\[ \int \frac {A+B x+C x^2}{(a+b x)^{7/2} \sqrt {c+d x} \sqrt {e+f x}} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (A \,b^{2}-a \left (b B -a C \right )\right ) \sqrt {d x +c}\, \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 (2 a^{3} C d f +a \,b^{2} \left (-8 A d f +B c f +B d e +10 c C e \right )-b^{3} \left (5 B c e -4 A \left (c f +d e \right )\right )+3 a^{2} b \left (B d f -2 C \left (c f +d e \right )\right )\right ) \sqrt {d x +c}\, \sqrt {f x +e}}{15 b \left (-a d +b c \right )^{2} \left (-a f +b e \right )^{2} \left (b x +a \right )^{\frac {3}{2}}}+\frac {2 \left (2 a^{4} C \,d^{2} f^{2}+a^{3} b d f \left (3 B d f -7 C \left (c f +d e \right )\right )-b^{4} \left (8 A \,d^{2} e^{2}-c d e \left (-7 A f +10 B e \right )+c^{2} \left (8 A \,f^{2}-10 B e f +15 C \,e^{2}\right )\right )-a \,b^{3} \left (d^{2} e \left (-23 A f +2 B e \right )-2 c^{2} f \left (-B f +5 C e \right )-c d \left (23 A \,f^{2}-33 B e f +10 C \,e^{2}\right )\right )-a^{2} b^{2} \left (C \left (3 c^{2} f^{2}-13 c d e f +3 d^{2} e^{2}\right )+d f \left (23 A d f -7 B \left (c f +d e \right )\right )\right )\right ) \sqrt {d x +c}\, \sqrt {f x +e}}{15 b \left (-a d +b c \right )^{3} \left (-a f +b e \right )^{3} \sqrt {b x +a}}+\frac {2 \left (2 a^{4} C \,d^{2} f^{2}+a^{3} b d f \left (3 B d f -7 C \left (c f +d e \right )\right )-b^{4} \left (8 A \,d^{2} e^{2}-c d e \left (-7 A f +10 B e \right )+c^{2} \left (8 A \,f^{2}-10 B e f +15 C \,e^{2}\right )\right )-a \,b^{3} \left (d^{2} e \left (-23 A f +2 B e \right )-2 c^{2} f \left (-B f +5 C e \right )-c d \left (23 A \,f^{2}-33 B e f +10 C \,e^{2}\right )\right )-a^{2} b^{2} \left (C \left (3 c^{2} f^{2}-13 c d e f +3 d^{2} e^{2}\right )+d f \left (23 A d f -7 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^{2} \left (a d -b c \right )^{\frac {5}{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 (a^{3} C d f \left (-c f +d e \right )+b^{3} \left (8 A \,d^{2} e^{2}-c d e \left (-3 A f +10 B e \right )+c^{2} \left (4 A \,f^{2}-5 B e f +15 C \,e^{2}\right )\right )+a \,b^{2} \left (d^{2} e \left (-19 A f +2 B e \right )-c^{2} f \left (-B f +20 C e \right )-c d \left (11 A \,f^{2}-27 B e f +10 C \,e^{2}\right )\right )-3 a^{2} b \left (d f \left (-5 A d f +3 B c f +2 B d e \right )-C \left (3 c^{2} f^{2}+c d e f +d^{2} e^{2}\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 {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^{2} \left (a d -b c \right )^{\frac {5}{2}} \left (-a f +b e \right )^{2} \sqrt {d x +c}\, \sqrt {f x +e}} \]
command
integrate((C*x^2+B*x+A)/(b*x+a)^(7/2)/(d*x+c)^(1/2)/(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 \[ {\rm integral}\left (\frac {{\left (C x^{2} + B x + A\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {f x + e}}{b^{4} d f x^{6} + a^{4} c e + {\left (b^{4} d e + {\left (b^{4} c + 4 \, a b^{3} d\right )} f\right )} x^{5} + {\left ({\left (b^{4} c + 4 \, a b^{3} d\right )} e + 2 \, {\left (2 \, a b^{3} c + 3 \, a^{2} b^{2} d\right )} f\right )} x^{4} + 2 \, {\left ({\left (2 \, a b^{3} c + 3 \, a^{2} b^{2} d\right )} e + {\left (3 \, a^{2} b^{2} c + 2 \, a^{3} b d\right )} f\right )} x^{3} + {\left (2 \, {\left (3 \, a^{2} b^{2} c + 2 \, a^{3} b d\right )} e + {\left (4 \, a^{3} b c + a^{4} d\right )} f\right )} x^{2} + {\left (a^{4} c f + {\left (4 \, a^{3} b c + a^{4} d\right )} e\right )} x}, x\right ) \]