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Integrate[Sqrt[4 - c*x^2]/Sqrt[x], x] ==
(2*Sqrt[x]*Sqrt[4 - c*x^2])/3 + (((8*I)/3)*Sqrt[2 - 8/(c*x^2)]*x* EllipticF[I*ArcSinh[(Sqrt[2]*Sqrt[-(1/Sqrt[c])])/ Sqrt[x]], -1])/(Sqrt[-(1/Sqrt[c])]* Sqrt[4 - c*x^2])



                      2
          Sqrt[4 - c x ]
Integrate[--------------, x] == 
             Sqrt[x]

                      2
2 Sqrt[x] Sqrt[4 - c x ]
------------------------ + 
           3
 
   8 I           8
  (--- Sqrt[2 - ----] x 
    3              2
                c x
 
                                           1
                         Sqrt[2] Sqrt[-(-------)]
                                        Sqrt[c]
     EllipticF[I ArcSinh[------------------------], 
                                 Sqrt[x]
 
                        1                  2
      -1]) / (Sqrt[-(-------)] Sqrt[4 - c x ])
                     Sqrt[c]

Time to compute: 0.42 second

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