4 added 316 characters in body
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About a 1 second.

NIntegrate[
  1/(px^2 + py^2 + 1)/((px - qx)^2 + (px - qy)^2 + 1)/(qx^2 + qy^2 + 1), 
  {px, 0, 10}, {py, 0, 10}, {qx, 0, 10}, {qy, 0, 10}, 
  Method -> {"LocalAdaptive", "SymbolicProcessing" -> 0}] // Timing

(*{1.45313, 2.84718}*)

or:

NIntegrate[1/(px^2 + py^2 + 1)/((px - qx)^2 + (px - qy)^2 + 1)/(qx^2 + qy^2 + 
  1), {px, 0, 10}, {py, 0, 10}, {qx, 0, 10}, {qy, 0, 10}, 
  PrecisionGoal -> 5, MinRecursion -> 1, Method -> {"LocalAdaptive", 
  "SymbolicProcessing" -> 0}] // AbsoluteTiming

(*{0.687526, 2.84731}*)

About a 1 second.

NIntegrate[
  1/(px^2 + py^2 + 1)/((px - qx)^2 + (px - qy)^2 + 1)/(qx^2 + qy^2 + 1), 
  {px, 0, 10}, {py, 0, 10}, {qx, 0, 10}, {qy, 0, 10}, 
  Method -> {"LocalAdaptive", "SymbolicProcessing" -> 0}] // Timing

(*{1.45313, 2.84718}*)

About a 1 second.

NIntegrate[
  1/(px^2 + py^2 + 1)/((px - qx)^2 + (px - qy)^2 + 1)/(qx^2 + qy^2 + 1), 
  {px, 0, 10}, {py, 0, 10}, {qx, 0, 10}, {qy, 0, 10}, 
  Method -> {"LocalAdaptive", "SymbolicProcessing" -> 0}] // Timing

(*{1.45313, 2.84718}*)

or:

NIntegrate[1/(px^2 + py^2 + 1)/((px - qx)^2 + (px - qy)^2 + 1)/(qx^2 + qy^2 + 
  1), {px, 0, 10}, {py, 0, 10}, {qx, 0, 10}, {qy, 0, 10}, 
  PrecisionGoal -> 5, MinRecursion -> 1, Method -> {"LocalAdaptive", 
  "SymbolicProcessing" -> 0}] // AbsoluteTiming

(*{0.687526, 2.84731}*)
3 Improved formatting
source | link

About a 1 second.

NIntegrate[1NIntegrate[
  1/(px^2 + py^2 + 1)/((px - qx)^2 + (px - qy)^2 + 1)/(qx^2 + qy^2 + 1), 
  {px, 0, 10}, {py, 0, 10}, {qx, 0, 10}, {qy, 0, 10}, 
  Method -> {"LocalAdaptive", "SymbolicProcessing" -> 0}] // Timing 

(*{1.45313, 2.84718}*)

About a 1 second.

NIntegrate[1/(px^2 + py^2 + 1)/((px - qx)^2 + (px - qy)^2 + 1)/(qx^2 + qy^2 + 1), {px, 0, 10}, {py, 0, 10}, {qx, 0, 10}, {qy, 0, 10}, 
Method -> {"LocalAdaptive", "SymbolicProcessing" -> 0}] // Timing
(*{1.45313, 2.84718}*)

About a 1 second.

NIntegrate[
  1/(px^2 + py^2 + 1)/((px - qx)^2 + (px - qy)^2 + 1)/(qx^2 + qy^2 + 1), 
  {px, 0, 10}, {py, 0, 10}, {qx, 0, 10}, {qy, 0, 10}, 
  Method -> {"LocalAdaptive", "SymbolicProcessing" -> 0}] // Timing 

(*{1.45313, 2.84718}*)
2 added 2 characters in body
source | link

About a 1 second.

NIntegrate[1/(px^2 + py^2 + 1)/((px - qx)^2 + (px - qy)^2 + 1)/(qx^2 + qy^2 + 1), {px, 0, 10}, {py, 0, 10}, {qx, 0, 10}, {qy, 0, 10}, 
Method -> {"LocalAdaptive", "SymbolicProcessing" -> 0}] // Timing
(*{1.45313, 2.84718}*)

About a 1 second.

NIntegrate[1/(px^2 + py^2 + 1)/((px - qx)^2 + (px - qy)^2 + 1)/(qx^2 + qy^2 + 1), {px, 0, 10}, {py, 0, 10}, {qx, 0, 10}, {qy, 0, 10}, 
Method -> {"LocalAdaptive", "SymbolicProcessing" -> 0}] // Timing
{1.45313, 2.84718}

About a 1 second.

NIntegrate[1/(px^2 + py^2 + 1)/((px - qx)^2 + (px - qy)^2 + 1)/(qx^2 + qy^2 + 1), {px, 0, 10}, {py, 0, 10}, {qx, 0, 10}, {qy, 0, 10}, 
Method -> {"LocalAdaptive", "SymbolicProcessing" -> 0}] // Timing
(*{1.45313, 2.84718}*)
1
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