\[ \int \frac {\sqrt {a+b x^2+c x^4}}{x \left (d+e x^2\right )} \, dx \]
Optimal antiderivative \[ -\frac {\arctanh \left (\frac {b \,x^{2}+2 a}{2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}\right ) \sqrt {a}}{2 d}+\frac {\arctanh \left (\frac {2 c \,x^{2}+b}{2 \sqrt {c}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}\right ) \sqrt {c}}{2 e}-\frac {\arctanh \left (\frac {b d -2 a e +\left (-b e +2 c d \right ) x^{2}}{2 \sqrt {a \,e^{2}-b d e +c \,d^{2}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}\right ) \sqrt {a \,e^{2}-b d e +c \,d^{2}}}{2 d e} \]
command
integrate((c*x^4+b*x^2+a)^(1/2)/x/(e*x^2+d),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \text {output too large to display} \]
Fricas 1.3.7 via sagemath 9.3 output \[ \text {Timed out} \]