*I need help with this problem*

Feb 2019
United Arab Emirates
Hello everyone,

So I started a mathematics research in calculus.
I was trying to find which quadratic equation that goes through point A(-5,5) and B(5,5) would give the minimum surface area of revolution.
As the general formula of a quadratic function is
After putting A and B into the general formula, I found that f(x)=ax^2-25a+5 would be the general formula for the quadratic equations that pass the two points.
Using the surface area of revolution formula
I ended up at following expression (check attachment please)

So if i solved the definite integral above it would give me an expression in a and then dA/da=0 and d^2A/da^2>0 would give me the a value at which the surface area is the minimum right?
However, the integration became too complicated and that is where I'm stuck.
I did get the answer to that integral but differentiating that again is overwhelming.
So, I'm asking if there was a way to simplify the method or the calculation.