\[ \int \frac {\sqrt {a+b x+c x^2} \left (A+B x+C x^2\right )}{(d+e x)^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (C \,d^{2}-e \left (-A e +B d \right )\right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{e \left (a \,e^{2}-b d e +c \,d^{2}\right ) \sqrt {e x +d}}-\frac {2 \left (b C \,e^{2} \left (-a e +b d \right )+c^{2} d \left (24 C \,d^{2}-5 e \left (-3 A e +4 B d \right )\right )+c e \left (a e \left (-5 B e +9 C d \right )-5 b \left (3 A \,e^{2}-4 B d e +5 C \,d^{2}\right )\right )+3 c \,e^{2} \left (5 B c d +b C d -\frac {6 c C \,d^{2}}{e}-5 A c e -a C e \right ) x \right ) \sqrt {e x +d}\, \sqrt {c \,x^{2}+b x +a}}{15 c \,e^{3} \left (a \,e^{2}-b d e +c \,d^{2}\right )}-\frac {\left (2 b^{2} C \,e^{2}+c e \left (-5 b B e -6 a C e +8 b C d \right )-c^{2} \left (48 C \,d^{2}-10 e \left (-3 A e +4 B d \right )\right )\right ) \EllipticE \left (\frac {\sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {2}}{2}, \sqrt {-\frac {2 e \sqrt {-4 a c +b^{2}}}{2 c d -e \left (b +\sqrt {-4 a c +b^{2}}\right )}}\right ) \sqrt {2}\, \sqrt {-4 a c +b^{2}}\, \sqrt {e x +d}\, \sqrt {-\frac {c \left (c \,x^{2}+b x +a \right )}{-4 a c +b^{2}}}}{15 c^{2} e^{4} \sqrt {c \,x^{2}+b x +a}\, \sqrt {\frac {c \left (e x +d \right )}{2 c d -e \left (b +\sqrt {-4 a c +b^{2}}\right )}}}+\frac {2 \left (b C \,e^{2} \left (-a e +b d \right )-2 c^{2} d \left (24 C \,d^{2}-5 e \left (-3 A e +4 B d \right )\right )-c e \left (2 a e \left (-5 B e +9 C d \right )-b \left (32 C \,d^{2}-5 e \left (-3 A e +5 B d \right )\right )\right )\right ) \EllipticF \left (\frac {\sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {2}}{2}, \sqrt {-\frac {2 e \sqrt {-4 a c +b^{2}}}{2 c d -e \left (b +\sqrt {-4 a c +b^{2}}\right )}}\right ) \sqrt {2}\, \sqrt {-4 a c +b^{2}}\, \sqrt {-\frac {c \left (c \,x^{2}+b x +a \right )}{-4 a c +b^{2}}}\, \sqrt {\frac {c \left (e x +d \right )}{2 c d -e \left (b +\sqrt {-4 a c +b^{2}}\right )}}}{15 c^{2} e^{4} \sqrt {e x +d}\, \sqrt {c \,x^{2}+b x +a}} \]
command
integrate((C*x^2+B*x+A)*(c*x^2+b*x+a)^(1/2)/(e*x+d)^(3/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {2 \, {\left ({\left (48 \, C c^{3} d^{4} - {\left (2 \, C b^{3} + 15 \, {\left (2 \, B a + A b\right )} c^{2} - {\left (9 \, C a b + 5 \, B b^{2}\right )} c\right )} x e^{4} - {\left ({\left (7 \, C b^{2} c - 30 \, A c^{3} - {\left (42 \, C a + 25 \, B b\right )} c^{2}\right )} d x + {\left (2 \, C b^{3} + 15 \, {\left (2 \, B a + A b\right )} c^{2} - {\left (9 \, C a b + 5 \, B b^{2}\right )} c\right )} d\right )} e^{3} - {\left (8 \, {\left (4 \, C b c^{2} + 5 \, B c^{3}\right )} d^{2} x + {\left (7 \, C b^{2} c - 30 \, A c^{3} - {\left (42 \, C a + 25 \, B b\right )} c^{2}\right )} d^{2}\right )} e^{2} + 8 \, {\left (6 \, C c^{3} d^{3} x - {\left (4 \, C b c^{2} + 5 \, B c^{3}\right )} d^{3}\right )} e\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 (48 \, C c^{3} d^{3} e - {\left (2 \, C b^{2} c - 30 \, A c^{3} - {\left (6 \, C a + 5 \, B b\right )} c^{2}\right )} x e^{4} - {\left (8 \, {\left (C b c^{2} + 5 \, B c^{3}\right )} d x + {\left (2 \, C b^{2} c - 30 \, A c^{3} - {\left (6 \, C a + 5 \, B b\right )} c^{2}\right )} d\right )} e^{3} + 8 \, {\left (6 \, C c^{3} d^{2} x - {\left (C b c^{2} + 5 \, B c^{3}\right )} d^{2}\right )} e^{2}\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 (24 \, C c^{3} d^{2} e^{2} - {\left (3 \, C c^{3} x^{2} - 15 \, A c^{3} + {\left (C b c^{2} + 5 \, B c^{3}\right )} x\right )} e^{4} + {\left (6 \, C c^{3} d x - {\left (C b c^{2} + 20 \, B c^{3}\right )} d\right )} e^{3}\right )} \sqrt {c x^{2} + b x + a} \sqrt {x e + d}\right )}}{45 \, {\left (c^{3} x e^{6} + c^{3} d e^{5}\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (C x^{2} + B x + A\right )} \sqrt {c x^{2} + b x + a} \sqrt {e x + d}}{e^{2} x^{2} + 2 \, d e x + d^{2}}, x\right ) \]