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Integrate[(a + b*x + c*x^2)^(5/2), x] ==
((b + 2*c*x)*Sqrt[a + x*(b + c*x)]* (15*b^4 - 40*b^3*c*x + 32*b*c^2*x* (13*a + 8*c*x^2) + 8*b^2*c*(-20*a + 11*c*x^2) + 16*c^2*(33*a^2 + 26*a*c*x^2 + 8*c^2*x^4)))/ (1536*c^3) - (5*(b^2 - 4*a*c)^3* Log[b + 2*c*x + 2*Sqrt[c]*Sqrt[a + x*(b + c*x)]])/ (1024*c^(7/2))



                        2 5/2
Integrate[(a + b x + c x )   , x] == 

((b + 2 c x) Sqrt[a + x (b + c x)] 
 
          4       3             2                2
     (15 b  - 40 b  c x + 32 b c  x (13 a + 8 c x ) + 
 
          2                  2
       8 b  c (-20 a + 11 c x ) + 
 
           2      2           2      2  4
       16 c  (33 a  + 26 a c x  + 8 c  x ))) / 
 
          3         2         3
   (1536 c ) - (5 (b  - 4 a c)  
 
     Log[b + 2 c x + 2 Sqrt[c] Sqrt[a + x (b + c x)]])
 
              7/2
     / (1024 c   )

Time to compute: 0.25 second

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