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


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



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

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

Time to compute: 0.08 second

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