three integrals sum to a $_{3}F_{2}$ value

Question

Let $K(x)$ and $E(x)$ denote complete elliptic integrals of the first and second kind. Let $$A=\frac{1024}{9\pi^{3}} \int\limits_{0}^{\infty} \,\frac{t\left( 8t^{4}+8t^{2}-1\right) E\left( i\,t\right) ^{2}}{\left( 2t^{2}+1\right) ^{5}}dt,$$
$$B=\frac{-2048}{9\pi^{3}} \int\limits_{0}^{\infty} \,\frac{t\left( 4t^{4}+3t^{2}-1\right) E\left( i\,t\right) K\left( i\,t\right) }{\left( 2t^{2}+1\right) ^{5}}dt,$$
$$C=\frac{1024}{9\pi^{3}} \int\limits_{0}^{\infty} \,\frac{t\left( 2t^{4}+t^{2}-1\right) K\left( i\,t\right) ^{2}}{\left( 2t^{2}+1\right) ^{5}}dt.$$
Prove that $A+B+C=_{3}F_{2}\left(-\tfrac{1}{2},\tfrac{1}{2},\tfrac{3}{2};1,2;1\right)$. If possible, find $A$, $B$ and $C$ separately as well.
This would solve an open problem in convex geometry (http://arxiv.org/abs/1204.3468). Thanks!