\(\(\( (*\ 
      our\ new\ function\ *) \)\(
\)\(\(c\_2[
          x] = \((x\^3 + 6  x)\) a[x];\)
 (*\ 
      cache\ some\ derivatives\ *) 

    \(d1 = D[c\_2[x], x];\)

    \(d2 = D[c\_2[x], {x, 2}];\)\)\)\)

d2 + (1/2) x d1 - (3/2) (Subscript[c, 2])[x] // Simplify

i1 = 
      Integrate[((-1)/2) (((24 + 18  (x^2) + (x^4))/(x (6 + (x^2))))), x]

Integrate[(1/(x^2)) (1/(((6 + (x^2)))^2)) (E^(((-(x^2))/4))), 
      x]