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