Malefic Milkman goes to the Moon!

To celebrate his recent boost in powdered milk profits, the Malefic Milkman has decided to take a trip to the Moon.

Two hours into his journey, he decides that his rocket is travelling too slowly, so he wants to double its speed. At this point, the force of gravitational attraction from the Moon and from the Earth are equal.

How long will it take him to reach his destination if he doubles his speed?

$\textbf{Details and Assumptions}$

The Earth and the Moon can be modelled as stationary point particles.

The distance between the Earth and the Moon is 384,000 km.

The mass of the Earth is $5.97 \times 10^{24}$ kg. The mass of the Moon is $7.35 \times 10^{22}$ kg.

The mass of the Malefic Milkman and his rocket is $1.75 \times 10^{4}$ kg.

The time of acceleration for the Malefic Milkman's rocket is negligible, and before he doubles it, the speed of his rocket is constant.

Note: Give your answer in minutes, to three significant figures.