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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: 0.19 second

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