\[ \int \frac {\sqrt {d+e x^2}}{x \left (a+b x^2+c x^4\right )} \, dx \]
Optimal antiderivative \[ -\frac {\arctanh \left (\frac {\sqrt {e \,x^{2}+d}}{\sqrt {d}}\right ) \sqrt {d}}{a}+\frac {\arctanh \left (\frac {\sqrt {2}\, \sqrt {c}\, \sqrt {e \,x^{2}+d}}{\sqrt {2 c d -e \left (b -\sqrt {-4 a c +b^{2}}\right )}}\right ) \sqrt {c}\, \left (b d -2 a e +d \sqrt {-4 a c +b^{2}}\right ) \sqrt {2}}{2 a \sqrt {-4 a c +b^{2}}\, \sqrt {2 c d -e \left (b -\sqrt {-4 a c +b^{2}}\right )}}-\frac {\arctanh \left (\frac {\sqrt {2}\, \sqrt {c}\, \sqrt {e \,x^{2}+d}}{\sqrt {2 c d -e \left (b +\sqrt {-4 a c +b^{2}}\right )}}\right ) \sqrt {c}\, \left (b d -2 a e -d \sqrt {-4 a c +b^{2}}\right ) \sqrt {2}}{2 a \sqrt {-4 a c +b^{2}}\, \sqrt {2 c d -e \left (b +\sqrt {-4 a c +b^{2}}\right )}} \]
command
integrate((e*x^2+d)^(1/2)/x/(c*x^4+b*x^2+a),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 \[ \text {Timed out} \]