\[ \int \frac {(A+B x) \sqrt {d+e x}}{\left (b x+c x^2\right )^{5/2}} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (A b -\left (-2 A c +b B \right ) x \right ) \sqrt {e x +d}}{3 b^{2} \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}-\frac {2 \left (b \left (-b e +c d \right ) \left (A b e -8 A c d +4 B b d \right )-c \left (16 A \,c^{2} d^{2}+b^{2} e \left (A e +7 B d \right )-8 b c d \left (2 A e +B d \right )\right ) x \right ) \sqrt {e x +d}}{3 b^{4} d \left (-b e +c d \right ) \sqrt {c \,x^{2}+b x}}-\frac {2 \left (16 A \,c^{2} d^{2}+b^{2} e \left (A e +7 B d \right )-8 b c d \left (2 A e +B d \right )\right ) \EllipticE \left (\frac {\sqrt {c}\, \sqrt {x}}{\sqrt {-b}}, \sqrt {\frac {b e}{c d}}\right ) \sqrt {c}\, \sqrt {x}\, \sqrt {1+\frac {c x}{b}}\, \sqrt {e x +d}}{3 \left (-b \right )^{\frac {7}{2}} d \left (-b e +c d \right ) \sqrt {1+\frac {e x}{d}}\, \sqrt {c \,x^{2}+b x}}+\frac {2 \left (16 A \,c^{2} d +3 b^{2} B e -8 b c \left (A e +B d \right )\right ) \EllipticF \left (\frac {\sqrt {c}\, \sqrt {x}}{\sqrt {-b}}, \sqrt {\frac {b e}{c d}}\right ) \sqrt {x}\, \sqrt {1+\frac {c x}{b}}\, \sqrt {1+\frac {e x}{d}}}{3 \left (-b \right )^{\frac {7}{2}} \sqrt {c}\, \sqrt {e x +d}\, \sqrt {c \,x^{2}+b x}} \]
command
integrate((B*x+A)*(e*x+d)^(1/2)/(c*x^2+b*x)^(5/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {2 \, {\left ({\left (8 \, {\left (B b c^{4} - 2 \, A c^{5}\right )} d^{3} x^{4} + 16 \, {\left (B b^{2} c^{3} - 2 \, A b c^{4}\right )} d^{3} x^{3} + 8 \, {\left (B b^{3} c^{2} - 2 \, A b^{2} c^{3}\right )} d^{3} x^{2} - {\left (A b^{3} c^{2} x^{4} + 2 \, A b^{4} c x^{3} + A b^{5} x^{2}\right )} e^{3} + 2 \, {\left ({\left (B b^{3} c^{2} - 3 \, A b^{2} c^{3}\right )} d x^{4} + 2 \, {\left (B b^{4} c - 3 \, A b^{3} c^{2}\right )} d x^{3} + {\left (B b^{5} - 3 \, A b^{4} c\right )} d x^{2}\right )} e^{2} - {\left ({\left (11 \, B b^{2} c^{3} - 24 \, A b c^{4}\right )} d^{2} x^{4} + 2 \, {\left (11 \, B b^{3} c^{2} - 24 \, A b^{2} c^{3}\right )} d^{2} x^{3} + {\left (11 \, B b^{4} c - 24 \, A b^{3} c^{2}\right )} d^{2} 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 ({\left (A b^{2} c^{3} x^{4} + 2 \, A b^{3} c^{2} x^{3} + A b^{4} c x^{2}\right )} e^{3} + {\left ({\left (7 \, B b^{2} c^{3} - 16 \, A b c^{4}\right )} d x^{4} + 2 \, {\left (7 \, B b^{3} c^{2} - 16 \, A b^{2} c^{3}\right )} d x^{3} + {\left (7 \, B b^{4} c - 16 \, A b^{3} c^{2}\right )} d x^{2}\right )} e^{2} - 8 \, {\left ({\left (B b c^{4} - 2 \, A c^{5}\right )} d^{2} x^{4} + 2 \, {\left (B b^{2} c^{3} - 2 \, A b c^{4}\right )} d^{2} x^{3} + {\left (B b^{3} c^{2} - 2 \, A b^{2} c^{3}\right )} d^{2} 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 ({\left (A b^{2} c^{3} x^{3} + 2 \, A b^{3} c^{2} x^{2} + A b^{4} c x\right )} e^{3} + {\left (A b^{4} c d + {\left (7 \, B b^{2} c^{3} - 16 \, A b c^{4}\right )} d x^{3} + {\left (11 \, B b^{3} c^{2} - 25 \, A b^{2} c^{3}\right )} d x^{2} + {\left (3 \, B b^{4} c - 7 \, A b^{3} c^{2}\right )} d x\right )} e^{2} - {\left (A b^{3} c^{2} d^{2} + 8 \, {\left (B b c^{4} - 2 \, A c^{5}\right )} d^{2} x^{3} + 12 \, {\left (B b^{2} c^{3} - 2 \, A b c^{4}\right )} d^{2} x^{2} + 3 \, {\left (B b^{3} c^{2} - 2 \, A b^{2} c^{3}\right )} d^{2} x\right )} e\right )} \sqrt {x e + d}\right )}}{9 \, {\left ({\left (b^{5} c^{3} d x^{4} + 2 \, b^{6} c^{2} d x^{3} + b^{7} c d x^{2}\right )} e^{2} - {\left (b^{4} c^{4} d^{2} x^{4} + 2 \, b^{5} c^{3} d^{2} x^{3} + b^{6} c^{2} d^{2} x^{2}\right )} e\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {c x^{2} + b x} {\left (B x + A\right )} \sqrt {e x + d}}{c^{3} x^{6} + 3 \, b c^{2} x^{5} + 3 \, b^{2} c x^{4} + b^{3} x^{3}}, x\right ) \]