Taken from the answer to In Parent chain choosing should one consider longest chain or chain with more substituents? by Loong.
From the Nomenclature of Organic Chemistry – IUPAC Recommendations and Preferred Names 2013 (Blue Book)
P-45.2.1 The preferred IUPAC name is based on the senior parent structure that has the maximum number of substituents cited as prefixes (other than ‘hydro/dehydro’) to the parent structure.
Therefore in this case, the diethyl and methyl make up 3 substituent groups whereas the ethyl and isopropyl group only make up 2 substituents. Therefore the chain that has the diethyl and methyl group needs to be taken as the parent chain here.
Therefore, the PIN of this compound is 3,3-diethyl-2-methylhexane.
As a side note: A brief set of the IUPAC rules can be found here