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Integrate[Sqrt[a + b*x]*Sqrt[c + d*x], x] ==
((b*c + a*d)/(4*b*d) + x/2)*Sqrt[a + b*x]* Sqrt[c + d*x] - ((b*c - a*d)^2*Log[b*c + a*d + 2*b*d*x + 2*Sqrt[b]*Sqrt[d]*Sqrt[a + b*x]*Sqrt[c + d*x]])/ (8*b^(3/2)*d^(3/2))



Integrate[Sqrt[a + b x] Sqrt[c + d x], x] == 

 b c + a d   x
(--------- + -) Sqrt[a + b x] Sqrt[c + d x] - 
   4 b d     2
 
              2
  ((b c - a d)  Log[b c + a d + 2 b d x + 
 
       2 Sqrt[b] Sqrt[d] Sqrt[a + b x] Sqrt[c + d x]])
 
           3/2  3/2
     / (8 b    d   )

Time to compute: 0.14 second

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