\[ \int \frac {1}{(d+e x)^{7/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx \]
Optimal antiderivative \[ \frac {1}{5 \left (-a \,e^{2}+c \,d^{2}\right ) \left (e x +d \right )^{\frac {7}{2}} \left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}\right )^{\frac {3}{2}}}+\frac {13 c d}{40 \left (-a \,e^{2}+c \,d^{2}\right )^{2} \left (e x +d \right )^{\frac {5}{2}} \left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}\right )^{\frac {3}{2}}}+\frac {143 c^{2} d^{2}}{240 \left (-a \,e^{2}+c \,d^{2}\right )^{3} \left (e x +d \right )^{\frac {3}{2}} \left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}\right )^{\frac {3}{2}}}+\frac {3003 c^{5} d^{5} e^{\frac {3}{2}} \arctan \left (\frac {\sqrt {e}\, \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}{\sqrt {-a \,e^{2}+c \,d^{2}}\, \sqrt {e x +d}}\right )}{128 \left (-a \,e^{2}+c \,d^{2}\right )^{\frac {15}{2}}}+\frac {429 c^{3} d^{3}}{320 \left (-a \,e^{2}+c \,d^{2}\right )^{4} \left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}\right )^{\frac {3}{2}} \sqrt {e x +d}}-\frac {1001 c^{4} d^{4} \sqrt {e x +d}}{320 \left (-a \,e^{2}+c \,d^{2}\right )^{5} \left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}\right )^{\frac {3}{2}}}-\frac {1001 c^{4} d^{4} e}{128 \left (-a \,e^{2}+c \,d^{2}\right )^{6} \sqrt {e x +d}\, \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}+\frac {3003 c^{5} d^{5} e \sqrt {e x +d}}{128 \left (-a \,e^{2}+c \,d^{2}\right )^{7} \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}} \]
command
integrate(1/(e*x+d)^(7/2)/(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}{1920} \, {\left (\frac {45045 \, c^{5} d^{5} \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^{7} d^{14} - 7 \, a c^{6} d^{12} e^{2} + 21 \, a^{2} c^{5} d^{10} e^{4} - 35 \, a^{3} c^{4} d^{8} e^{6} + 35 \, a^{4} c^{3} d^{6} e^{8} - 21 \, a^{5} c^{2} d^{4} e^{10} + 7 \, a^{6} c d^{2} e^{12} - a^{7} e^{14}\right )} \sqrt {c d^{2} e - a e^{3}}} - \frac {1280 \, {\left (c^{6} d^{7} e^{2} - a c^{5} d^{5} e^{4} - 18 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )} c^{5} d^{5} e\right )}}{{\left (c^{7} d^{14} - 7 \, a c^{6} d^{12} e^{2} + 21 \, a^{2} c^{5} d^{10} e^{4} - 35 \, a^{3} c^{4} d^{8} e^{6} + 35 \, a^{4} c^{3} d^{6} e^{8} - 21 \, a^{5} c^{2} d^{4} e^{10} + 7 \, a^{6} c d^{2} e^{12} - a^{7} e^{14}\right )} {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}}} + \frac {{\left (35595 \, \sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}} c^{9} d^{13} e^{5} - 142380 \, \sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}} a c^{8} d^{11} e^{7} + 121310 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} c^{8} d^{11} e^{4} + 213570 \, \sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}} a^{2} c^{7} d^{9} e^{9} - 363930 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} a c^{7} d^{9} e^{6} + 160384 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {5}{2}} c^{7} d^{9} e^{3} - 142380 \, \sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}} a^{3} c^{6} d^{7} e^{11} + 363930 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} a^{2} c^{6} d^{7} e^{8} - 320768 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {5}{2}} a c^{6} d^{7} e^{5} + 96290 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {7}{2}} c^{6} d^{7} e^{2} + 35595 \, \sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}} a^{4} c^{5} d^{5} e^{13} - 121310 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} a^{3} c^{5} d^{5} e^{10} + 160384 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {5}{2}} a^{2} c^{5} d^{5} e^{7} - 96290 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {7}{2}} a c^{5} d^{5} e^{4} + 22005 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {9}{2}} c^{5} d^{5} e\right )} e^{\left (-5\right )}}{{\left (c^{7} d^{14} - 7 \, a c^{6} d^{12} e^{2} + 21 \, a^{2} c^{5} d^{10} e^{4} - 35 \, a^{3} c^{4} d^{8} e^{6} + 35 \, a^{4} c^{3} d^{6} e^{8} - 21 \, a^{5} c^{2} d^{4} e^{10} + 7 \, a^{6} c d^{2} e^{12} - a^{7} e^{14}\right )} {\left (x e + d\right )}^{5} c^{5} d^{5}}\right )} e \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: TypeError} \]________________________________________________________________________________________