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



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

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

Time to compute: 0.51 second

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