\[ \int \frac {\sqrt {c+d x} \sqrt {e+f x} \left (A+B x+C x^2\right )}{(a+b x)^{7/2}} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (A \,b^{2}-a \left (b B -a C \right )\right ) \left (d x +c \right )^{\frac {3}{2}} \left (f x +e \right )^{\frac {3}{2}}}{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 (6 a^{3} C d f +a \,b^{2} \left (-4 A d f +3 B c f +3 B d e +10 c C e \right )-b^{3} \left (5 B c e -2 A \left (c f +d e \right )\right )-a^{2} b \left (B d f +8 C \left (c f +d e \right )\right )\right ) \left (f x +e \right )^{\frac {3}{2}} \sqrt {d x +c}}{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 (24 a^{3} C \,d^{2} f -a^{2} b d \left (4 B d f +41 c C f +23 C d e \right )-b^{3} \left (15 c^{2} C e -2 A \,d^{2} e +c d \left (A f +5 B e \right )\right )+a \,b^{2} \left (15 c^{2} C f +d^{2} \left (-A f +3 B e \right )+c \left (6 B d f +40 C d e \right )\right )\right ) \sqrt {d x +c}\, \sqrt {f x +e}}{15 b^{3} \left (-a d +b c \right )^{2} \left (-a f +b e \right ) \sqrt {b x +a}}+\frac {2 \left (48 a^{4} C \,d^{2} f^{2}-8 a^{3} b d f \left (B d f +11 C \left (c f +d e \right )\right )-b^{4} \left (2 A \,d^{2} e^{2}-c d e \left (2 A f +5 B e \right )-c^{2} \left (-2 A \,f^{2}+5 B e f +30 C \,e^{2}\right )\right )-a \,b^{3} \left (d^{2} e \left (-2 A f +3 B e \right )+c^{2} f \left (3 B f +70 C e \right )+2 c d \left (-A \,f^{2}+11 B e f +35 C \,e^{2}\right )\right )+a^{2} b^{2} \left (2 C \left (19 c^{2} f^{2}+81 c d e f +19 d^{2} e^{2}\right )-d f \left (2 A d f -13 B \left (c f +d 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^{4} \left (a d -b c \right )^{\frac {3}{2}} \left (-a f +b e \right )^{2} \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 (24 a^{3} C \,d^{2} f -a^{2} b d \left (4 B d f +41 c C f +23 C d e \right )-b^{3} \left (15 c^{2} C e -2 A \,d^{2} e +c d \left (A f +5 B e \right )\right )+a \,b^{2} \left (15 c^{2} C f +d^{2} \left (-A f +3 B e \right )+c \left (6 B d f +40 C d e \right )\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 {\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^{4} \left (a d -b c \right )^{\frac {3}{2}} \left (-a f +b e \right ) \sqrt {d}\, \sqrt {d x +c}\, \sqrt {f x +e}} \]
command
integrate((C*x^2+B*x+A)*(d*x+c)^(1/2)*(f*x+e)^(1/2)/(b*x+a)^(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 \[ {\rm integral}\left (\frac {{\left (C x^{2} + B x + A\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {f x + e}}{b^{4} x^{4} + 4 \, a b^{3} x^{3} + 6 \, a^{2} b^{2} x^{2} + 4 \, a^{3} b x + a^{4}}, x\right ) \]