\[ \int \frac {(d+e x)^{7/2}}{\left (a+b x+c x^2\right )^{5/2}} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (e x +d \right )^{\frac {5}{2}} \left (b d -2 a e +\left (-b e +2 c d \right ) x \right )}{3 \left (-4 a c +b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}+\frac {2 \left (8 b c d \left (3 a \,e^{2}+c \,d^{2}\right )-4 a c e \left (5 a \,e^{2}+3 c \,d^{2}\right )-b^{2} \left (-a \,e^{3}+9 c \,d^{2} e \right )+\left (-b e +2 c d \right ) \left (8 c^{2} d^{2}-b^{2} e^{2}-4 c e \left (-3 a e +2 b d \right )\right ) x \right ) \sqrt {e x +d}}{3 c \left (-4 a c +b^{2}\right )^{2} \sqrt {c \,x^{2}+b x +a}}-\frac {2 \left (-b e +2 c d \right ) \left (4 c^{2} d^{2}-b^{2} e^{2}-4 c e \left (-2 a e +b d \right )\right ) \EllipticE \left (\frac {\sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {2}}{2}, \sqrt {-\frac {2 e \sqrt {-4 a c +b^{2}}}{2 c d -e \left (b +\sqrt {-4 a c +b^{2}}\right )}}\right ) \sqrt {2}\, \sqrt {e x +d}\, \sqrt {-\frac {c \left (c \,x^{2}+b x +a \right )}{-4 a c +b^{2}}}}{3 c^{2} \left (-4 a c +b^{2}\right )^{\frac {3}{2}} \sqrt {c \,x^{2}+b x +a}\, \sqrt {\frac {c \left (e x +d \right )}{2 c d -e \left (b +\sqrt {-4 a c +b^{2}}\right )}}}+\frac {2 \left (a \,e^{2}-b d e +c \,d^{2}\right ) \left (16 c^{2} d^{2}-b^{2} e^{2}-4 c e \left (-5 a e +4 b d \right )\right ) \EllipticF \left (\frac {\sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {2}}{2}, \sqrt {-\frac {2 e \sqrt {-4 a c +b^{2}}}{2 c d -e \left (b +\sqrt {-4 a c +b^{2}}\right )}}\right ) \sqrt {2}\, \sqrt {-\frac {c \left (c \,x^{2}+b x +a \right )}{-4 a c +b^{2}}}\, \sqrt {\frac {c \left (e x +d \right )}{2 c d -e \left (b +\sqrt {-4 a c +b^{2}}\right )}}}{3 c^{2} \left (-4 a c +b^{2}\right )^{\frac {3}{2}} \sqrt {e x +d}\, \sqrt {c \,x^{2}+b x +a}} \]
command
integrate((e*x+d)^(7/2)/(c*x^2+b*x+a)^(5/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \text {output too large to display} \]
Fricas 1.3.7 via sagemath 9.3 output \[ {\rm integral}\left (\frac {{\left (e^{3} x^{3} + 3 \, d e^{2} x^{2} + 3 \, d^{2} e x + d^{3}\right )} \sqrt {c x^{2} + b x + a} \sqrt {e x + d}}{c^{3} x^{6} + 3 \, b c^{2} x^{5} + 3 \, {\left (b^{2} c + a c^{2}\right )} x^{4} + 3 \, a^{2} b x + {\left (b^{3} + 6 \, a b c\right )} x^{3} + a^{3} + 3 \, {\left (a b^{2} + a^{2} c\right )} x^{2}}, x\right ) \]