\[ \int \frac {A+B x+C x^2}{(d+e x)^{3/2} \sqrt {a+b x+c x^2}} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (C \,d^{2}-e \left (-A e +B d \right )\right ) \sqrt {c \,x^{2}+b x +a}}{e \left (a \,e^{2}-b d e +c \,d^{2}\right ) \sqrt {e x +d}}-\frac {\left (C e \left (-a e +b d \right )-c \left (2 C \,d^{2}-e \left (-A e +B d \right )\right )\right ) \EllipticE \left (\frac {\sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {2}}{2}, \sqrt {-\frac {2 e \sqrt {-4 a c +b^{2}}}{2 c d -e \left (b +\sqrt {-4 a c +b^{2}}\right )}}\right ) \sqrt {2}\, \sqrt {-4 a c +b^{2}}\, \sqrt {e x +d}\, \sqrt {-\frac {c \left (c \,x^{2}+b x +a \right )}{-4 a c +b^{2}}}}{c \,e^{2} \left (a \,e^{2}-b d e +c \,d^{2}\right ) \sqrt {c \,x^{2}+b x +a}\, \sqrt {\frac {c \left (e x +d \right )}{2 c d -e \left (b +\sqrt {-4 a c +b^{2}}\right )}}}-\frac {2 \left (-B e +2 C d \right ) \EllipticF \left (\frac {\sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {2}}{2}, \sqrt {-\frac {2 e \sqrt {-4 a c +b^{2}}}{2 c d -e \left (b +\sqrt {-4 a c +b^{2}}\right )}}\right ) \sqrt {2}\, \sqrt {-4 a c +b^{2}}\, \sqrt {-\frac {c \left (c \,x^{2}+b x +a \right )}{-4 a c +b^{2}}}\, \sqrt {\frac {c \left (e x +d \right )}{2 c d -e \left (b +\sqrt {-4 a c +b^{2}}\right )}}}{c \,e^{2} \sqrt {e x +d}\, \sqrt {c \,x^{2}+b x +a}} \]
command
integrate((C*x^2+B*x+A)/(e*x+d)^(3/2)/(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 (2 \, C c^{2} d^{4} + {\left (C a b - {\left (3 \, B a - A b\right )} c\right )} x e^{4} - {\left ({\left (C b^{2} + 2 \, A c^{2} - 2 \, {\left (2 \, C a + B b\right )} c\right )} d x - {\left (C a b - {\left (3 \, B a - A b\right )} c\right )} d\right )} e^{3} - {\left ({\left (2 \, C b c + B c^{2}\right )} d^{2} x + {\left (C b^{2} + 2 \, A c^{2} - 2 \, {\left (2 \, C a + B b\right )} c\right )} d^{2}\right )} e^{2} + {\left (2 \, C c^{2} d^{3} x - {\left (2 \, C b c + B c^{2}\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 (2 \, C c^{2} d^{3} e + {\left (C a c + A c^{2}\right )} x e^{4} - {\left ({\left (C b c + B c^{2}\right )} d x - {\left (C a c + A c^{2}\right )} d\right )} e^{3} + {\left (2 \, C c^{2} d^{2} x - {\left (C b c + B c^{2}\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 (C c^{2} d^{2} e^{2} - B c^{2} d e^{3} + A c^{2} e^{4}\right )} \sqrt {c x^{2} + b x + a} \sqrt {x e + d}\right )}}{3 \, {\left (c^{3} d^{3} e^{3} + a c^{2} x e^{6} - {\left (b c^{2} d x - a c^{2} d\right )} e^{5} + {\left (c^{3} d^{2} x - b c^{2} d^{2}\right )} e^{4}\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (C x^{2} + B x + A\right )} \sqrt {c x^{2} + b x + a} \sqrt {e x + d}}{c e^{2} x^{4} + {\left (2 \, c d e + b e^{2}\right )} x^{3} + a d^{2} + {\left (c d^{2} + 2 \, b d e + a e^{2}\right )} x^{2} + {\left (b d^{2} + 2 \, a d e\right )} x}, x\right ) \]