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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)^(-2^(-1) - 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: 0.06 second

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