\[ \int \frac {(d+e x)^{15/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx \]
Optimal antiderivative \[ \frac {21 e^{2} \left (-a \,e^{2}+c \,d^{2}\right ) \left (e x +d \right )^{\frac {3}{2}}}{4 c^{4} d^{4}}+\frac {63 e^{2} \left (e x +d \right )^{\frac {5}{2}}}{20 c^{3} d^{3}}-\frac {9 e \left (e x +d \right )^{\frac {7}{2}}}{4 c^{2} d^{2} \left (c d x +a e \right )}-\frac {\left (e x +d \right )^{\frac {9}{2}}}{2 c d \left (c d x +a e \right )^{2}}-\frac {63 e^{2} \left (-a \,e^{2}+c \,d^{2}\right )^{\frac {5}{2}} \arctanh \left (\frac {\sqrt {c}\, \sqrt {d}\, \sqrt {e x +d}}{\sqrt {-a \,e^{2}+c \,d^{2}}}\right )}{4 c^{\frac {11}{2}} d^{\frac {11}{2}}}+\frac {63 e^{2} \left (-a \,e^{2}+c \,d^{2}\right )^{2} \sqrt {e x +d}}{4 c^{5} d^{5}} \]
command
integrate((e*x+d)^(15/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {63 \, {\left (c^{3} d^{6} e^{2} - 3 \, a c^{2} d^{4} e^{4} + 3 \, a^{2} c d^{2} e^{6} - a^{3} e^{8}\right )} \arctan \left (\frac {\sqrt {x e + d} c d}{\sqrt {-c^{2} d^{3} + a c d e^{2}}}\right )}{4 \, \sqrt {-c^{2} d^{3} + a c d e^{2}} c^{5} d^{5}} - \frac {17 \, {\left (x e + d\right )}^{\frac {3}{2}} c^{4} d^{7} e^{2} - 15 \, \sqrt {x e + d} c^{4} d^{8} e^{2} - 51 \, {\left (x e + d\right )}^{\frac {3}{2}} a c^{3} d^{5} e^{4} + 60 \, \sqrt {x e + d} a c^{3} d^{6} e^{4} + 51 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{2} c^{2} d^{3} e^{6} - 90 \, \sqrt {x e + d} a^{2} c^{2} d^{4} e^{6} - 17 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{3} c d e^{8} + 60 \, \sqrt {x e + d} a^{3} c d^{2} e^{8} - 15 \, \sqrt {x e + d} a^{4} e^{10}}{4 \, {\left ({\left (x e + d\right )} c d - c d^{2} + a e^{2}\right )}^{2} c^{5} d^{5}} + \frac {2 \, {\left ({\left (x e + d\right )}^{\frac {5}{2}} c^{12} d^{12} e^{2} + 5 \, {\left (x e + d\right )}^{\frac {3}{2}} c^{12} d^{13} e^{2} + 30 \, \sqrt {x e + d} c^{12} d^{14} e^{2} - 5 \, {\left (x e + d\right )}^{\frac {3}{2}} a c^{11} d^{11} e^{4} - 60 \, \sqrt {x e + d} a c^{11} d^{12} e^{4} + 30 \, \sqrt {x e + d} a^{2} c^{10} d^{10} e^{6}\right )}}{5 \, c^{15} d^{15}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Timed out} \]________________________________________________________________________________________