\[ \int \frac {A+B x}{\sqrt {d+e x} \left (a+b x+c x^2\right )^{3/2}} \, dx \]
Optimal antiderivative \[ \frac {2 \left (a B \left (-b e +2 c d \right )-A \left (2 a c e -b^{2} e +b c d \right )+c \left (A b e -2 A c d -2 a B e +B b d \right ) x \right ) \sqrt {e x +d}}{\left (-4 a c +b^{2}\right ) \left (a \,e^{2}-b d e +c \,d^{2}\right ) \sqrt {c \,x^{2}+b x +a}}-\frac {\left (A b e -2 A c d -2 a B e +B b d \right ) \EllipticE \left (\frac {\sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {2}}{2}, \sqrt {-\frac {2 e \sqrt {-4 a c +b^{2}}}{2 c d -e \left (b +\sqrt {-4 a c +b^{2}}\right )}}\right ) \sqrt {2}\, \sqrt {e x +d}\, \sqrt {-\frac {c \left (c \,x^{2}+b x +a \right )}{-4 a c +b^{2}}}}{\left (a \,e^{2}-b d e +c \,d^{2}\right ) \sqrt {-4 a c +b^{2}}\, \sqrt {c \,x^{2}+b x +a}\, \sqrt {\frac {c \left (e x +d \right )}{2 c d -e \left (b +\sqrt {-4 a c +b^{2}}\right )}}}+\frac {2 \left (-2 A c +b B \right ) \EllipticF \left (\frac {\sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {2}}{2}, \sqrt {-\frac {2 e \sqrt {-4 a c +b^{2}}}{2 c d -e \left (b +\sqrt {-4 a c +b^{2}}\right )}}\right ) \sqrt {2}\, \sqrt {-\frac {c \left (c \,x^{2}+b x +a \right )}{-4 a c +b^{2}}}\, \sqrt {\frac {c \left (e x +d \right )}{2 c d -e \left (b +\sqrt {-4 a c +b^{2}}\right )}}}{c \sqrt {-4 a c +b^{2}}\, \sqrt {e x +d}\, \sqrt {c \,x^{2}+b x +a}} \]
command
integrate((B*x+A)/(e*x+d)^(1/2)/(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 ({\left (B b c^{2} - 2 \, A c^{3}\right )} d^{2} x^{2} + {\left (B b^{2} c - 2 \, A b c^{2}\right )} d^{2} x + {\left (B a b c - 2 \, A a c^{2}\right )} d^{2} + {\left (B a^{2} b + A a b^{2} - 6 \, A a^{2} c - {\left (6 \, A a c^{2} - {\left (B a b + A b^{2}\right )} c\right )} x^{2} + {\left (B a b^{2} + A b^{3} - 6 \, A a b c\right )} x\right )} e^{2} - 2 \, {\left ({\left (B b^{2} c - {\left (2 \, B a + A b\right )} c^{2}\right )} d x^{2} + {\left (B b^{3} - {\left (2 \, B a b + A b^{2}\right )} c\right )} d x + {\left (B a b^{2} - {\left (2 \, B a^{2} + A a b\right )} c\right )} d\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 ({\left (2 \, B a - A b\right )} c^{2} x^{2} + {\left (2 \, B a b - A b^{2}\right )} c x + {\left (2 \, B a^{2} - A a b\right )} c\right )} e^{2} - {\left ({\left (B b c^{2} - 2 \, A c^{3}\right )} d x^{2} + {\left (B b^{2} c - 2 \, A b c^{2}\right )} d x + {\left (B a b c - 2 \, A a c^{2}\right )} d\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 (2 \, A a c^{2} + {\left (2 \, B a - A b\right )} c^{2} x + {\left (B a b - A b^{2}\right )} c\right )} e^{2} - {\left ({\left (2 \, B a - A b\right )} c^{2} d + {\left (B b c^{2} - 2 \, A c^{3}\right )} d x\right )} e\right )} \sqrt {x e + d}\right )}}{3 \, {\left ({\left (a^{2} b^{2} c - 4 \, a^{3} c^{2} + {\left (a b^{2} c^{2} - 4 \, a^{2} c^{3}\right )} x^{2} + {\left (a b^{3} c - 4 \, a^{2} b c^{2}\right )} x\right )} e^{3} - {\left ({\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} d x^{2} + {\left (b^{4} c - 4 \, a b^{2} c^{2}\right )} d x + {\left (a b^{3} c - 4 \, a^{2} b c^{2}\right )} d\right )} e^{2} + {\left ({\left (b^{2} c^{3} - 4 \, a c^{4}\right )} d^{2} x^{2} + {\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} d^{2} x + {\left (a b^{2} c^{2} - 4 \, a^{2} c^{3}\right )} d^{2}\right )} e\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {c x^{2} + b x + a} {\left (B x + A\right )} \sqrt {e x + d}}{c^{2} e x^{5} + {\left (c^{2} d + 2 \, b c e\right )} x^{4} + {\left (2 \, b c d + {\left (b^{2} + 2 \, a c\right )} e\right )} x^{3} + a^{2} d + {\left (2 \, a b e + {\left (b^{2} + 2 \, a c\right )} d\right )} x^{2} + {\left (2 \, a b d + a^{2} e\right )} x}, x\right ) \]