∫(x + 5)/(x^2 + 9) dx =

∫x/(x^2 + 9) dx + ∫5/(x^2 + 9) dx =

1/2∫2x/(x^2 + 9) dx + ∫5/(x^2 + 9) dx =

ln|x^2 + 9)|/2 + 5arctan(x/3)/3 + c

∫(x + 5)/(x^2 + 9) dx

log(3) + (5 x)/9 + x^2/18 - (5 x^3)/243 + O(x^4)

(Taylor series)

Break it in 2 integrals

x * dx / (x^2 + 9)  +  5 * dx / (x^2 + 9)

u = x^2 + 9

du = 2x * dx

(1/2) * du / u  +  5 * dx / (9 + x^2)

x = 3 * tan(t)

dx = 3 * sec(t)^2 * dt

(1/2) * du / u  +  5 * 3 * sec(t)^2 * dt / (9 + 9 * tan(t)^2) =>

(1/2) * du / u  +  5 * 3 * sec(t)^2 * dt / (9 * sec(t)^2) =>

(1/2) * du / u  +  (5/3) * dt

Integrate

(1/2) * ln|u| + (5/3) * t + C =>

(1/2) * ln|x^2 + 9| + (5/3) * arctan(x/3) + C