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