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-((x^(1 - m)*Sqrt[1 + x^2/a^2]*Hypergeometric2F1[1/2, 1/2 - m/2, 3/2 - m/2, -(x^2/a^2)])/((-1 + m)*Sqrt[a^2 + x^2]))


Integrate[1/(x^m*Sqrt[a^2 + x^2]), x] ==
-((x^(1 - m)*Sqrt[1 + x^2/a^2]*Hypergeometric2F1[1/2, 1/2 - m/2, 3/2 - m/2, -(x^2/a^2)])/ ((-1 + m)*Sqrt[a^2 + x^2]))



                 1
Integrate[----------------, x] == 
           m       2    2
          x  Sqrt[a  + x ]

                    2
    1 - m          x
-((x      Sqrt[1 + --] 
                    2
                   a
 
                                            2
                        1  1   m  3   m    x
      Hypergeometric2F1[-, - - -, - - -, -(--)]) / 
                        2  2   2  2   2     2
                                           a
 
                    2    2
    ((-1 + m) Sqrt[a  + x ]))

Time to compute: 0.08 second

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