\[ \int \frac {1}{\sqrt {d+e x} \left (a+b x+c x^2\right )^2} \, dx \]
Optimal antiderivative \[ -\frac {\left (b c d -b^{2} e +2 a c e +c \left (-b e +2 c d \right ) x \right ) \sqrt {e x +d}}{\left (-4 a c +b^{2}\right ) \left (a \,e^{2}-b d e +c \,d^{2}\right ) \left (c \,x^{2}+b x +a \right )}+\frac {\arctanh \left (\frac {\sqrt {2}\, \sqrt {c}\, \sqrt {e x +d}}{\sqrt {2 c d -e \left (b -\sqrt {-4 a c +b^{2}}\right )}}\right ) \sqrt {c}\, \left (8 c^{2} d^{2}-b \,e^{2} \left (b +\sqrt {-4 a c +b^{2}}\right )-2 c e \left (4 b d -6 a e -d \sqrt {-4 a c +b^{2}}\right )\right ) \sqrt {2}}{2 \left (-4 a c +b^{2}\right )^{\frac {3}{2}} \left (a \,e^{2}-b d e +c \,d^{2}\right ) \sqrt {2 c d -e \left (b -\sqrt {-4 a c +b^{2}}\right )}}-\frac {\arctanh \left (\frac {\sqrt {2}\, \sqrt {c}\, \sqrt {e x +d}}{\sqrt {2 c d -e \left (b +\sqrt {-4 a c +b^{2}}\right )}}\right ) \sqrt {c}\, \left (8 c^{2} d^{2}-b \,e^{2} \left (b -\sqrt {-4 a c +b^{2}}\right )-2 c e \left (4 b d -6 a e +d \sqrt {-4 a c +b^{2}}\right )\right ) \sqrt {2}}{2 \left (-4 a c +b^{2}\right )^{\frac {3}{2}} \left (a \,e^{2}-b d e +c \,d^{2}\right ) \sqrt {2 c d -e \left (b +\sqrt {-4 a c +b^{2}}\right )}} \]
command
integrate(1/(e*x+d)^(1/2)/(c*x^2+b*x+a)^2,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 \[ \text {Timed out} \]