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