\[ \int \frac {(d+e x)^3}{(f+g x)^2 \left (d^2-e^2 x^2\right )^{7/2}} \, dx \]
Optimal antiderivative \[ \frac {4 d e \left (e x +d \right )}{5 \left (d g +e f \right )^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {e \left (5 d \left (-3 d g +e f \right )-e \left (21 d g +e f \right ) x \right )}{15 d \left (d g +e f \right )^{3} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {e \,g^{3} \left (-3 d g +4 e f \right ) \arctan \! \left (\frac {e^{2} f x +d^{2} g}{\sqrt {-d^{2} g^{2}+e^{2} f^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}\right )}{\left (-d g +e f \right ) \left (d g +e f \right )^{4} \sqrt {-d^{2} g^{2}+e^{2} f^{2}}}+\frac {e \left (45 d^{3} g^{2}+e \left (57 d^{2} g^{2}+14 d e f g +2 e^{2} f^{2}\right ) x \right )}{15 d^{3} \left (d g +e f \right )^{4} \sqrt {-e^{2} x^{2}+d^{2}}}+\frac {g^{4} \sqrt {-e^{2} x^{2}+d^{2}}}{\left (-d g +e f \right ) \left (d g +e f \right )^{4} \left (g x +f \right )} \]
command
integrate((e*x+d)^3/(g*x+f)^2/(-e^2*x^2+d^2)^(7/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \text {Timed out} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {output too large to display} \]