Evaluate `int sqrt(x^(2)+2x+5)dx`

Explanation

If we recognize that `(x^3//3) - x^2 + 5x` is an antiderivative of `x^2 – 2x + 5`, we can evaluate the integral as

If we do not recognize the antiderivative right away, we can generate it term by term with the sum and difference Rule :
`int(x^2 - 2x + 5)dx = intx^2 dx = int x^2 dx - int2xdx + int5dx`
`= (x^3)/(3) + C_(1) - x^2 + C_(2) + 5x + C_(3)`
This formula is more complicated than it needs to be. If we combine `C_1, C_2 and C_3` into a single constant `C = C_1 + C_2 + C_3`, the formula simplifies to
`(x^3)/3 - x^2 + 5x + C`
and still gives all the antiderivatives there are. For this reason we recommend that you go right to the final form even if you elect to integrate term by term. Write
`int(x^2 - 2x + 5)dx = intx^2 dx - int 2xdx + int 5 dx`
`= (x^3)/3 - x^2 + 5x + C`.
Find the simplest antiderivative you can for each part add the constant at the end.