How to integrate $\int \frac{1}{\sqrt{1+29x^2+100x^4}}dx$ and $\int \frac{1}{\sqrt{1-2x^2-8x^4}}dx$ using elliptic functions? How to integrate $$\int \frac{1}{\sqrt{1+29x^2+100x^4}}dx$$ and $$\int\frac{1}{\sqrt{1-2x^2-8x^4}}dx$$  using elliptic functions?
I have tried to use them, but I got incorrect formula $$\frac{1}{\sqrt{2}}F(arctan(\sqrt{2}x)∣3)$$ for the first one (second argument should be less or equal 1).
Could anyone solve it?
Thanks

I'm not totally sure how to do it by hand, but using the identity
$$\int\frac{1}{\sqrt{a+bx^2+cx^4}}\mathrm{d}x=-\frac{i \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^2}{\sqrt{b^2-4 a c}+b}} \sqrt{\frac{2 c
   x^2}{b-\sqrt{b^2-4 a c}}+1} F\left(i \sinh ^{-1}\left(\sqrt{2}
   \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} x\right)|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a
   c}}\right)}{\sqrt{2} \sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}} \sqrt{a+b x^2+c x^4}}$$
one obtains 
$$\int\frac{1}{\sqrt{1+29x^2+100x^4}}\mathrm{d}x=-\frac{i \sqrt{4 x^2+1} \sqrt{25 x^2+1} F\left(i \sinh ^{-1}(5 x)|\frac{4}{25}\right)}{5
   \sqrt{100 x^4+29 x^2+1}}$$
and
$$\int\frac{1}{\sqrt{1-2x^2-8x^4}}\mathrm{d}x=\frac{1}{2} F\left(\sin ^{-1}(2 x)|-\frac{1}{2}\right)$$
which agrees with numerical experiments.
