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