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

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

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

                2    2
  2 + 2 m      a  + x
(x        Sqrt[-------] 
                  2
                 a
 
                                          2
                      1                  x
    Hypergeometric2F1[-, 1 + m, 2 + m, -(--)]) / 
                      2                   2
                                         a
 
                   2    2
  (2 (1 + m) Sqrt[a  + x ])
Time to compute: 0.06 second
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