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Integrate[1/(x*(a^2 + x^2)^(3/2)), x] ==
(a/Sqrt[a^2 + x^2] + Log[a*x] - Log[2*a^3*(a + Sqrt[a^2 + x^2])])/a^3

                1
Integrate[--------------, x] == 
              2    2 3/2
          x (a  + x )

       a
(------------- + Log[a x] - 
       2    2
 Sqrt[a  + x ]
 
           3            2    2        3
    Log[2 a  (a + Sqrt[a  + x ])]) / a