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


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



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

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

Time to compute: 0.07 second

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