\[ \int \frac {1}{\sqrt {d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx \]
Optimal antiderivative \[ \frac {35 c^{2} d^{2} e^{\frac {3}{2}} \arctan \left (\frac {\sqrt {e}\, \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}{\sqrt {-a \,e^{2}+c \,d^{2}}\, \sqrt {e x +d}}\right )}{4 \left (-a \,e^{2}+c \,d^{2}\right )^{\frac {9}{2}}}+\frac {1}{2 \left (-a \,e^{2}+c \,d^{2}\right ) \left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}\right )^{\frac {3}{2}} \sqrt {e x +d}}-\frac {7 c d \sqrt {e x +d}}{6 \left (-a \,e^{2}+c \,d^{2}\right )^{2} \left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}\right )^{\frac {3}{2}}}-\frac {35 c d e}{12 \left (-a \,e^{2}+c \,d^{2}\right )^{3} \sqrt {e x +d}\, \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}+\frac {35 c^{2} d^{2} e \sqrt {e x +d}}{4 \left (-a \,e^{2}+c \,d^{2}\right )^{4} \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}} \]
command
integrate(1/(e*x+d)^(1/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {1}{12} \, {\left (\frac {105 \, c^{2} d^{2} \arctan \left (\frac {\sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}}}{\sqrt {c d^{2} e - a e^{3}}}\right ) e}{{\left (c^{4} d^{8} - 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} - 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}\right )} \sqrt {c d^{2} e - a e^{3}}} - \frac {8 \, {\left (c^{3} d^{4} e^{2} - a c^{2} d^{2} e^{4} - 9 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )} c^{2} d^{2} e\right )}}{{\left (c^{4} d^{8} - 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} - 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}\right )} {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}}} + \frac {3 \, {\left (13 \, \sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}} c^{3} d^{4} e^{2} - 13 \, \sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}} a c^{2} d^{2} e^{4} + 11 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} c^{2} d^{2} e\right )} e^{\left (-2\right )}}{{\left (c^{4} d^{8} - 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} - 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}\right )} {\left (x e + d\right )}^{2} c^{2} d^{2}}\right )} e \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: TypeError} \]________________________________________________________________________________________