\[17x^{2}+22y^{2}+\displaystyle \left(\int^{\infty}_{0} \frac{1}{2t^{4}+t^{2}+1} \ dt \right)x \\ +\displaystyle \left(\int^{1}_{0} 2z(1-z)e^{2z(1-z)} \ dz \right)y+ \left(25!\right)!=0\]

Find the eccentricity of the ellipse satisfying the equation above. Give your answer up to three decimal places.

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