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Integrate[Sqrt[x + Sqrt[a^2 + x^2]]/x, x] ==
-((Sqrt[a^2 + x^2]*(x + Sqrt[a^2 + x^2])* (-2*Sqrt[x + Sqrt[a^2 + x^2]] + 2*Sqrt[a]*ArcTan[Sqrt[x + Sqrt[a^2 + x^2]]/ Sqrt[a]] - Sqrt[a]* Log[-Sqrt[a] + Sqrt[x + Sqrt[a^2 + x^2]]] + Sqrt[a]*Log[Sqrt[a] + Sqrt[x + Sqrt[a^2 + x^2]]]))/ (a^2 + x*(x + Sqrt[a^2 + x^2])))



                         2    2
          Sqrt[x + Sqrt[a  + x ]]
Integrate[-----------------------, x] == 
                     x

         2    2             2    2
-((Sqrt[a  + x ] (x + Sqrt[a  + x ]) 
 
                         2    2
      (-2 Sqrt[x + Sqrt[a  + x ]] + 
 
                                        2    2
                         Sqrt[x + Sqrt[a  + x ]]
        2 Sqrt[a] ArcTan[-----------------------] - 
                                 Sqrt[a]
 
        Sqrt[a] Log[-Sqrt[a] + 
 
                          2    2
           Sqrt[x + Sqrt[a  + x ]]] + 
 
                                             2    2
        Sqrt[a] Log[Sqrt[a] + Sqrt[x + Sqrt[a  + x ]]]
 
               2                2    2
        )) / (a  + x (x + Sqrt[a  + x ])))

Time to compute: 0.48 second

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