\[ \int \frac {1}{(d+e x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx \]
Optimal antiderivative \[ \frac {1}{3 \left (-a \,e^{2}+c \,d^{2}\right ) \left (e x +d \right )^{\frac {3}{2}} \left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}\right )^{\frac {3}{2}}}+\frac {105 c^{3} d^{3} e^{\frac {3}{2}} \arctan \left (\frac {\sqrt {e}\, \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}{\sqrt {-a \,e^{2}+c \,d^{2}}\, \sqrt {e x +d}}\right )}{8 \left (-a \,e^{2}+c \,d^{2}\right )^{\frac {11}{2}}}+\frac {3 c d}{4 \left (-a \,e^{2}+c \,d^{2}\right )^{2} \left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}\right )^{\frac {3}{2}} \sqrt {e x +d}}-\frac {7 c^{2} d^{2} \sqrt {e x +d}}{4 \left (-a \,e^{2}+c \,d^{2}\right )^{3} \left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}\right )^{\frac {3}{2}}}-\frac {35 c^{2} d^{2} e}{8 \left (-a \,e^{2}+c \,d^{2}\right )^{4} \sqrt {e x +d}\, \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}+\frac {105 c^{3} d^{3} e \sqrt {e x +d}}{8 \left (-a \,e^{2}+c \,d^{2}\right )^{5} \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}} \]
command
integrate(1/(e*x+d)^(3/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {1}{24} \, {\left (\frac {315 \, c^{3} d^{3} \arctan \left (\frac {\sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}}}{\sqrt {c d^{2} e - a e^{3}}}\right ) e}{{\left (c^{5} d^{10} - 5 \, a c^{4} d^{8} e^{2} + 10 \, a^{2} c^{3} d^{6} e^{4} - 10 \, a^{3} c^{2} d^{4} e^{6} + 5 \, a^{4} c d^{2} e^{8} - a^{5} e^{10}\right )} \sqrt {c d^{2} e - a e^{3}}} - \frac {16 \, c^{7} d^{11} e^{5} - 64 \, a c^{6} d^{9} e^{7} - 144 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )} c^{6} d^{9} e^{4} + 96 \, a^{2} c^{5} d^{7} e^{9} + 432 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )} a c^{5} d^{7} e^{6} - 693 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{2} c^{5} d^{7} e^{3} - 64 \, a^{3} c^{4} d^{5} e^{11} - 432 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )} a^{2} c^{4} d^{5} e^{8} + 1386 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{2} a c^{4} d^{5} e^{5} - 840 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{3} c^{4} d^{5} e^{2} + 16 \, a^{4} c^{3} d^{3} e^{13} + 144 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )} a^{3} c^{3} d^{3} e^{10} - 693 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{2} a^{2} c^{3} d^{3} e^{7} + 840 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{3} a c^{3} d^{3} e^{4} - 315 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{4} c^{3} d^{3} e}{{\left (c^{5} d^{10} - 5 \, a c^{4} d^{8} e^{2} + 10 \, a^{2} c^{3} d^{6} e^{4} - 10 \, a^{3} c^{2} d^{4} e^{6} + 5 \, a^{4} c d^{2} e^{8} - a^{5} e^{10}\right )} {\left (\sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}} c d^{2} e - \sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}} a e^{3} + {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}}\right )}^{3}}\right )} e \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: TypeError} \]________________________________________________________________________________________