Recently, I determined the volume of the solid of revolution created by rotating the parabola, $y=ax^2$, about the axis of revolution, $y=bx$, as a function of $a$ and $b$, where both variables are positive, real numbers to be
$$V=\frac{\pi b^5}{30a^3\sqrt{1+b^2}}\tag{1}$$
I proved this several different ways, but I got stuck at the conclusion of my application of the disk method. The final, integrable equation that I derived was
$$V=\int_0^{\frac{b}{a}\sqrt{1+b^2}}\pi\left(-\frac{1}{b}x-\frac{\sqrt{1+b^2}}{2ab^2}+\frac{\sqrt{1+b^2}}{2ab^2}\sqrt{4abx\sqrt{1+b^2}+1}\right)^2dx\tag{2}$$
I happen to know that equation 2 does simplify to equation 1 (thank you, Desmos). However, I could not think of a way of performing this simplification without a full-blown expansion of the integrand, separation at the plus/minus signs, integration, returning of limits, and simplification, which (I began but did not finish) took a head-exploding amount of care.
In effect, my question is: What is the most conservative, simplest, best way of evaluating equation 2? All solutions are welcome, but one that uses relatively elementary calculus concepts would be best (I'm a junior in high school taking AP BC Calculus).
Thanks!
Use computer algebra (e.g., Mathematica) which gives a very simple answer:
$$\frac{\pi b^5}{30 \sqrt{b^2+1}}$$