\[ \int \sqrt {a+c x^4} \, dx \]
Optimal antiderivative \[ \frac {x \sqrt {c \,x^{4}+a}}{3}+\frac {a^{\frac {3}{4}} \sqrt {\frac {\cos \left (4 \arctan \left (\frac {c^{\frac {1}{4}} x}{a^{\frac {1}{4}}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (2 \arctan \left (\frac {c^{\frac {1}{4}} x}{a^{\frac {1}{4}}}\right )\right ), \frac {\sqrt {2}}{2}\right ) \left (\sqrt {a}+x^{2} \sqrt {c}\right ) \sqrt {\frac {c \,x^{4}+a}{\left (\sqrt {a}+x^{2} \sqrt {c}\right )^{2}}}}{3 \cos \left (2 \arctan \left (\frac {c^{\frac {1}{4}} x}{a^{\frac {1}{4}}}\right )\right ) c^{\frac {1}{4}} \sqrt {c \,x^{4}+a}} \]
command
integrate((c*x^4+a)^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {2}{3} \, \sqrt {c} \left (-\frac {a}{c}\right )^{\frac {3}{4}} {\rm ellipticF}\left (\frac {\left (-\frac {a}{c}\right )^{\frac {1}{4}}}{x}, -1\right ) + \frac {1}{3} \, \sqrt {c x^{4} + a} x \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\sqrt {c x^{4} + a}, x\right ) \]