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Integrate[x^(-1 - 2*m)/Sqrt[a^2 + x^2], x] ==
(Sqrt[a^2 + x^2]* (-(a^2*(-1 + m)*Hypergeometric2F1[-1/2, -m, 1 - m, -(x^2/a^2)]) + m*x^2*Hypergeometric2F1[1/2, 1 - m, 2 - m, -(x^2/a^2)]))/ (2*a^4*(-1 + m)*m*x^(2*m)*Sqrt[1 + x^2/a^2])



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

       2    2      2
(Sqrt[a  + x ] (-(a  (-1 + m) 
 
                                               2
                             1                x
         Hypergeometric2F1[-(-), -m, 1 - m, -(--)]) + 
                             2                 2
                                              a
 
                                                 2
         2                   1                  x
      m x  Hypergeometric2F1[-, 1 - m, 2 - m, -(--)]))
                             2                   2
                                                a
 
                                      2
          4             2 m          x
    / (2 a  (-1 + m) m x    Sqrt[1 + --])
                                      2
                                     a

Time to compute: 0.15 second

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