Let the GP be $a_n=ar^{n-1}$ with $a>0$ and $r>1$ since the terms are increasing positive numbers.

We are given:

1. Product of 4th and 6th terms is $9$: $a_4a_6=(ar^3)(ar^5)=a^2r^8=9$ But $a_5=ar^4$ so $a_5^2=a^2r^8=9 \,\Rightarrow\, a_5=3$ (since all terms are positive).

2. Sum of 5th and 7th terms is $24$: $a_5+a_7=ar^4+ar^6=ar^4(1+r^2)=24$ Using $a_5=ar^4=3$, $3(1+r^2)=24$ $1+r^2=8$ $r^2=7$ Since $r>1$, $r=\sqrt{7}$

Now compute the required expression: $a_1a_9+a_2a_4a_9+a_5+a_7$

Step 1: Find $a_1a_9$

$a_1a_9=a\cdot ar^8=a^2r^8=9$

Step 2: Find $a_2a_4a_9$

$a_2a_4a_9=(ar)(ar^3)(ar^8)=a^3r^{12}$ Rewrite as $a^3r^{12}=(ar^4)^3=a_5^3=3^3=27$

Step 3: Find $a_5+a_7$

This is already given: $a_5+a_7=24$

Step 4: Add all parts

$a_1a_9+a_2a_4a_9+a_5+a_7=9+27+24=60$

Therefore, the required integer is $\boxed{60}$