - (405 + 86 I) y^3 - (42 - 
             2 I) x y^3 + (2 + 42 I) y^4 + 932 y z - 
          42 x y z - (8 + 86 I) y^2 z + 2 I x y^2 z + 2 y^3 z - 
          470 y z^2 + 21 x y z^2 + (4 + 43 I) y^2 z^2 - I x y^2 z^2 - 
          y^3 z^2 + 4 y z^3 - y z^4))/((20 I + I x + y) Sqrt[
        443 - (2 + 42 I) y - 2 z + z^2])])/(
    160 y (-443 + (2 + 42 I) y + 2 z - z^2) Sqrt[
     443 - (2 + 42 I) y - 2 z + z^2]) + (
    I Log[(320 I y (-363 + (2 + 38 I) y + 2 z - z^2) Sqrt[
        3 - 2 x + x^2 - 2 y + y^2 - 2 z + z^2])/(-20 + x + I y) + (
       320 (6171 y - 
          6897 x y + (692 - 1009 I) y^2 + (38 + 
             1085 I) x y^2 - (405 + 74 I) y^3 + (38 - 
             2 I) x y^3 + (2 + 38 I) y^4 + 692 y z + 
          38 x y z - (8 + 74 I) y^2 z - 2 I x y^2 z + 2 y^3 z - 
          350 y z^2 - 19 x y z^2 + (4 + 37 I) y^2 z^2 + I x y^2 z^2 - 
          y^3 z^2 + 4 y z^3 - y z^4))/((20 I - I x + y) Sqrt[
        363 - (2 + 38 I) y - 2 z + z^2])])/(
    160 y (-363 + (2 + 38 I) y + 2 z - z^2) Sqrt[
     363 - (2 + 38 I) y - 2 z + z^2]) + 
    Log[-((320 y (-443 I + (42 + 2 I) y + 2 I z - I z^2) Sqrt[
        3 - 2 x + x^2 - 2 y + y^2 - 2 z + z^2])/(20 + x + I y)) + (
      320 (10189 y - 
         9303 x y - (932 - 1409 I) y^2 + (42 - 1325 I) x y^2 + (405 - 
            86 I) y^3 + (42 + 2 I) x y^3 - (2 - 42 I) y^4 - 932 y z + 
         42 x y z + (8 - 86 I) y^2 z + 2 I x y^2 z - 2 y^3 z + 
         470 y z^2 - 21 x y z^2 - (4 - 43 I) y^2 z^2 - I x y^2 z^2 + 
         y^3 z^2 - 4 y z^3 + y z^4))/((-20 I - I x + y) Sqrt[
       443 - (2 - 42 I) y - 2 z + z^2])]/(
    160 y (-443 I + (42 + 2 I) y + 2 I z - I z^2) Sqrt[
     443 - (2 - 42 I) y - 2 z + z^2]))

I ignored the whole result as soon as I say those imaginary parts in it.... -.-

In[1]:= r0 = {1, 2, 3};

In[5]:= NIntegrate[
 Norm[{x, y, z} - r0], {x, -5, 5}, {y, -5, 5}, {z, -5, 5}]

Out[5]= 5870.68