Answer
$${\text{False}}$$
Work Step by Step
$$\eqalign{
& \int {\frac{{3{x^4} + 5}}{{{{\left( {{x^2} + 1} \right)}^2}}}dx} \cr
& {\text{False, the integrand }}\frac{{3{x^4} + 5}}{{{{\left( {{x^2} + 1} \right)}^2}}}{\text{ is not a proper rational function}} \cr
& {\text{because if we expand the expression }}{\left( {{x^2} + 1} \right)^2}{\text{ we got }} \cr
& {x^4} + 2{x^2} + 1,{\text{ then}} \cr
& \frac{{3{x^4} + 5}}{{{{\left( {{x^2} + 1} \right)}^2}}} = \frac{{3{x^4} + 5}}{{{x^4} + 2{x^2} + 1}} \cr
& {\text{The numerator and denominator have the same degree, so the}} \cr
& {\text{integrand is not a proper rational function}}{\text{.}} \cr} $$
