I am interested in evaluating some elliptic integrals, and I have not been able to secure a reference to do exactly what I need. Most of the references I've found seem to focus on reducing more general elliptic integrals to Legendre form, but leave out the part about actually dealing with complete elliptic integrals. In particular, I am interested in the following toy problem, which is to show the following: $$\displaystyle \int_0^\infty \frac{dx}{\sqrt{x(x+2)(x+3)}} = \sqrt{2}K(-1/2),$$ where $K(k)$ is the complete elliptic integral of the first kind. The above result is due to Wolfram Alpha.

I tried the obvious substitution which is $x = \tan \theta$, and after some labour we obtain the integral

$$\displaystyle \int_0^{\pi/2} \frac{2 d \theta}{\sqrt{7 \sin 2\theta + 5 \sin^2 2 \theta + 5 \sin 2 \theta \cos 2 \theta}},$$ which again can be checked to evaluate to $\sqrt{2}K(-1/2)$, although in this case Wolfram only gave the numerical value and not the closed form. Further, this last one does not look like what the 'correct' form should be, which is

$$\displaystyle \int_0^{\pi/2} \frac{\sqrt{2} d \theta}{\sqrt{1 - (1/4)\sin^2 \theta}}.$$

Based on some data I got from playing around with Wolfram, I suspect that the following is true: Suppose that $0 < a < b$. Then

$$\displaystyle \int_0^\infty \frac{dx}{\sqrt{x(x+a)(x+b)}} = \frac{2 K(1 - b/a)}{\sqrt{a}}.$$

Any help would be much appreciated, I apologize if this problem is in fact trivial or well-known.