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

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

                  2 1/2 + m
Integrate[(c + a x )       , x] == 

                            2
          2 1/2 + m      a x  -(1/2) - m
x (c + a x )        (1 + ----)           
                          c
 
                                         2
                    1    1       3    a x
  Hypergeometric2F1[-, -(-) - m, -, -(----)]
                    2    2       2     c
Time to compute: 0.05 second
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