\[ \int \sqrt {d+e x} \sqrt {a+b x+c x^2} \left (A+B x+C x^2\right ) \, dx \]
Optimal antiderivative \[ \frac {2 C \left (e x +d \right )^{\frac {3}{2}} \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{9 c e}-\frac {2 \left (-3 B c e +2 C b e +2 C c d \right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} \sqrt {e x +d}}{21 c^{2} e}+\frac {2 \left (8 b^{3} C \,e^{3}-3 b c \,e^{2} \left (4 b B e -a C e +b C d \right )+c^{3} d \left (8 C \,d^{2}-3 e \left (-7 A e +4 B d \right )\right )+3 c^{2} e \left (a e \left (-5 B e +C d \right )-b \left (-7 A \,e^{2}-2 B d e +C \,d^{2}\right )\right )+3 c e \left (8 b^{2} C \,e^{2}-c e \left (12 b B e +7 a C e +b C d \right )-c^{2} \left (2 C \,d^{2}-3 e \left (7 A e +B d \right )\right )\right ) x \right ) \sqrt {e x +d}\, \sqrt {c \,x^{2}+b x +a}}{315 c^{3} e^{3}}+\frac {\left (2 \left (4 c^{2} d^{2}-b^{2} e^{2}-\frac {3 c e \left (-2 a e +b d \right )}{2}\right ) \left (8 b^{2} C \,e^{2}-c e \left (12 b B e +7 a C e +b C d \right )-c^{2} \left (2 C \,d^{2}-3 e \left (7 A e +B d \right )\right )\right )-5 c e \left (-b e +2 c d \right ) \left (6 b^{2} C d e +c e \left (21 A c d -3 a B e -5 a C d \right )+b \left (2 a C \,e^{2}-c d \left (9 B e +C d \right )\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}}}}{315 c^{4} e^{4} \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 (a \,e^{2}-b d e +c \,d^{2}\right ) \left (8 b^{3} C \,e^{3}-3 c^{2} e^{2} \left (-7 A b e -10 a B e +B b d +2 a C d \right )+3 b c \,e^{2} \left (-4 b B e -9 a C e +b C d \right )-2 c^{3} d \left (8 C \,d^{2}-3 e \left (-7 A e +4 B d \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 )}}}{315 c^{4} e^{4} \sqrt {e x +d}\, \sqrt {c \,x^{2}+b x +a}} \]
command
integrate((e*x+d)^(1/2)*(C*x^2+B*x+A)*(c*x^2+b*x+a)^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {2 \, {\left ({\left (16 \, C c^{5} d^{5} - 8 \, {\left (2 \, C b c^{4} + 3 \, B c^{5}\right )} d^{4} e - {\left (5 \, C b^{2} c^{3} - 42 \, A c^{5} - 3 \, {\left (10 \, C a + 9 \, B b\right )} c^{4}\right )} d^{3} e^{2} - {\left (5 \, C b^{3} c^{2} + 3 \, {\left (22 \, B a + 21 \, A b\right )} c^{4} - 3 \, {\left (7 \, C a b + 4 \, B b^{2}\right )} c^{3}\right )} d^{2} e^{3} - {\left (16 \, C b^{4} c - 378 \, A a c^{4} + 3 \, {\left (22 \, C a^{2} + 41 \, B a b + 21 \, A b^{2}\right )} c^{3} - 3 \, {\left (28 \, C a b^{2} + 9 \, B b^{3}\right )} c^{2}\right )} d e^{4} + {\left (16 \, C b^{5} - 9 \, {\left (10 \, B a^{2} + 21 \, A a b\right )} c^{3} + 3 \, {\left (41 \, C a^{2} b + 41 \, B a b^{2} + 14 \, A b^{3}\right )} c^{2} - 24 \, {\left (4 \, C a b^{3} + B b^{4}\right )} c\right )} e^{5}\right )} \sqrt {c} e^{\frac {1}{2}} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + {\left (b^{2} - 3 \, a c\right )} e^{2}\right )} e^{\left (-2\right )}}{3 \, c^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, {\left (b^{2} c - 6 \, a c^{2}\right )} d e^{2} + {\left (2 \, b^{3} - 9 \, a b c\right )} e^{3}\right )} e^{\left (-3\right )}}{27 \, c^{3}}, \frac {{\left (c d + {\left (3 \, c x + b\right )} e\right )} e^{\left (-1\right )}}{3 \, c}\right ) + 3 \, {\left (16 \, C c^{5} d^{4} e - 8 \, {\left (C b c^{4} + 3 \, B c^{5}\right )} d^{3} e^{2} - 3 \, {\left (2 \, C b^{2} c^{3} - 14 \, A c^{5} - {\left (6 \, C a + 5 \, B b\right )} c^{4}\right )} d^{2} e^{3} - {\left (8 \, C b^{3} c^{2} + 6 \, {\left (8 \, B a + 7 \, A b\right )} c^{4} - 15 \, {\left (2 \, C a b + B b^{2}\right )} c^{3}\right )} d e^{4} + {\left (16 \, C b^{4} c - 126 \, A a c^{4} + 3 \, {\left (14 \, C a^{2} + 29 \, B a b + 14 \, A b^{2}\right )} c^{3} - 24 \, {\left (3 \, C a b^{2} + B b^{3}\right )} c^{2}\right )} e^{5}\right )} \sqrt {c} e^{\frac {1}{2}} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + {\left (b^{2} - 3 \, a c\right )} e^{2}\right )} e^{\left (-2\right )}}{3 \, c^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, {\left (b^{2} c - 6 \, a c^{2}\right )} d e^{2} + {\left (2 \, b^{3} - 9 \, a b c\right )} e^{3}\right )} e^{\left (-3\right )}}{27 \, c^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + {\left (b^{2} - 3 \, a c\right )} e^{2}\right )} e^{\left (-2\right )}}{3 \, c^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, {\left (b^{2} c - 6 \, a c^{2}\right )} d e^{2} + {\left (2 \, b^{3} - 9 \, a b c\right )} e^{3}\right )} e^{\left (-3\right )}}{27 \, c^{3}}, \frac {{\left (c d + {\left (3 \, c x + b\right )} e\right )} e^{\left (-1\right )}}{3 \, c}\right )\right ) + 3 \, {\left (8 \, C c^{5} d^{3} e^{2} + {\left (35 \, C c^{5} x^{3} + 8 \, C b^{3} c^{2} + 3 \, {\left (10 \, B a + 7 \, A b\right )} c^{4} - 3 \, {\left (9 \, C a b + 4 \, B b^{2}\right )} c^{3} + 5 \, {\left (C b c^{4} + 9 \, B c^{5}\right )} x^{2} - {\left (6 \, C b^{2} c^{3} - 63 \, A c^{5} - {\left (14 \, C a + 9 \, B b\right )} c^{4}\right )} x\right )} e^{5} + {\left (5 \, C c^{5} d x^{2} + {\left (2 \, C b c^{4} + 9 \, B c^{5}\right )} d x - {\left (3 \, C b^{2} c^{3} - 21 \, A c^{5} - 2 \, {\left (4 \, C a + 3 \, B b\right )} c^{4}\right )} d\right )} e^{4} - 3 \, {\left (2 \, C c^{5} d^{2} x + {\left (C b c^{4} + 4 \, B c^{5}\right )} d^{2}\right )} e^{3}\right )} \sqrt {c x^{2} + b x + a} \sqrt {x e + d}\right )} e^{\left (-5\right )}}{945 \, c^{5}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left ({\left (C x^{2} + B x + A\right )} \sqrt {c x^{2} + b x + a} \sqrt {e x + d}, x\right ) \]