\[ \int \frac {(d+e x)^{3/2} \left (A+B x+C x^2\right )}{\sqrt {a+b x+c x^2}} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (-7 B c e +6 C b e +2 C c d \right ) \left (e x +d \right )^{\frac {3}{2}} \sqrt {c \,x^{2}+b x +a}}{35 c^{2} e}+\frac {2 C \left (e x +d \right )^{\frac {5}{2}} \sqrt {c \,x^{2}+b x +a}}{7 c e}+\frac {2 \left (24 b^{2} C \,e^{2}-c e \left (28 b B e +25 a C e +15 b C d \right )-c^{2} \left (6 C \,d^{2}-7 e \left (5 A e +3 B d \right )\right )\right ) \sqrt {e x +d}\, \sqrt {c \,x^{2}+b x +a}}{105 c^{3} e}-\frac {\left (48 b^{3} C \,e^{3}-8 b c \,e^{2} \left (7 b B e +13 a C e +9 b C d \right )+c^{3} d \left (6 C \,d^{2}-7 e \left (20 A e +3 B d \right )\right )+c^{2} e \left (a e \left (63 B e +82 C d \right )+b \left (70 A \,e^{2}+91 B d e +12 C \,d^{2}\right )\right )\right ) \EllipticE \left (\frac {\sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {2}}{2}, \sqrt {-\frac {2 e \sqrt {-4 a c +b^{2}}}{2 c d -e \left (b +\sqrt {-4 a c +b^{2}}\right )}}\right ) \sqrt {2}\, \sqrt {-4 a c +b^{2}}\, \sqrt {e x +d}\, \sqrt {-\frac {c \left (c \,x^{2}+b x +a \right )}{-4 a c +b^{2}}}}{105 c^{4} e^{2} \sqrt {c \,x^{2}+b x +a}\, \sqrt {\frac {c \left (e x +d \right )}{2 c d -e \left (b +\sqrt {-4 a c +b^{2}}\right )}}}-\frac {2 \left (a \,e^{2}-b d e +c \,d^{2}\right ) \left (24 b^{2} C \,e^{2}-c e \left (28 b B e +25 a C e +15 b C d \right )-c^{2} \left (6 C \,d^{2}-7 e \left (5 A e +3 B d \right )\right )\right ) \EllipticF \left (\frac {\sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {2}}{2}, \sqrt {-\frac {2 e \sqrt {-4 a c +b^{2}}}{2 c d -e \left (b +\sqrt {-4 a c +b^{2}}\right )}}\right ) \sqrt {2}\, \sqrt {-4 a c +b^{2}}\, \sqrt {-\frac {c \left (c \,x^{2}+b x +a \right )}{-4 a c +b^{2}}}\, \sqrt {\frac {c \left (e x +d \right )}{2 c d -e \left (b +\sqrt {-4 a c +b^{2}}\right )}}}{105 c^{4} e^{2} \sqrt {e x +d}\, \sqrt {c \,x^{2}+b x +a}} \]
command
integrate((e*x+d)^(3/2)*(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 (6 \, C c^{4} d^{4} + 3 \, {\left (3 \, C b c^{3} - 7 \, B c^{4}\right )} d^{3} e + {\left (39 \, C b^{2} c^{2} + 175 \, A c^{4} - {\left (71 \, C a + 56 \, B b\right )} c^{3}\right )} d^{2} e^{2} - {\left (96 \, C b^{3} c + 7 \, {\left (27 \, B a + 25 \, A b\right )} c^{3} - {\left (260 \, C a b + 119 \, B b^{2}\right )} c^{2}\right )} d e^{3} + {\left (48 \, C b^{4} - 105 \, A a c^{3} + {\left (75 \, C a^{2} + 147 \, B a b + 70 \, A b^{2}\right )} c^{2} - 8 \, {\left (22 \, C a b^{2} + 7 \, B b^{3}\right )} c\right )} e^{4}\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 (6 \, C c^{4} d^{3} e + 3 \, {\left (4 \, C b c^{3} - 7 \, B c^{4}\right )} d^{2} e^{2} - {\left (72 \, C b^{2} c^{2} + 140 \, A c^{4} - {\left (82 \, C a + 91 \, B b\right )} c^{3}\right )} d e^{3} + {\left (48 \, C b^{3} c + 7 \, {\left (9 \, B a + 10 \, A b\right )} c^{3} - 8 \, {\left (13 \, C a b + 7 \, B b^{2}\right )} c^{2}\right )} e^{4}\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 (3 \, C c^{4} d^{2} e^{2} + {\left (15 \, C c^{4} x^{2} + 24 \, C b^{2} c^{2} + 35 \, A c^{4} - {\left (25 \, C a + 28 \, B b\right )} c^{3} - 3 \, {\left (6 \, C b c^{3} - 7 \, B c^{4}\right )} x\right )} e^{4} + 3 \, {\left (8 \, C c^{4} d x - {\left (11 \, C b c^{3} - 14 \, B c^{4}\right )} d\right )} e^{3}\right )} \sqrt {c x^{2} + b x + a} \sqrt {x e + d}\right )} e^{\left (-3\right )}}{315 \, c^{5}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (C e x^{3} + {\left (C d + B e\right )} x^{2} + A d + {\left (B d + A e\right )} x\right )} \sqrt {e x + d}}{\sqrt {c x^{2} + b x + a}}, x\right ) \]