\[ \int \frac {\sqrt {c+d x} \left (A+B x+C x^2\right )}{(a+b x)^{7/2} \sqrt {e+f x}} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (A \,b^{2}-a \left (b B -a C \right )\right ) \left (d x +c \right )^{\frac {3}{2}} \sqrt {f x +e}}{5 b \left (-a d +b c \right ) \left (-a f +b e \right ) \left (b x +a \right )^{\frac {5}{2}}}+\frac {2 \left (4 a^{3} C d f -b^{3} \left (-4 A c f -2 A d e +5 B c e \right )+a \,b^{2} \left (-6 A d f +B c f +3 B d e +10 c C e \right )-a^{2} b \left (-B d f +6 c C f +8 C d e \right )\right ) \sqrt {d x +c}\, \sqrt {f x +e}}{15 b^{2} \left (-a d +b c \right ) \left (-a f +b e \right )^{2} \left (b x +a \right )^{\frac {3}{2}}}-\frac {2 \left (8 a^{4} C \,d^{2} f^{2}-a^{3} b d f \left (-2 B d f +13 c C f +23 C d e \right )-b^{4} \left (2 A \,d^{2} e^{2}-c d e \left (-3 A f +5 B e \right )-c^{2} \left (8 A \,f^{2}-10 B e f +15 C \,e^{2}\right )\right )-a^{2} b^{2} \left (d f \left (-3 A d f +2 B c f +7 B d e \right )-C \left (3 c^{2} f^{2}+37 c d e f +23 d^{2} e^{2}\right )\right )-a \,b^{3} \left (d^{2} e \left (-7 A f +3 B e \right )+2 c^{2} f \left (-B f +5 C e \right )+c d \left (40 C \,e^{2}-13 f \left (-A f +B e \right )\right )\right )\right ) \sqrt {d x +c}\, \sqrt {f x +e}}{15 b^{2} \left (-a d +b c \right )^{2} \left (-a f +b e \right )^{3} \sqrt {b x +a}}+\frac {2 \left (8 a^{4} C \,d^{2} f^{2}-a^{3} b d f \left (-2 B d f +13 c C f +23 C d e \right )-b^{4} \left (2 A \,d^{2} e^{2}-c d e \left (-3 A f +5 B e \right )-c^{2} \left (8 A \,f^{2}-10 B e f +15 C \,e^{2}\right )\right )-a^{2} b^{2} \left (d f \left (-3 A d f +2 B c f +7 B d e \right )-C \left (3 c^{2} f^{2}+37 c d e f +23 d^{2} e^{2}\right )\right )-a \,b^{3} \left (d^{2} e \left (-7 A f +3 B e \right )+2 c^{2} f \left (-B f +5 C e \right )+c d \left (40 C \,e^{2}-13 f \left (-A f +B e \right )\right )\right )\right ) \EllipticE \left (\frac {\sqrt {d}\, \sqrt {b x +a}}{\sqrt {a d -b c}}, \sqrt {\frac {\left (-a d +b c \right ) f}{d \left (-a f +b e \right )}}\right ) \sqrt {d}\, \sqrt {\frac {b \left (d x +c \right )}{-a d +b c}}\, \sqrt {f x +e}}{15 b^{3} \left (a d -b c \right )^{\frac {3}{2}} \left (-a f +b e \right )^{3} \sqrt {d x +c}\, \sqrt {\frac {b \left (f x +e \right )}{-a f +b e}}}+\frac {2 \left (-c f +d e \right ) \left (4 a^{3} C d f -b^{3} \left (-4 A c f -2 A d e +5 B c e \right )+a \,b^{2} \left (-6 A d f +B c f +3 B d e +10 c C e \right )-a^{2} b \left (-B d f +6 c C f +8 C d e \right )\right ) \EllipticF \left (\frac {\sqrt {d}\, \sqrt {b x +a}}{\sqrt {a d -b c}}, \sqrt {\frac {\left (-a d +b c \right ) f}{d \left (-a f +b e \right )}}\right ) \sqrt {d}\, \sqrt {\frac {b \left (d x +c \right )}{-a d +b c}}\, \sqrt {\frac {b \left (f x +e \right )}{-a f +b e}}}{15 b^{3} \left (a d -b c \right )^{\frac {3}{2}} \left (-a f +b e \right )^{2} \sqrt {d x +c}\, \sqrt {f x +e}} \]
command
integrate((C*x^2+B*x+A)*(d*x+c)^(1/2)/(b*x+a)^(7/2)/(f*x+e)^(1/2),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 \[ {\rm integral}\left (\frac {{\left (C x^{2} + B x + A\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {f x + e}}{b^{4} f x^{5} + a^{4} e + {\left (b^{4} e + 4 \, a b^{3} f\right )} x^{4} + 2 \, {\left (2 \, a b^{3} e + 3 \, a^{2} b^{2} f\right )} x^{3} + 2 \, {\left (3 \, a^{2} b^{2} e + 2 \, a^{3} b f\right )} x^{2} + {\left (4 \, a^{3} b e + a^{4} f\right )} x}, x\right ) \]