\[ \int \frac {\sqrt {d+e x}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx \]
Optimal antiderivative \[ \frac {3 e \arctanh \left (\frac {\sqrt {c}\, \sqrt {d}\, \sqrt {e x +d}}{\sqrt {-a \,e^{2}+c \,d^{2}}}\right ) \sqrt {c}\, \sqrt {d}}{\left (-a \,e^{2}+c \,d^{2}\right )^{\frac {5}{2}}}-\frac {3 e}{\left (-a \,e^{2}+c \,d^{2}\right )^{2} \sqrt {e x +d}}-\frac {1}{\left (-a \,e^{2}+c \,d^{2}\right ) \left (c d x +a e \right ) \sqrt {e x +d}} \]
command
integrate((e*x+d)^(1/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^2,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {3 \, c d \arctan \left (\frac {\sqrt {x e + d} c d}{\sqrt {-c^{2} d^{3} + a c d e^{2}}}\right ) e}{{\left (c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} \sqrt {-c^{2} d^{3} + a c d e^{2}}} - \frac {3 \, {\left (x e + d\right )} c d e - 2 \, c d^{2} e + 2 \, a e^{3}}{{\left (c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} {\left ({\left (x e + d\right )}^{\frac {3}{2}} c d - \sqrt {x e + d} c d^{2} + \sqrt {x e + d} a e^{2}\right )}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Timed out} \]________________________________________________________________________________________