\[ \int \frac {(d+e x)^{13/2} (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx \]
Optimal antiderivative \[ \frac {2 \left (-b e g +c d g +c e f \right ) \left (e x +d \right )^{\frac {13}{2}}}{3 c \,e^{2} \left (-b e +2 c d \right ) \left (d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}\right )^{\frac {3}{2}}}+\frac {128 \left (-b e +2 c d \right )^{2} \left (-10 b e g +13 c d g +7 c e f \right ) \left (e x +d \right )^{\frac {3}{2}}}{105 c^{5} e^{2} \sqrt {d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}}}+\frac {32 \left (-b e +2 c d \right ) \left (-10 b e g +13 c d g +7 c e f \right ) \left (e x +d \right )^{\frac {5}{2}}}{105 c^{4} e^{2} \sqrt {d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}}}+\frac {16 \left (-10 b e g +13 c d g +7 c e f \right ) \left (e x +d \right )^{\frac {7}{2}}}{105 c^{3} e^{2} \sqrt {d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}}}+\frac {2 \left (-10 b e g +13 c d g +7 c e f \right ) \left (e x +d \right )^{\frac {9}{2}}}{21 c^{2} e^{2} \left (-b e +2 c d \right ) \sqrt {d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}}}-\frac {256 \left (-b e +2 c d \right )^{3} \left (-10 b e g +13 c d g +7 c e f \right ) \sqrt {e x +d}}{105 c^{6} e^{2} \sqrt {d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}}} \]
command
integrate((e*x+d)^(13/2)*(g*x+f)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \text {output too large to display} \]
Giac 1.7.0 via sagemath 9.3 output \[ \text {Timed out} \]_______________________________________________________