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