Good game

gistfile1.hs

import Numeric.AD

assert = print

eval = id

main = do
    let a x = 2 * x^2 + 3
    let b = eval a 1
    assert (b == 5)         -- Prints `True`

    let d = diff a
    assert (eval d 1 == 4)  -- Prints `True`

    -- Of course, this is silly, we can just get rid of `eval` and `assert`
    -- completely ...
    print (     a 1)        -- Prints `5`
    print (diff a 1)        -- Prints `4`

    -- Your symbolic integration is undefined: you need to specify integration
    -- bounds

say poly = 3x^4 + 5x^3+2x^2+7x+8
then encode poly as 2 vectors
poly = (3 5 2 7 1) (4 3 2 1 0)
diff would just be dot product
( 3_4 5_3 2_2 7_1 1) (3 2 1 0 0)
int would be similar too..