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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: 0.17 second

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