Simplify the integrand:
[tex]\dfrac{4x^4 + 3x^2 + 5x}{x^2} = 4x^2 + 3 + \dfrac5x[/tex]
Now integrate one term at a time.
[tex]\displaystyle \int 4x^2 \, dx = \frac43 x^3 + C[/tex]
[tex]\displaystyle \int 3 \, dx = 3x + C[/tex]
[tex]\displaystyle \int \frac5x \, dx = 5 \ln|x| + C[/tex]
Putting everything together,
[tex]\displaystyle \int \frac{4x^4 + 3x^2 + 5x}{x^2} \, dx = \int \left(4x^2 + 3 + \frac5x\right) \, dx = \boxed{\frac43 x^3 + 3x + 5 \ln|x| + C}[/tex]
