\[ \int \frac {A+B x}{(d+e x)^{3/2} \left (b x+c x^2\right )^{5/2}} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (A b \left (-b e +c d \right )+c \left (2 A c d -b \left (A e +B d \right )\right ) x \right )}{3 b^{2} d \left (-b e +c d \right ) \left (c \,x^{2}+b x \right )^{\frac {3}{2}} \sqrt {e x +d}}+\frac {\frac {2 b \left (-b e +c d \right ) \left (8 A \,c^{2} d^{2}+b^{2} e \left (-4 A e +3 B d \right )-b c d \left (3 A e +4 B d \right )\right )}{3}+\frac {2 c \left (16 A \,c^{3} d^{3}+15 b^{2} B c \,d^{2} e -b^{3} e^{2} \left (-4 A e +3 B d \right )-8 b \,c^{2} d^{2} \left (3 A e +B d \right )\right ) x}{3}}{b^{4} d^{2} \left (-b e +c d \right )^{2} \sqrt {e x +d}\, \sqrt {c \,x^{2}+b x}}-\frac {2 \left (16 A \,c^{4} d^{4}-b^{3} c d \,e^{2} \left (-7 A e +9 B d \right )+2 b^{4} e^{3} \left (-4 A e +3 B d \right )-8 b \,c^{3} d^{3} \left (4 A e +B d \right )+b^{2} c^{2} d^{2} e \left (9 A e +19 B d \right )\right ) \EllipticE \left (\frac {\sqrt {c}\, \sqrt {x}}{\sqrt {-b}}, \sqrt {\frac {b e}{c d}}\right ) \sqrt {c}\, \sqrt {x}\, \sqrt {1+\frac {c x}{b}}\, \sqrt {e x +d}}{3 \left (-b \right )^{\frac {7}{2}} d^{3} \left (-b e +c d \right )^{3} \sqrt {1+\frac {e x}{d}}\, \sqrt {c \,x^{2}+b x}}+\frac {2 \left (16 A \,c^{3} d^{3}+15 b^{2} B c \,d^{2} e -b^{3} e^{2} \left (-4 A e +3 B d \right )-8 b \,c^{2} d^{2} \left (3 A e +B d \right )\right ) \EllipticF \left (\frac {\sqrt {c}\, \sqrt {x}}{\sqrt {-b}}, \sqrt {\frac {b e}{c d}}\right ) \sqrt {c}\, \sqrt {x}\, \sqrt {1+\frac {c x}{b}}\, \sqrt {1+\frac {e x}{d}}}{3 \left (-b \right )^{\frac {7}{2}} d^{2} \left (-b e +c d \right )^{2} \sqrt {e x +d}\, \sqrt {c \,x^{2}+b x}}+\frac {2 e \left (16 A \,c^{4} d^{4}-b^{3} c d \,e^{2} \left (-7 A e +9 B d \right )-8 b \,c^{3} d^{3} \left (4 A e +B d \right )+b^{2} c^{2} d^{2} e \left (9 A e +19 B d \right )+b^{4} \left (-8 A \,e^{4}+6 B d \,e^{3}\right )\right ) \sqrt {c \,x^{2}+b x}}{3 b^{4} d^{3} \left (-b e +c d \right )^{3} \sqrt {e x +d}} \]
command
integrate((B*x+A)/(e*x+d)^(3/2)/(c*x^2+b*x)^(5/2),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 \[ {\rm integral}\left (\frac {\sqrt {c x^{2} + b x} {\left (B x + A\right )} \sqrt {e x + d}}{c^{3} e^{2} x^{8} + b^{3} d^{2} x^{3} + {\left (2 \, c^{3} d e + 3 \, b c^{2} e^{2}\right )} x^{7} + {\left (c^{3} d^{2} + 6 \, b c^{2} d e + 3 \, b^{2} c e^{2}\right )} x^{6} + {\left (3 \, b c^{2} d^{2} + 6 \, b^{2} c d e + b^{3} e^{2}\right )} x^{5} + {\left (3 \, b^{2} c d^{2} + 2 \, b^{3} d e\right )} x^{4}}, x\right ) \]