\[ \int \frac {A+B x}{\sqrt {d+e x} \left (b x+c x^2\right )^{5/2}} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (A b \left (-b e +c d \right )+c \left (2 A c d -b \left (A e +B d \right )\right ) x \right ) \sqrt {e x +d}}{3 b^{2} d \left (-b e +c d \right ) \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}+\frac {2 \left (b \left (-b e +c d \right ) \left (8 A \,c^{2} d^{2}+b^{2} e \left (-2 A e +3 B d \right )-b c d \left (5 A e +4 B d \right )\right )+c \left (16 A \,c^{3} d^{3}-b^{3} e^{2} \left (-2 A e +3 B d \right )-8 b \,c^{2} d^{2} \left (3 A e +B d \right )+b^{2} c d e \left (4 A e +13 B d \right )\right ) x \right ) \sqrt {e x +d}}{3 b^{4} d^{2} \left (-b e +c d \right )^{2} \sqrt {c \,x^{2}+b x}}-\frac {2 \left (16 A \,c^{3} d^{3}-b^{3} e^{2} \left (-2 A e +3 B d \right )-8 b \,c^{2} d^{2} \left (3 A e +B d \right )+b^{2} c d e \left (4 A e +13 B d \right )\right ) \EllipticE \left (\frac {\sqrt {c}\, \sqrt {x}}{\sqrt {-b}}, \sqrt {\frac {b e}{c d}}\right ) \sqrt {c}\, \sqrt {x}\, \sqrt {1+\frac {c x}{b}}\, \sqrt {e x +d}}{3 \left (-b \right )^{\frac {7}{2}} d^{2} \left (-b e +c d \right )^{2} \sqrt {1+\frac {e x}{d}}\, \sqrt {c \,x^{2}+b x}}+\frac {2 \left (16 A \,c^{2} d^{2}+b^{2} e \left (-A e +9 B d \right )-8 b c d \left (2 A e +B d \right )\right ) \EllipticF \left (\frac {\sqrt {c}\, \sqrt {x}}{\sqrt {-b}}, \sqrt {\frac {b e}{c d}}\right ) \sqrt {c}\, \sqrt {x}\, \sqrt {1+\frac {c x}{b}}\, \sqrt {1+\frac {e x}{d}}}{3 \left (-b \right )^{\frac {7}{2}} d \left (-b e +c d \right ) \sqrt {e x +d}\, \sqrt {c \,x^{2}+b x}} \]
command
integrate((B*x+A)/(c*x^2+b*x)^(5/2)/(e*x+d)^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {2 \, {\left ({\left (8 \, {\left (B b c^{5} - 2 \, A c^{6}\right )} d^{4} x^{4} + 16 \, {\left (B b^{2} c^{4} - 2 \, A b c^{5}\right )} d^{4} x^{3} + 8 \, {\left (B b^{3} c^{3} - 2 \, A b^{2} c^{4}\right )} d^{4} x^{2} - 2 \, {\left (A b^{4} c^{2} x^{4} + 2 \, A b^{5} c x^{3} + A b^{6} x^{2}\right )} e^{4} + 3 \, {\left ({\left (B b^{4} c^{2} - A b^{3} c^{3}\right )} d x^{4} + 2 \, {\left (B b^{5} c - A b^{4} c^{2}\right )} d x^{3} + {\left (B b^{6} - A b^{5} c\right )} d x^{2}\right )} e^{3} + {\left ({\left (8 \, B b^{3} c^{3} - 13 \, A b^{2} c^{4}\right )} d^{2} x^{4} + 2 \, {\left (8 \, B b^{4} c^{2} - 13 \, A b^{3} c^{3}\right )} d^{2} x^{3} + {\left (8 \, B b^{5} c - 13 \, A b^{4} c^{2}\right )} d^{2} x^{2}\right )} e^{2} - {\left ({\left (17 \, B b^{2} c^{4} - 32 \, A b c^{5}\right )} d^{3} x^{4} + 2 \, {\left (17 \, B b^{3} c^{3} - 32 \, A b^{2} c^{4}\right )} d^{3} x^{3} + {\left (17 \, B b^{4} c^{2} - 32 \, A b^{3} c^{3}\right )} d^{3} x^{2}\right )} e\right )} \sqrt {c} e^{\frac {1}{2}} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} 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 \, b^{2} c d e^{2} + 2 \, b^{3} 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 (2 \, {\left (A b^{3} c^{3} x^{4} + 2 \, A b^{4} c^{2} x^{3} + A b^{5} c x^{2}\right )} e^{4} - {\left ({\left (3 \, B b^{3} c^{3} - 4 \, A b^{2} c^{4}\right )} d x^{4} + 2 \, {\left (3 \, B b^{4} c^{2} - 4 \, A b^{3} c^{3}\right )} d x^{3} + {\left (3 \, B b^{5} c - 4 \, A b^{4} c^{2}\right )} d x^{2}\right )} e^{3} + {\left ({\left (13 \, B b^{2} c^{4} - 24 \, A b c^{5}\right )} d^{2} x^{4} + 2 \, {\left (13 \, B b^{3} c^{3} - 24 \, A b^{2} c^{4}\right )} d^{2} x^{3} + {\left (13 \, B b^{4} c^{2} - 24 \, A b^{3} c^{3}\right )} d^{2} x^{2}\right )} e^{2} - 8 \, {\left ({\left (B b c^{5} - 2 \, A c^{6}\right )} d^{3} x^{4} + 2 \, {\left (B b^{2} c^{4} - 2 \, A b c^{5}\right )} d^{3} x^{3} + {\left (B b^{3} c^{3} - 2 \, A b^{2} c^{4}\right )} d^{3} x^{2}\right )} e\right )} \sqrt {c} e^{\frac {1}{2}} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} 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 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )} e^{\left (-3\right )}}{27 \, c^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} 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 \, b^{2} c d e^{2} + 2 \, b^{3} 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} {\left (2 \, {\left (A b^{3} c^{3} x^{3} + 2 \, A b^{4} c^{2} x^{2} + A b^{5} c x\right )} e^{4} - {\left (A b^{5} c d + {\left (3 \, B b^{3} c^{3} - 4 \, A b^{2} c^{4}\right )} d x^{3} + {\left (6 \, B b^{4} c^{2} - 7 \, A b^{3} c^{3}\right )} d x^{2} + {\left (3 \, B b^{5} c - 2 \, A b^{4} c^{2}\right )} d x\right )} e^{3} + {\left (2 \, A b^{4} c^{2} d^{2} + {\left (13 \, B b^{2} c^{4} - 24 \, A b c^{5}\right )} d^{2} x^{3} + {\left (20 \, B b^{3} c^{3} - 37 \, A b^{2} c^{4}\right )} d^{2} x^{2} + 2 \, {\left (3 \, B b^{4} c^{2} - 5 \, A b^{3} c^{3}\right )} d^{2} x\right )} e^{2} - {\left (A b^{3} c^{3} d^{3} + 8 \, {\left (B b c^{5} - 2 \, A c^{6}\right )} d^{3} x^{3} + 12 \, {\left (B b^{2} c^{4} - 2 \, A b c^{5}\right )} d^{3} x^{2} + 3 \, {\left (B b^{3} c^{3} - 2 \, A b^{2} c^{4}\right )} d^{3} x\right )} e\right )} \sqrt {x e + d}\right )}}{9 \, {\left ({\left (b^{6} c^{3} d^{2} x^{4} + 2 \, b^{7} c^{2} d^{2} x^{3} + b^{8} c d^{2} x^{2}\right )} e^{3} - 2 \, {\left (b^{5} c^{4} d^{3} x^{4} + 2 \, b^{6} c^{3} d^{3} x^{3} + b^{7} c^{2} d^{3} x^{2}\right )} e^{2} + {\left (b^{4} c^{5} d^{4} x^{4} + 2 \, b^{5} c^{4} d^{4} x^{3} + b^{6} c^{3} d^{4} x^{2}\right )} e\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {c x^{2} + b x} {\left (B x + A\right )} \sqrt {e x + d}}{c^{3} e x^{7} + b^{3} d x^{3} + {\left (c^{3} d + 3 \, b c^{2} e\right )} x^{6} + 3 \, {\left (b c^{2} d + b^{2} c e\right )} x^{5} + {\left (3 \, b^{2} c d + b^{3} e\right )} x^{4}}, x\right ) \]