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