So How do you increase muscle size scientifically? The answer is by understanding the two types of muscle growth: Sarcoplasmic growth and Myofibrillar growth. When you lift with high reps and high volume, you increase the volume of fluid inside of the muscle which results in a quick increase in size. When you lift for lower reps the growth comes from an actual increase in muscle tissue. The best way is to maximize both types of growth.

High Reps and High Fatigue for an Increase in Fluids Within the Muscle
I know it sounds weird to increase the fluid (sarcoplasm) within a muscle, but this is legitimate growth. Our bodies or comprised largely of water so this is a legitimate way to get larger muscles. This sounds like an extremely temporary state, but that would be false. This is a good way to add size and as long as you train hard in the proper rep range, you will maintain this size.
Lower Reps and High Tension Result in Actual Muscle Fiber Growth
So why not just aim to increase muscle tissue and not worry about sarcoplasm? Well, this growth happens really slowly. If you just did low rep training it would take years to see a noticeable increase in muscle size. The biggest benefit is that muscle fiber growth (myofibrillar growth) adds density and makes the muscles firm. It also creates better definition when you increase muscle size. Another benefit is that this also makes the muscles much stronger than just increasing size through fluid increase.
A Scientific Approach is to Simply Alternate Between Both Types of Training
What you want to do is spend a period of 8-12 weeks alternating between the two types of muscle growth. The reason you don’t want to do high reps and high volume at the same time you do your lower rep strength training is that it confuses your body. When you try to be a “jack of all trades” your body responds by not adapting to either style of training very well.
For Those Who Like to Keep It Simple
If you aren’t rushed to increase muscle size as quickly as possible, you can focus in the middle rep range year round. The popular 5 sets of 5 reps routine works extremely well. You will experience slow and steady increases in muscle. The nice thing is that you will also gain strength the whole time as well. Pick 2-3 exercises per body part and do 5 sets of 5 reps. Don’t rest too long, but long enough to be able to use some relatively heavy weights. This is how you increase muscle size systematically.