\[ \int \frac {x^3 \sqrt {d^2-e^2 x^2}}{(d+e x)^4} \, dx \]
Optimal antiderivative \[ \frac {d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{5 e^{4} \left (e x +d \right )^{4}}-\frac {14 d \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{15 e^{4} \left (e x +d \right )^{3}}-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{e^{4} \left (e x +d \right )^{2}}+\frac {4 d \arctan \left (\frac {e x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{e^{4}}+\frac {8 d \sqrt {-e^{2} x^{2}+d^{2}}}{e^{4} \left (e x +d \right )} \]
command
integrate(x^3*(-e^2*x^2+d^2)^(1/2)/(e*x+d)^4,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ 4 \, d \arcsin \left (\frac {x e}{d}\right ) e^{\left (-4\right )} \mathrm {sgn}\left (d\right ) + \sqrt {-x^{2} e^{2} + d^{2}} e^{\left (-4\right )} - \frac {2 \, {\left (\frac {335 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} d e^{\left (-2\right )}}{x} + \frac {505 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} d e^{\left (-4\right )}}{x^{2}} + \frac {285 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} d e^{\left (-6\right )}}{x^{3}} + \frac {60 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{4} d e^{\left (-8\right )}}{x^{4}} + 79 \, d\right )} e^{\left (-4\right )}}{15 \, {\left (\frac {{\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} e^{\left (-2\right )}}{x} + 1\right )}^{5}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________