\[ \int \frac {(d+e x)^{13/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx \]
Optimal antiderivative \[ \frac {35 e^{2} \left (e x +d \right )^{\frac {3}{2}}}{12 c^{3} d^{3}}-\frac {7 e \left (e x +d \right )^{\frac {5}{2}}}{4 c^{2} d^{2} \left (c d x +a e \right )}-\frac {\left (e x +d \right )^{\frac {7}{2}}}{2 c d \left (c d x +a e \right )^{2}}-\frac {35 e^{2} \left (-a \,e^{2}+c \,d^{2}\right )^{\frac {3}{2}} \arctanh \left (\frac {\sqrt {c}\, \sqrt {d}\, \sqrt {e x +d}}{\sqrt {-a \,e^{2}+c \,d^{2}}}\right )}{4 c^{\frac {9}{2}} d^{\frac {9}{2}}}+\frac {35 e^{2} \left (-a \,e^{2}+c \,d^{2}\right ) \sqrt {e x +d}}{4 c^{4} d^{4}} \]
command
integrate((e*x+d)^(13/2)/(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
\[ \frac {35 \, {\left (c^{2} d^{4} e^{2} - 2 \, a c d^{2} e^{4} + a^{2} e^{6}\right )} \arctan \left (\frac {\sqrt {x e + d} c d}{\sqrt {-c^{2} d^{3} + a c d e^{2}}}\right )}{4 \, \sqrt {-c^{2} d^{3} + a c d e^{2}} c^{4} d^{4}} - \frac {13 \, {\left (x e + d\right )}^{\frac {3}{2}} c^{3} d^{5} e^{2} - 11 \, \sqrt {x e + d} c^{3} d^{6} e^{2} - 26 \, {\left (x e + d\right )}^{\frac {3}{2}} a c^{2} d^{3} e^{4} + 33 \, \sqrt {x e + d} a c^{2} d^{4} e^{4} + 13 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{2} c d e^{6} - 33 \, \sqrt {x e + d} a^{2} c d^{2} e^{6} + 11 \, \sqrt {x e + d} a^{3} e^{8}}{4 \, {\left ({\left (x e + d\right )} c d - c d^{2} + a e^{2}\right )}^{2} c^{4} d^{4}} + \frac {2 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} c^{6} d^{6} e^{2} + 9 \, \sqrt {x e + d} c^{6} d^{7} e^{2} - 9 \, \sqrt {x e + d} a c^{5} d^{5} e^{4}\right )}}{3 \, c^{9} d^{9}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Timed out} \]________________________________________________________________________________________