Trouble integrating $\int{\sqrt{1+x^{2}+x^{4}}} dx$ I'm confronted with the below integral:

$$\int{\sqrt{1+x^{2}+x^{4}}} dx$$

which has appeared within integrating a helio-central magnetic field.

The numerical evaluation just did work for our application. But I am curious to find a way for analytical calculation of it.

Does anybody have any idea?...

Best
Mathematica says:

$$
\frac{x^5+x^3+(-1)^{5/6} \sqrt{3 \sqrt[3]{-1} x^2+3} \sqrt{1-(-1)^{2/3} x^2} F\left(i
   \sinh ^{-1}\left((-1)^{5/6} x\right)|(-1)^{2/3}\right)+\sqrt[3]{-1} \sqrt{\sqrt[3]{-1}
   x^2+1} \sqrt{1-(-1)^{2/3} x^2} E\left(i \sinh ^{-1}\left((-1)^{5/6}
   x\right)|(-1)^{2/3}\right)+x}{3 \sqrt{x^4+x^2+1}},
$$ where $E, F$ are elliptic functions.
Mathematica says:

$$
\frac{x^5+x^3+(-1)^{5/6} \sqrt{3 \sqrt[3]{-1} x^2+3} \sqrt{1-(-1)^{2/3} x^2} F\left(i
   \sinh ^{-1}\left((-1)^{5/6} x\right)|(-1)^{2/3}\right)+\sqrt[3]{-1} \sqrt{\sqrt[3]{-1}
   x^2+1} \sqrt{1-(-1)^{2/3} x^2} E\left(i \sinh ^{-1}\left((-1)^{5/6}
   x\right)|(-1)^{2/3}\right)+x}{3 \sqrt{x^4+x^2+1}},
$$ where $E, F$ are elliptic functions.