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