\[ \int \frac {A+B x+C x^2}{(d+e x)^{5/2} \sqrt {a+b x+c x^2}} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (C \,d^{2}-e \left (-A e +B d \right )\right ) \sqrt {c \,x^{2}+b x +a}}{3 e \left (a \,e^{2}-b d e +c \,d^{2}\right ) \left (e x +d \right )^{\frac {3}{2}}}+\frac {2 \left (c d \left (2 C \,d^{2}+e \left (-4 A e +B d \right )\right )+e \left (3 a e \left (-B e +2 C d \right )-b \left (-2 A \,e^{2}-B d e +4 C \,d^{2}\right )\right )\right ) \sqrt {c \,x^{2}+b x +a}}{3 e \left (a \,e^{2}-b d e +c \,d^{2}\right )^{2} \sqrt {e x +d}}-\frac {\left (c d \left (2 C \,d^{2}+e \left (-4 A e +B d \right )\right )+e \left (3 a e \left (-B e +2 C d \right )-b \left (-2 A \,e^{2}-B d e +4 C \,d^{2}\right )\right )\right ) \EllipticE \left (\frac {\sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {2}}{2}, \sqrt {-\frac {2 e \sqrt {-4 a c +b^{2}}}{2 c d -e \left (b +\sqrt {-4 a c +b^{2}}\right )}}\right ) \sqrt {2}\, \sqrt {-4 a c +b^{2}}\, \sqrt {e x +d}\, \sqrt {-\frac {c \left (c \,x^{2}+b x +a \right )}{-4 a c +b^{2}}}}{3 e^{2} \left (a \,e^{2}-b d e +c \,d^{2}\right )^{2} \sqrt {c \,x^{2}+b x +a}\, \sqrt {\frac {c \left (e x +d \right )}{2 c d -e \left (b +\sqrt {-4 a c +b^{2}}\right )}}}-\frac {2 \left (3 C e \left (-a e +b d \right )-c \left (2 C \,d^{2}+e \left (-A e +B d \right )\right )\right ) \EllipticF \left (\frac {\sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {2}}{2}, \sqrt {-\frac {2 e \sqrt {-4 a c +b^{2}}}{2 c d -e \left (b +\sqrt {-4 a c +b^{2}}\right )}}\right ) \sqrt {2}\, \sqrt {-4 a c +b^{2}}\, \sqrt {-\frac {c \left (c \,x^{2}+b x +a \right )}{-4 a c +b^{2}}}\, \sqrt {\frac {c \left (e x +d \right )}{2 c d -e \left (b +\sqrt {-4 a c +b^{2}}\right )}}}{3 c \,e^{2} \left (a \,e^{2}-b d e +c \,d^{2}\right ) \sqrt {e x +d}\, \sqrt {c \,x^{2}+b x +a}} \]
command
integrate((C*x^2+B*x+A)/(e*x+d)^(5/2)/(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^{6} + {\left (9 \, C a^{2} - 3 \, B a b + 2 \, A b^{2} - 3 \, A a c\right )} x^{2} e^{6} - {\left ({\left (12 \, C a b - B b^{2} - {\left (9 \, B a - 5 \, A b\right )} c\right )} d x^{2} - 2 \, {\left (9 \, C a^{2} - 3 \, B a b + 2 \, A b^{2} - 3 \, A a c\right )} d x\right )} e^{5} + {\left ({\left (5 \, C b^{2} + 5 \, A c^{2} + {\left (3 \, C a - 4 \, B b\right )} c\right )} d^{2} x^{2} - 2 \, {\left (12 \, C a b - B b^{2} - {\left (9 \, B a - 5 \, A b\right )} c\right )} d^{2} x + {\left (9 \, C a^{2} - 3 \, B a b + 2 \, A b^{2} - 3 \, A a c\right )} d^{2}\right )} e^{4} - {\left ({\left (5 \, C b c - B c^{2}\right )} d^{3} x^{2} - 2 \, {\left (5 \, C b^{2} + 5 \, A c^{2} + {\left (3 \, C a - 4 \, B b\right )} c\right )} d^{3} x + {\left (12 \, C a b - B b^{2} - {\left (9 \, B a - 5 \, A b\right )} c\right )} d^{3}\right )} e^{3} + {\left (2 \, C c^{2} d^{4} x^{2} - 2 \, {\left (5 \, C b c - B c^{2}\right )} d^{4} x + {\left (5 \, C b^{2} + 5 \, A c^{2} + {\left (3 \, C a - 4 \, B b\right )} c\right )} d^{4}\right )} e^{2} + {\left (4 \, C c^{2} d^{5} x - {\left (5 \, C b c - B c^{2}\right )} d^{5}\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^{5} e - {\left (3 \, B a - 2 \, A b\right )} c x^{2} e^{6} - {\left (2 \, {\left (3 \, B a - 2 \, A b\right )} c d x + {\left (4 \, A c^{2} - {\left (6 \, C a + B b\right )} c\right )} d x^{2}\right )} e^{5} - {\left ({\left (4 \, C b c - B c^{2}\right )} d^{2} x^{2} + {\left (3 \, B a - 2 \, A b\right )} c d^{2} + 2 \, {\left (4 \, A c^{2} - {\left (6 \, C a + B b\right )} c\right )} d^{2} x\right )} e^{4} + {\left (2 \, C c^{2} d^{3} x^{2} - 2 \, {\left (4 \, C b c - B c^{2}\right )} d^{3} x - {\left (4 \, A c^{2} - {\left (6 \, C a + B b\right )} c\right )} d^{3}\right )} e^{3} + {\left (4 \, C c^{2} d^{4} x - {\left (4 \, C b c - B c^{2}\right )} d^{4}\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^{4} e^{2} - {\left (A a c + {\left (3 \, B a - 2 \, A b\right )} c x\right )} e^{6} - {\left ({\left (2 \, B a - 3 \, A b\right )} c d + {\left (4 \, A c^{2} - {\left (6 \, C a + B b\right )} c\right )} d x\right )} e^{5} - {\left ({\left (4 \, C b c - B c^{2}\right )} d^{2} x - 5 \, {\left (C a c - A c^{2}\right )} d^{2}\right )} e^{4} + {\left (2 \, C c^{2} d^{3} x - {\left (3 \, C b c - 2 \, B c^{2}\right )} d^{3}\right )} e^{3}\right )} \sqrt {c x^{2} + b x + a} \sqrt {x e + d}\right )}}{9 \, {\left (c^{3} d^{6} e^{3} + a^{2} c x^{2} e^{9} - 2 \, {\left (a b c d x^{2} - a^{2} c d x\right )} e^{8} - {\left (4 \, a b c d^{2} x - a^{2} c d^{2} - {\left (b^{2} c + 2 \, a c^{2}\right )} d^{2} x^{2}\right )} e^{7} - 2 \, {\left (b c^{2} d^{3} x^{2} + a b c d^{3} - {\left (b^{2} c + 2 \, a c^{2}\right )} d^{3} x\right )} e^{6} + {\left (c^{3} d^{4} x^{2} - 4 \, b c^{2} d^{4} x + {\left (b^{2} c + 2 \, a c^{2}\right )} d^{4}\right )} e^{5} + 2 \, {\left (c^{3} d^{5} x - b c^{2} d^{5}\right )} e^{4}\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (C x^{2} + B x + A\right )} \sqrt {c x^{2} + b x + a} \sqrt {e x + d}}{c e^{3} x^{5} + {\left (3 \, c d e^{2} + b e^{3}\right )} x^{4} + a d^{3} + {\left (3 \, c d^{2} e + 3 \, b d e^{2} + a e^{3}\right )} x^{3} + {\left (c d^{3} + 3 \, b d^{2} e + 3 \, a d e^{2}\right )} x^{2} + {\left (b d^{3} + 3 \, a d^{2} e\right )} x}, x\right ) \]