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(-2*Sqrt[4 - 25/x^2]*x*EllipticF[ArcSin[Sqrt[5/2]/Sqrt[x]], -1])/Sqrt[250 - 40*x^2]

Integrate[1/(Sqrt[x]*Sqrt[25 - 4*x^2]), x] ==
(-2*Sqrt[4 - 25/x^2]*x*EllipticF[ ArcSin[Sqrt[5/2]/Sqrt[x]], -1])/Sqrt[250 - 40*x^2] 

                     1
Integrate[-----------------------, x] == 
                               2
          Sqrt[x] Sqrt[25 - 4 x ]

                                        5
                                   Sqrt[-]
            25                          2
-2 Sqrt[4 - --] x EllipticF[ArcSin[-------], -1]
             2                     Sqrt[x]
            x
------------------------------------------------
                              2
               Sqrt[250 - 40 x ]
Time to compute: 0.12 second
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