\[ \int \frac {1}{\sqrt {d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx \]
Optimal antiderivative \[ -\frac {5 e}{3 \left (-a \,e^{2}+c \,d^{2}\right )^{2} \left (e x +d \right )^{\frac {3}{2}}}-\frac {1}{\left (-a \,e^{2}+c \,d^{2}\right ) \left (c d x +a e \right ) \left (e x +d \right )^{\frac {3}{2}}}+\frac {5 c^{\frac {3}{2}} d^{\frac {3}{2}} e \arctanh \left (\frac {\sqrt {c}\, \sqrt {d}\, \sqrt {e x +d}}{\sqrt {-a \,e^{2}+c \,d^{2}}}\right )}{\left (-a \,e^{2}+c \,d^{2}\right )^{\frac {7}{2}}}-\frac {5 c d e}{\left (-a \,e^{2}+c \,d^{2}\right )^{3} \sqrt {e x +d}} \]
command
integrate(1/(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
\[ -\frac {5 \, c^{2} d^{2} \arctan \left (\frac {\sqrt {x e + d} c d}{\sqrt {-c^{2} d^{3} + a c d e^{2}}}\right ) e}{{\left (c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6}\right )} \sqrt {-c^{2} d^{3} + a c d e^{2}}} - \frac {\sqrt {x e + d} c^{2} d^{2} e}{{\left (c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6}\right )} {\left ({\left (x e + d\right )} c d - c d^{2} + a e^{2}\right )}} - \frac {2 \, {\left (6 \, {\left (x e + d\right )} c d e + c d^{2} e - a e^{3}\right )}}{3 \, {\left (c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6}\right )} {\left (x e + d\right )}^{\frac {3}{2}}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Timed out} \]________________________________________________________________________________________