In[1]:= S[t_] :=
 Integrate[
  1/((1 + x^2)^(1/4) (y - x)^(3/2)) -
   1/((1 + y^2)^(1/4) (y - x)^(3/2)), {x, 0, t}, {y, t, Infinity}]

In[2]:= u[x_] := (1 + x^2)^(-1/4)

In[6]:= K[x_] := D[((1 + t^2)^(5/4)/t)*u'[x]*S[t], t]

In[7]:= K[x]/(u[x]^3)


During evaluation of In[7]:= Refine::fas: Warning: one or more assumptions evaluated to False.

Out[7]= (1 + x^2)^(
 3/4) (-(1/(2 t (1 + x^2)^(5/4)))(1 + t^2)^(5/4) x !(
*SubscriptBox[([PartialD]), (t)](
*SubsuperscriptBox[([Integral]), (t),
([Infinity])]If[((((Re[t] !=
               0 && ((((Im[y] >= 0 && Im[t] <= 0)) ||
*FractionBox[(Im[y]), (Im[t])] >= 1 ||