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