\[ \int \frac {(d+e x)^{11/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx \]
Optimal antiderivative \[ -\frac {5 e \left (e x +d \right )^{\frac {3}{2}}}{4 c^{2} d^{2} \left (c d x +a e \right )}-\frac {\left (e x +d \right )^{\frac {5}{2}}}{2 c d \left (c d x +a e \right )^{2}}-\frac {15 e^{2} \arctanh \left (\frac {\sqrt {c}\, \sqrt {d}\, \sqrt {e x +d}}{\sqrt {-a \,e^{2}+c \,d^{2}}}\right ) \sqrt {-a \,e^{2}+c \,d^{2}}}{4 c^{\frac {7}{2}} d^{\frac {7}{2}}}+\frac {15 e^{2} \sqrt {e x +d}}{4 c^{3} d^{3}} \]
command
integrate((e*x+d)^(11/2)/(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
\[ \frac {15 \, {\left (c d^{2} e^{2} - a e^{4}\right )} \arctan \left (\frac {\sqrt {x e + d} c d}{\sqrt {-c^{2} d^{3} + a c d e^{2}}}\right )}{4 \, \sqrt {-c^{2} d^{3} + a c d e^{2}} c^{3} d^{3}} + \frac {2 \, \sqrt {x e + d} e^{2}}{c^{3} d^{3}} - \frac {9 \, {\left (x e + d\right )}^{\frac {3}{2}} c^{2} d^{3} e^{2} - 7 \, \sqrt {x e + d} c^{2} d^{4} e^{2} - 9 \, {\left (x e + d\right )}^{\frac {3}{2}} a c d e^{4} + 14 \, \sqrt {x e + d} a c d^{2} e^{4} - 7 \, \sqrt {x e + d} a^{2} e^{6}}{4 \, {\left ({\left (x e + d\right )} c d - c d^{2} + a e^{2}\right )}^{2} c^{3} d^{3}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: TypeError} \]________________________________________________________________________________________