First: there is not such thing as "the antiderivative of $f(x)$". There are generally infinitely many different functions that are antiderivatives of $f(x)$. What you can talk about is either "the family of antiderivatives" or "the most general antiderivative", or else "an antiderivative". It seems clear you are looking for "the most general antiderivative."
Remember that the derivative of $\ln(x)$ is $\frac{1}{x}$. And the derivative of $kf(x)$ with $k$ a constant is $kf'(x)$.
So to get a derivative equal to $-\frac{8}{x} = -8\left(\frac{1}{x}\right)$, you can take $-8f(x)$, where $f(x)$ is a function whose derivative is $\frac{1}{x}$. For example.... $\ln(x)$.
(In fact, the most general function whose derivative is $\frac{1}{x}$ is $\ln|x|+C$, because $(\ln(-x))' = \frac{1}{-x}(-x)' = \frac{1}{x}$, using the Chain Rule.)
So we can, and should, take $-8\ln|x|$ to get the $-\frac{8}{x}$ part of the most general antiderivative. For $\frac{3}{5}$, we can take $\frac{3}{5}x$. And finally, a sum of antiderivatives is an antiderivative of a sum because the derivative of a sum is the sum of antiderivatives; so we take the sum of the two parts, and add the constant to get the general antiderivative. We have: $\underbrace{\frac{3}{5}x}_{\text{antiderivative of }\frac{3}{4}} -\underbrace{8\ln|x|}_{\text{antiderivative of }\frac{8}{x}} + C.$