Does the anti derivative of $\int \frac{1}{ax+b} dx$ have multiple answers?

Question

$$∫ \dfrac{1}{ax+b} dx = \dfrac{1}{a}\log_{e}{|ax+b|} + c$$

but if you factor out $1/a$ from the integral first then the anti derivative becomes

$$\cfrac{1}{a}\log_{e}\left|x+\dfrac{b}{a}\right| + c$$

which is correct? If both why?