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