\[ \int \frac {1}{\sqrt {d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx \]
Optimal antiderivative \[ \frac {63 e^{2}}{20 \left (-a \,e^{2}+c \,d^{2}\right )^{3} \left (e x +d \right )^{\frac {5}{2}}}-\frac {1}{2 \left (-a \,e^{2}+c \,d^{2}\right ) \left (c d x +a e \right )^{2} \left (e x +d \right )^{\frac {5}{2}}}+\frac {9 e}{4 \left (-a \,e^{2}+c \,d^{2}\right )^{2} \left (c d x +a e \right ) \left (e x +d \right )^{\frac {5}{2}}}+\frac {21 c d \,e^{2}}{4 \left (-a \,e^{2}+c \,d^{2}\right )^{4} \left (e x +d \right )^{\frac {3}{2}}}-\frac {63 c^{\frac {5}{2}} d^{\frac {5}{2}} e^{2} \arctanh \left (\frac {\sqrt {c}\, \sqrt {d}\, \sqrt {e x +d}}{\sqrt {-a \,e^{2}+c \,d^{2}}}\right )}{4 \left (-a \,e^{2}+c \,d^{2}\right )^{\frac {11}{2}}}+\frac {63 c^{2} d^{2} e^{2}}{4 \left (-a \,e^{2}+c \,d^{2}\right )^{5} \sqrt {e x +d}} \]
command
integrate(1/(e*x+d)^(1/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {63 \, c^{3} d^{3} \arctan \left (\frac {\sqrt {x e + d} c d}{\sqrt {-c^{2} d^{3} + a c d e^{2}}}\right ) e^{2}}{4 \, {\left (c^{5} d^{10} - 5 \, a c^{4} d^{8} e^{2} + 10 \, a^{2} c^{3} d^{6} e^{4} - 10 \, a^{3} c^{2} d^{4} e^{6} + 5 \, a^{4} c d^{2} e^{8} - a^{5} e^{10}\right )} \sqrt {-c^{2} d^{3} + a c d e^{2}}} + \frac {15 \, {\left (x e + d\right )}^{\frac {3}{2}} c^{4} d^{4} e^{2} - 17 \, \sqrt {x e + d} c^{4} d^{5} e^{2} + 17 \, \sqrt {x e + d} a c^{3} d^{3} e^{4}}{4 \, {\left (c^{5} d^{10} - 5 \, a c^{4} d^{8} e^{2} + 10 \, a^{2} c^{3} d^{6} e^{4} - 10 \, a^{3} c^{2} d^{4} e^{6} + 5 \, a^{4} c d^{2} e^{8} - a^{5} e^{10}\right )} {\left ({\left (x e + d\right )} c d - c d^{2} + a e^{2}\right )}^{2}} + \frac {2 \, {\left (30 \, {\left (x e + d\right )}^{2} c^{2} d^{2} e^{2} + 5 \, {\left (x e + d\right )} c^{2} d^{3} e^{2} + c^{2} d^{4} e^{2} - 5 \, {\left (x e + d\right )} a c d e^{4} - 2 \, a c d^{2} e^{4} + a^{2} e^{6}\right )}}{5 \, {\left (c^{5} d^{10} - 5 \, a c^{4} d^{8} e^{2} + 10 \, a^{2} c^{3} d^{6} e^{4} - 10 \, a^{3} c^{2} d^{4} e^{6} + 5 \, a^{4} c d^{2} e^{8} - a^{5} e^{10}\right )} {\left (x e + d\right )}^{\frac {5}{2}}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________