\[ \int \frac {(A+B x) (d+e x)^{5/2}}{\left (a+b x+c x^2\right )^{3/2}} \, dx \]
Optimal antiderivative \[ \frac {2 \left (e x +d \right )^{\frac {3}{2}} \left (2 a c \left (A e +B d \right )-b \left (A c d +a B e \right )-\left (b^{2} B e -b c \left (A e +B d \right )+2 c \left (A c d -a B e \right )\right ) x \right )}{c \left (-4 a c +b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}+\frac {2 e \left (4 b^{2} B e -3 b c \left (A e +B d \right )+2 c \left (3 A c d -5 a B e \right )\right ) \sqrt {e x +d}\, \sqrt {c \,x^{2}+b x +a}}{3 c^{2} \left (-4 a c +b^{2}\right )}-\frac {\left (8 b^{3} B \,e^{2}-b^{2} c e \left (6 A e +13 B d \right )-2 c^{2} \left (-9 A a \,e^{2}+3 A c \,d^{2}-20 a B d e \right )+b c \left (6 A c d e -29 a B \,e^{2}+3 B c \,d^{2}\right )\right ) \EllipticE \left (\frac {\sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {2}}{2}, \sqrt {-\frac {2 e \sqrt {-4 a c +b^{2}}}{2 c d -e \left (b +\sqrt {-4 a c +b^{2}}\right )}}\right ) \sqrt {2}\, \sqrt {e x +d}\, \sqrt {-\frac {c \left (c \,x^{2}+b x +a \right )}{-4 a c +b^{2}}}}{3 c^{3} \sqrt {-4 a c +b^{2}}\, \sqrt {c \,x^{2}+b x +a}\, \sqrt {\frac {c \left (e x +d \right )}{2 c d -e \left (b +\sqrt {-4 a c +b^{2}}\right )}}}-\frac {2 \left (a \,e^{2}-b d e +c \,d^{2}\right ) \left (4 b^{2} B e -3 b c \left (A e +B d \right )+2 c \left (3 A c d -5 a B e \right )\right ) \EllipticF \left (\frac {\sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {2}}{2}, \sqrt {-\frac {2 e \sqrt {-4 a c +b^{2}}}{2 c d -e \left (b +\sqrt {-4 a c +b^{2}}\right )}}\right ) \sqrt {2}\, \sqrt {-\frac {c \left (c \,x^{2}+b x +a \right )}{-4 a c +b^{2}}}\, \sqrt {\frac {c \left (e x +d \right )}{2 c d -e \left (b +\sqrt {-4 a c +b^{2}}\right )}}}{3 c^{3} \sqrt {-4 a c +b^{2}}\, \sqrt {e x +d}\, \sqrt {c \,x^{2}+b x +a}} \]
command
integrate((B*x+A)*(e*x+d)^(5/2)/(c*x^2+b*x+a)^(3/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {2 \, {\left ({\left (3 \, {\left (B b c^{4} - 2 \, A c^{5}\right )} d^{3} x^{2} + 3 \, {\left (B b^{2} c^{3} - 2 \, A b c^{4}\right )} d^{3} x + 3 \, {\left (B a b c^{3} - 2 \, A a c^{4}\right )} d^{3} + {\left (8 \, B a b^{4} + 3 \, {\left (10 \, B a^{3} + 9 \, A a^{2} b\right )} c^{2} + {\left (8 \, B b^{4} c + 3 \, {\left (10 \, B a^{2} + 9 \, A a b\right )} c^{3} - {\left (41 \, B a b^{2} + 6 \, A b^{3}\right )} c^{2}\right )} x^{2} - {\left (41 \, B a^{2} b^{2} + 6 \, A a b^{3}\right )} c + {\left (8 \, B b^{5} + 3 \, {\left (10 \, B a^{2} b + 9 \, A a b^{2}\right )} c^{2} - {\left (41 \, B a b^{3} + 6 \, A b^{4}\right )} c\right )} x\right )} e^{3} - {\left ({\left (17 \, B b^{3} c^{2} + 54 \, A a c^{4} - {\left (77 \, B a b + 9 \, A b^{2}\right )} c^{3}\right )} d x^{2} + {\left (17 \, B b^{4} c + 54 \, A a b c^{3} - {\left (77 \, B a b^{2} + 9 \, A b^{3}\right )} c^{2}\right )} d x + {\left (17 \, B a b^{3} c + 54 \, A a^{2} c^{3} - {\left (77 \, B a^{2} b + 9 \, A a b^{2}\right )} c^{2}\right )} d\right )} e^{2} + {\left ({\left (8 \, B b^{2} c^{3} - {\left (50 \, B a - 9 \, A b\right )} c^{4}\right )} d^{2} x^{2} + {\left (8 \, B b^{3} c^{2} - {\left (50 \, B a b - 9 \, A b^{2}\right )} c^{3}\right )} d^{2} x + {\left (8 \, B a b^{2} c^{2} - {\left (50 \, B a^{2} - 9 \, A a b\right )} c^{3}\right )} d^{2}\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 ({\left (8 \, B a b^{3} c + 18 \, A a^{2} c^{3} - {\left (29 \, B a^{2} b + 6 \, A a b^{2}\right )} c^{2} + {\left (8 \, B b^{3} c^{2} + 18 \, A a c^{4} - {\left (29 \, B a b + 6 \, A b^{2}\right )} c^{3}\right )} x^{2} + {\left (8 \, B b^{4} c + 18 \, A a b c^{3} - {\left (29 \, B a b^{2} + 6 \, A b^{3}\right )} c^{2}\right )} x\right )} e^{3} - {\left ({\left (13 \, B b^{2} c^{3} - 2 \, {\left (20 \, B a + 3 \, A b\right )} c^{4}\right )} d x^{2} + {\left (13 \, B b^{3} c^{2} - 2 \, {\left (20 \, B a b + 3 \, A b^{2}\right )} c^{3}\right )} d x + {\left (13 \, B a b^{2} c^{2} - 2 \, {\left (20 \, B a^{2} + 3 \, A a b\right )} c^{3}\right )} d\right )} e^{2} + 3 \, {\left ({\left (B b c^{4} - 2 \, A c^{5}\right )} d^{2} x^{2} + {\left (B b^{2} c^{3} - 2 \, A b c^{4}\right )} d^{2} x + {\left (B a b c^{3} - 2 \, A a c^{4}\right )} d^{2}\right )} e\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 \, \sqrt {c x^{2} + b x + a} {\left ({\left (4 \, B a b^{2} c^{2} - {\left (10 \, B a^{2} + 3 \, A a b\right )} c^{3} + {\left (B b^{2} c^{3} - 4 \, B a c^{4}\right )} x^{2} + {\left (4 \, B b^{3} c^{2} + 6 \, A a c^{4} - {\left (13 \, B a b + 3 \, A b^{2}\right )} c^{3}\right )} x\right )} e^{3} - 6 \, {\left ({\left (B b^{2} c^{3} - {\left (2 \, B a + A b\right )} c^{4}\right )} d x + {\left (B a b c^{3} - 2 \, A a c^{4}\right )} d\right )} e^{2} + 3 \, {\left ({\left (2 \, B a - A b\right )} c^{4} d^{2} + {\left (B b c^{4} - 2 \, A c^{5}\right )} d^{2} x\right )} e\right )} \sqrt {x e + d}\right )} e^{\left (-1\right )}}{9 \, {\left (a b^{2} c^{4} - 4 \, a^{2} c^{5} + {\left (b^{2} c^{5} - 4 \, a c^{6}\right )} x^{2} + {\left (b^{3} c^{4} - 4 \, a b c^{5}\right )} x\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (B e^{2} x^{3} + A d^{2} + {\left (2 \, B d e + A e^{2}\right )} x^{2} + {\left (B d^{2} + 2 \, A d e\right )} x\right )} \sqrt {c x^{2} + b x + a} \sqrt {e x + d}}{c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x + {\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}}, x\right ) \]