\[ \int \frac {(d+e x)^{3/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx \]
Optimal antiderivative \[ -\frac {15 e^{2} \arctanh \left (\frac {\sqrt {c}\, \sqrt {d}\, \sqrt {e x +d}}{\sqrt {-a \,e^{2}+c \,d^{2}}}\right ) \sqrt {c}\, \sqrt {d}}{4 \left (-a \,e^{2}+c \,d^{2}\right )^{\frac {7}{2}}}+\frac {15 e^{2}}{4 \left (-a \,e^{2}+c \,d^{2}\right )^{3} \sqrt {e x +d}}-\frac {1}{2 \left (-a \,e^{2}+c \,d^{2}\right ) \left (c d x +a e \right )^{2} \sqrt {e x +d}}+\frac {5 e}{4 \left (-a \,e^{2}+c \,d^{2}\right )^{2} \left (c d x +a e \right ) \sqrt {e x +d}} \]
command
integrate((e*x+d)^(3/2)/(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
\[ \frac {15 \, c d \arctan \left (\frac {\sqrt {x e + d} c d}{\sqrt {-c^{2} d^{3} + a c d e^{2}}}\right ) e^{2}}{4 \, {\left (c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6}\right )} \sqrt {-c^{2} d^{3} + a c d e^{2}}} + \frac {2 \, e^{2}}{{\left (c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6}\right )} \sqrt {x e + d}} + \frac {7 \, {\left (x e + d\right )}^{\frac {3}{2}} c^{2} d^{2} e^{2} - 9 \, \sqrt {x e + d} c^{2} d^{3} e^{2} + 9 \, \sqrt {x e + d} a c d e^{4}}{4 \, {\left (c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6}\right )} {\left ({\left (x e + d\right )} c d - c d^{2} + a e^{2}\right )}^{2}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________