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