\[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x)^{9/2}} \, dx \]
Optimal antiderivative \[ \frac {c^{3} d^{3} \arctan \left (\frac {\sqrt {e}\, \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}{\sqrt {-a \,e^{2}+c \,d^{2}}\, \sqrt {e x +d}}\right )}{8 e^{\frac {3}{2}} \left (-a \,e^{2}+c \,d^{2}\right )^{\frac {5}{2}}}-\frac {\sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}{3 e \left (e x +d \right )^{\frac {7}{2}}}+\frac {c d \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}{12 e \left (-a \,e^{2}+c \,d^{2}\right ) \left (e x +d \right )^{\frac {5}{2}}}+\frac {c^{2} d^{2} \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}{8 e \left (-a \,e^{2}+c \,d^{2}\right )^{2} \left (e x +d \right )^{\frac {3}{2}}} \]
command
integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(e*x+d)^(9/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {{\left (\frac {3 \, c^{4} d^{4} \arctan \left (\frac {\sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}}}{\sqrt {c d^{2} e - a e^{3}}}\right ) e}{{\left (c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} \sqrt {c d^{2} e - a e^{3}}} - \frac {{\left (3 \, \sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}} c^{6} d^{8} e^{3} - 6 \, \sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}} a c^{5} d^{6} e^{5} - 8 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} c^{5} d^{6} e^{2} + 3 \, \sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}} a^{2} c^{4} d^{4} e^{7} + 8 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} a c^{4} d^{4} e^{4} - 3 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {5}{2}} c^{4} d^{4} e\right )} e^{\left (-3\right )}}{{\left (c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} {\left (x e + d\right )}^{3} c^{3} d^{3}}\right )} e^{\left (-2\right )}}{24 \, c d} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int \frac {\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{{\left (e x + d\right )}^{\frac {9}{2}}}\,{d x} \]________________________________________________________________________________________