\[ \int \frac {c+d x}{\sqrt {a-b x^4}} \, dx \]
Optimal antiderivative \[ \frac {d \arctan \left (\frac {x^{2} \sqrt {b}}{\sqrt {-b \,x^{4}+a}}\right )}{2 \sqrt {b}}+\frac {a^{\frac {1}{4}} c \EllipticF \left (\frac {b^{\frac {1}{4}} x}{a^{\frac {1}{4}}}, i\right ) \sqrt {1-\frac {b \,x^{4}}{a}}}{b^{\frac {1}{4}} \sqrt {-b \,x^{4}+a}} \]
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
\[ \frac {4 \, \sqrt {-b} b c \left (\frac {a}{b}\right )^{\frac {3}{4}} {\rm ellipticF}\left (\frac {\left (\frac {a}{b}\right )^{\frac {1}{4}}}{x}, -1\right ) - a \sqrt {-b} d \log \left (2 \, b x^{4} - 2 \, \sqrt {-b x^{4} + a} \sqrt {-b} x^{2} - a\right )}{4 \, a b} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-\frac {\sqrt {-b x^{4} + a} {\left (d x + c\right )}}{b x^{4} - a}, x\right ) \]