\[ \int \frac {(d+e x)^{13/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx \]
Optimal antiderivative \[ \frac {3 e \left (-a \,e^{2}+c \,d^{2}\right )^{2} \left (e x +d \right )^{\frac {3}{2}}}{c^{4} d^{4}}+\frac {9 e \left (-a \,e^{2}+c \,d^{2}\right ) \left (e x +d \right )^{\frac {5}{2}}}{5 c^{3} d^{3}}+\frac {9 e \left (e x +d \right )^{\frac {7}{2}}}{7 c^{2} d^{2}}-\frac {\left (e x +d \right )^{\frac {9}{2}}}{c d \left (c d x +a e \right )}-\frac {9 e \left (-a \,e^{2}+c \,d^{2}\right )^{\frac {7}{2}} \arctanh \left (\frac {\sqrt {c}\, \sqrt {d}\, \sqrt {e x +d}}{\sqrt {-a \,e^{2}+c \,d^{2}}}\right )}{c^{\frac {11}{2}} d^{\frac {11}{2}}}+\frac {9 e \left (-a \,e^{2}+c \,d^{2}\right )^{3} \sqrt {e x +d}}{c^{5} d^{5}} \]
command
integrate((e*x+d)^(13/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^2,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {9 \, {\left (c^{4} d^{8} e - 4 \, a c^{3} d^{6} e^{3} + 6 \, a^{2} c^{2} d^{4} e^{5} - 4 \, a^{3} c d^{2} e^{7} + a^{4} e^{9}\right )} \arctan \left (\frac {\sqrt {x e + d} c d}{\sqrt {-c^{2} d^{3} + a c d e^{2}}}\right )}{\sqrt {-c^{2} d^{3} + a c d e^{2}} c^{5} d^{5}} - \frac {\sqrt {x e + d} c^{4} d^{8} e - 4 \, \sqrt {x e + d} a c^{3} d^{6} e^{3} + 6 \, \sqrt {x e + d} a^{2} c^{2} d^{4} e^{5} - 4 \, \sqrt {x e + d} a^{3} c d^{2} e^{7} + \sqrt {x e + d} a^{4} e^{9}}{{\left ({\left (x e + d\right )} c d - c d^{2} + a e^{2}\right )} c^{5} d^{5}} + \frac {2 \, {\left (5 \, {\left (x e + d\right )}^{\frac {7}{2}} c^{12} d^{12} e + 14 \, {\left (x e + d\right )}^{\frac {5}{2}} c^{12} d^{13} e + 35 \, {\left (x e + d\right )}^{\frac {3}{2}} c^{12} d^{14} e + 140 \, \sqrt {x e + d} c^{12} d^{15} e - 14 \, {\left (x e + d\right )}^{\frac {5}{2}} a c^{11} d^{11} e^{3} - 70 \, {\left (x e + d\right )}^{\frac {3}{2}} a c^{11} d^{12} e^{3} - 420 \, \sqrt {x e + d} a c^{11} d^{13} e^{3} + 35 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{2} c^{10} d^{10} e^{5} + 420 \, \sqrt {x e + d} a^{2} c^{10} d^{11} e^{5} - 140 \, \sqrt {x e + d} a^{3} c^{9} d^{9} e^{7}\right )}}{35 \, c^{14} d^{14}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Timed out} \]________________________________________________________________________________________