Tips on solving $a^2 (a + 2 b) (a + b^2 + b)^{\frac{3}{2}}>b^3 (a^2 + a + b)^{\frac{3}{2}}$ $$a^2 (a + 2 b) (a + b^2 + b)^{\frac{3}{2}}>b^3 (a^2 + a + b)^{\frac{3}{2}}$$
This is true with $a>b>0$, according to Wolfram Alpha, but I am not able to prove this.

I to simplify using the fact that $a^2(a+2b)=a^3+2ba^2>3b^3$, and then proving a stronger result:
$$3b^3(a + b^2 + b)^{\frac{3}{2}}>b^3 (a^2 + a + b)^{\frac{3}{2}}$$
$$\iff 3(a + b^2 + b)>(a^2 + a + b)$$
$$\iff 3a + 3b^2 + 3b>a^2 + a + b$$
But this is not true, e.g. $a=100, b=1$.

Try moving all the terms raised to the power of 3/2 onto one side of the equation only, and move all other terms to the other side of the equation. Then square both sides so that the power of 3/2 becomes 3, and use algebra from there.
