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

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

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

       c   -(1/2) - m         2 1/2 + m
((1 + ----)           (c + a x )        
         2
      a x
 
                        1         1       1
    Hypergeometric2F1[-(-) - m, -(-) - m, - - m, 
                        2         2       2
 
        c
     -(----)]) / (1 + 2 m)
          2
       a x
Time to compute: 0.10 second
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