<s>[INST] 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. $a=100, b=1$.
 [/INST] We prove the stronger result that comes from replacing $a+2b$ on the left by $a$
Note that for $0 \lt b \lt a, \frac 1b \gt \frac 1a$ and $\frac a{b^2} \gt \frac b{a^2}$  Then
$$1+\frac 1b+\frac a{b^2} \gt 1+\frac 1a+\frac b{a^2}\\
\sqrt{1+\frac 1b+\frac a{b^2}} \gt  \sqrt{1+\frac 1a+\frac b{a^2}}\\
ab\sqrt{1+\frac 1b+\frac a{b^2}} \gt ab \sqrt{1+\frac 1a+\frac b{a^2}}\\
a\sqrt{a+b^2+b} \gt b\sqrt{a^2+a+b}\\
a^3(a+b^2+b)^{3/2} \gt b^3(a^2+a+b)^{3/2}\\
a^2(a+2b)(a+b^2+b)^{3/2}\gt b^3(a^2+a+b)^{3/2}$$
</s>