\[ \int \frac {\sqrt {a+b x+c x^2} \left (A+B x+C x^2\right )}{(d+e x)^{7/2}} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (C \,d^{2}-e \left (-A e +B d \right )\right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{5 e \left (a \,e^{2}-b d e +c \,d^{2}\right ) \left (e x +d \right )^{\frac {5}{2}}}-\frac {2 \left (c^{2} d^{3} \left (24 C \,d^{2}-e \left (A e +4 B d \right )\right )+e^{2} \left (15 b^{2} C \,d^{3}+5 a^{2} e^{2} \left (B e +C d \right )-a b e \left (2 A \,e^{2}+3 B d e +22 C \,d^{2}\right )\right )-c d e \left (b d \left (A \,e^{2}-6 B d e +41 C \,d^{2}\right )-a e \left (7 A \,e^{2}-7 B d e +37 C \,d^{2}\right )\right )+e \left (5 c^{2} d^{2} \left (6 C \,d^{2}-e \left (A e +B d \right )\right )+e^{2} \left (15 a^{2} C \,e^{2}-5 a b e \left (-B e +8 C d \right )+b^{2} \left (-2 A \,e^{2}-3 B d e +23 C \,d^{2}\right )\right )-c e \left (5 b d \left (-A \,e^{2}-2 B d e +11 C \,d^{2}\right )-a e \left (3 A \,e^{2}-13 B d e +53 C \,d^{2}\right )\right )\right ) x \right ) \sqrt {c \,x^{2}+b x +a}}{15 e^{3} \left (a \,e^{2}-b d e +c \,d^{2}\right )^{2} \left (e x +d \right )^{\frac {3}{2}}}+\frac {\left (2 c^{2} d^{2} \left (24 C \,d^{2}-e \left (A e +4 B d \right )\right )+e^{2} \left (30 a^{2} C \,e^{2}-5 a b e \left (-B e +14 C d \right )+b^{2} \left (-2 A \,e^{2}-3 B d e +38 C \,d^{2}\right )\right )-c e \left (b d \left (-2 A \,e^{2}-13 B d e +88 C \,d^{2}\right )-2 a e \left (3 A \,e^{2}-8 B d e +43 C \,d^{2}\right )\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 {-4 a c +b^{2}}\, \sqrt {e x +d}\, \sqrt {-\frac {c \left (c \,x^{2}+b x +a \right )}{-4 a c +b^{2}}}}{15 e^{4} \left (a \,e^{2}-b d e +c \,d^{2}\right )^{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 (15 b C \,e^{2} \left (-a e +b d \right )+2 c^{2} d \left (24 C \,d^{2}-e \left (A e +4 B d \right )\right )+c e \left (10 a e \left (-B e +5 C d \right )-b \left (-A \,e^{2}-9 B d e +64 C \,d^{2}\right )\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 {-4 a c +b^{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 )}}}{15 c \,e^{4} \left (a \,e^{2}-b d e +c \,d^{2}\right ) \sqrt {e x +d}\, \sqrt {c \,x^{2}+b x +a}} \]
command
integrate((C*x^2+B*x+A)*(c*x^2+b*x+a)^(1/2)/(e*x+d)^(7/2),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 \[ {\rm integral}\left (\frac {{\left (C x^{2} + B x + A\right )} \sqrt {c x^{2} + b x + a} \sqrt {e x + d}}{e^{4} x^{4} + 4 \, d e^{3} x^{3} + 6 \, d^{2} e^{2} x^{2} + 4 \, d^{3} e x + d^{4}}, x\right ) \]