Dueling functions

$f\left( x \right) =\int _{ x }^{ { x }^{ 2 } }{ \left( { t }^{ 2 }+3 \right) } d\left( g\left( t \right) \right) \\ g\left( x \right) ={ x }^{ 2 }\ln { x }$

Given the following composition of functions, $x$ can be expressed in the form ${ e }^{ -d }$ for $x>0$ . Under which real values of $d$ is $f\left( x \right)$ decreasing?

Details and Assumptions

$e$ denotes Euler's number, the base of the natural logarithm.

---

This problem is original.

Picture credits: F-16 Releases Four Flares , Wikipedia