\[ \int \frac {(a+b x)^{3/2} \sqrt {c+d x} \left (A+B x+C x^2\right )}{\sqrt {e+f x}} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (4 a C d f +b \left (-9 B d f +6 c C f +8 C d e \right )\right ) \left (b x +a \right )^{\frac {3}{2}} \left (d x +c \right )^{\frac {3}{2}} \sqrt {f x +e}}{63 b \,d^{2} f^{2}}+\frac {2 C \left (b x +a \right )^{\frac {5}{2}} \left (d x +c \right )^{\frac {3}{2}} \sqrt {f x +e}}{9 b d f}-\frac {2 \left (7 b d f \left (-9 A b d f +a c C f +3 a C d e +5 b c C e \right )-\left (-3 a d f +4 b c f +6 b d e \right ) \left (4 a C d f +b \left (-9 B d f +6 c C f +8 C d e \right )\right )\right ) \left (d x +c \right )^{\frac {3}{2}} \sqrt {b x +a}\, \sqrt {f x +e}}{315 b \,d^{3} f^{3}}-\frac {2 \left (5 b d f \left (7 a d f \left (-9 A b d f +a c C f +3 a C d e +5 b c C e \right )-\left (a c f +3 a d e +3 b c e \right ) \left (4 a C d f +b \left (-9 B d f +6 c C f +8 C d e \right )\right )\right )+2 \left (\frac {a d f}{2}-b \left (c f +2 d e \right )\right ) \left (7 b d f \left (-9 A b d f +a c C f +3 a C d e +5 b c C e \right )-\left (-3 a d f +4 b c f +6 b d e \right ) \left (4 a C d f +b \left (-9 B d f +6 c C f +8 C d e \right )\right )\right )\right ) \sqrt {b x +a}\, \sqrt {d x +c}\, \sqrt {f x +e}}{945 b^{2} d^{3} f^{4}}+\frac {2 \left (8 a^{4} C \,d^{4} f^{4}+a^{3} b \,d^{3} f^{3} \left (-18 B d f -7 c C f +11 C d e \right )-3 a^{2} b^{2} d^{2} f^{2} \left (3 d f \left (-7 A d f -3 B c f +4 B d e \right )-C \left (-3 c^{2} f^{2}-5 c d e f +9 d^{2} e^{2}\right )\right )-a \,b^{3} d f \left (2 C \left (-16 c^{3} f^{3}-18 c^{2} d e \,f^{2}-33 c \,d^{2} e^{2} f +92 d^{3} e^{3}\right )+3 d f \left (7 A d f \left (-7 c f +13 d e \right )-B \left (-19 c^{2} f^{2}-29 c d e f +72 d^{2} e^{2}\right )\right )\right )+b^{4} \left (C \left (-16 c^{4} f^{4}-16 c^{3} d e \,f^{3}-21 c^{2} d^{2} e^{2} f^{2}-40 c \,d^{3} e^{3} f +128 d^{4} e^{4}\right )+3 d f \left (7 A d f \left (-2 c^{2} f^{2}-3 c d e f +8 d^{2} e^{2}\right )-B \left (-8 c^{3} f^{3}-9 c^{2} d e \,f^{2}-16 c \,d^{2} e^{2} f +48 d^{3} e^{3}\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 {a d -b c}\, \sqrt {\frac {b \left (d x +c \right )}{-a d +b c}}\, \sqrt {f x +e}}{315 b^{3} d^{\frac {7}{2}} f^{5} \sqrt {d x +c}\, \sqrt {\frac {b \left (f x +e \right )}{-a f +b e}}}+\frac {2 \left (-a f +b e \right ) \left (-c f +d e \right ) \left (4 a^{3} C \,d^{3} f^{3}+3 a^{2} b \,d^{2} f^{2} \left (-3 B d f -c C f +3 C d e \right )-3 a \,b^{2} d f \left (3 d f \left (-21 A d f +3 B c f +16 B d e \right )-5 C \left (c^{2} f^{2}+2 c d e f +8 d^{2} e^{2}\right )\right )-b^{3} \left (C \left (8 c^{3} f^{3}+15 c^{2} d e \,f^{2}+24 c \,d^{2} e^{2} f +128 d^{3} e^{3}\right )+3 d f \left (7 A d f \left (c f +8 d e \right )-4 B \left (c^{2} f^{2}+2 c d e f +12 d^{2} e^{2}\right )\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 {a d -b c}\, \sqrt {\frac {b \left (d x +c \right )}{-a d +b c}}\, \sqrt {\frac {b \left (f x +e \right )}{-a f +b e}}}{315 b^{3} d^{\frac {7}{2}} f^{5} \sqrt {d x +c}\, \sqrt {f x +e}} \]
command
integrate((b*x+a)^(3/2)*(C*x^2+B*x+A)*(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 \[ {\rm integral}\left (\frac {{\left (C b x^{3} + {\left (C a + B b\right )} x^{2} + A a + {\left (B a + A b\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{\sqrt {f x + e}}, x\right ) \]