\[ \int \frac {\sqrt {d+e x}}{\left (a+b x+c x^2\right )^3} \, dx \]
Optimal antiderivative \[ -\frac {\left (2 c x +b \right ) \sqrt {e x +d}}{2 \left (-4 a c +b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{2}}-\frac {\left (13 b^{2} c d e -4 a \,c^{2} d e -b^{3} e^{2}-4 b c \left (2 a \,e^{2}+3 c \,d^{2}\right )-c \left (24 c^{2} d^{2}+b^{2} e^{2}-4 c e \left (-5 a e +6 b d \right )\right ) x \right ) \sqrt {e x +d}}{4 \left (-4 a c +b^{2}\right )^{2} \left (a \,e^{2}-b d e +c \,d^{2}\right ) \left (c \,x^{2}+b x +a \right )}-\frac {\arctanh \left (\frac {\sqrt {2}\, \sqrt {c}\, \sqrt {e x +d}}{\sqrt {2 c d -e \left (b -\sqrt {-4 a c +b^{2}}\right )}}\right ) \sqrt {c}\, \left (96 c^{3} d^{3}+b^{2} e^{3} \left (b +\sqrt {-4 a c +b^{2}}\right )-8 c^{2} d e \left (18 b d -13 a e -3 d \sqrt {-4 a c +b^{2}}\right )+2 c \,e^{2} \left (23 b^{2} d +10 a e \sqrt {-4 a c +b^{2}}-2 b \left (13 a e +6 d \sqrt {-4 a c +b^{2}}\right )\right )\right ) \sqrt {2}}{8 \left (-4 a c +b^{2}\right )^{\frac {5}{2}} \left (a \,e^{2}-b d e +c \,d^{2}\right ) \sqrt {2 c d -e \left (b -\sqrt {-4 a c +b^{2}}\right )}}+\frac {\arctanh \left (\frac {\sqrt {2}\, \sqrt {c}\, \sqrt {e x +d}}{\sqrt {2 c d -e \left (b +\sqrt {-4 a c +b^{2}}\right )}}\right ) \sqrt {c}\, \left (96 c^{3} d^{3}+b^{2} e^{3} \left (b -\sqrt {-4 a c +b^{2}}\right )-8 c^{2} d e \left (18 b d -13 a e +3 d \sqrt {-4 a c +b^{2}}\right )+2 c \,e^{2} \left (23 b^{2} d -26 a b e +12 b d \sqrt {-4 a c +b^{2}}-10 a e \sqrt {-4 a c +b^{2}}\right )\right ) \sqrt {2}}{8 \left (-4 a c +b^{2}\right )^{\frac {5}{2}} \left (a \,e^{2}-b d e +c \,d^{2}\right ) \sqrt {2 c d -e \left (b +\sqrt {-4 a c +b^{2}}\right )}} \]
command
integrate((e*x+d)^(1/2)/(c*x^2+b*x+a)^3,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 \[ \text {Timed out} \]