Big O notation simplification from sum

On calculating the convexity of an optimization problem, I am getting a term $$O(\sqrt{n+m}(n)^3)$$. Here both $$m$$ and $$n$$ are parameters. Is there any way I can simplify this term to write it as a product?
I know that if time complexity $$T(n,m)=O(\sqrt{n+m}(n)^3)$$, then $$\exists C,M$$ such that $$\vert T(n,m) \vert \leq C\vert \sqrt{n+m}(n)^3 \vert$$ when either $$n,m \geq M$$. I need the simplification for the interpretation of the algorithm. (Like, if time complexity is $$O(n)$$, then if $$n$$ is multiplied by 10, then the time complexity is also multiplied by 10. But it seems such an interpretation is not possible in the above case, as $$m$$ and $$n$$ are independent terms in summation).

• As $\sqrt{n}<n^3\sqrt{n+m}$ and $\sqrt{m}<n^3\sqrt{n+m}$, then you cannot estimate $n^3\sqrt{n+m}$ with upper bound dependent only on one variable. Commented Jun 21, 2022 at 21:40
• I assume that $n/m$ means "both $n$ and $m$", and not the quotient ?
– user16034
Commented Jun 22, 2022 at 8:40
• @YvesDaoust I meant n or m. Commented Jun 22, 2022 at 8:57
• Better write $n,m\ge M$. (And I doubt that the or interpretation be correct.)
– user16034
Commented Jun 22, 2022 at 9:12

Presumably you are after a substitution like$$\sqrt{n+m}\,n^3\le f(n)\,g(m),$$ or equivalently

$$n+m\le p(n)\,q(m).$$

If you freeze $$m$$, then $$p(n)=\Omega(n)$$ must hold (and similarly $$q(m)=\Omega(m)$$). So I see no better solution than

$$\sqrt{n+m}\,n^3\le\sqrt{nm}\,n^3= n^{7/2}m^{1/2}.$$

If you multiply $$m$$ by $$10$$, the magnitude of the term varies by a factor

$$\sqrt{\frac{n+10m}{n+m}}=\sqrt{\frac{\frac nm+10}{\frac nm+1}}.$$

That factor stays in range $$\left(1,\sqrt{10}\right)$$ and varies relatively little as a function of $$\frac nm$$.

If you multiply $$n$$ by $$10$$, the magnitude varies by $$1000$$ times a similar factor.

So it is not very wrong to explain the complexity as "roughly $$n^{7/2}$$".