Evaluate: `int(2x+3)/(sqrt(x^2+4x+1))dx`

Answer

`2sqrt(x^(2)+4x+1)-log|x+2+sqrt(x^(2)+4x+1)|+C`

Explanation

`I=int(2x+3)/(sqrt(x^(2)+4x+1))dx=int((2x+4)-1)/(sqrt(x^(2)+4x+1))dx`
`=int(2x+4)/(sqrt(x^(2)+4x+1))dx-int(1)/(sqrt(x^(2)+4x+1))dx`
`=int(dt)/(sqrt(t))-int(1)/(sqrt((x+2)^(2)-(sqrt(3))^(2)))dx," where "t=x^(2)+4x+1`
`=2sqrt(t)-log|(x+2)+sqrt(x^(2)+4x+1)|+C`
`=2sqrt(x^(2)+4x+1)-log|x+2+sqrt(x^(2)+4x+1)|+C`