\[ \int \frac {(A+B x) \left (b x+c x^2\right )^{3/2}}{(d+e x)^{5/2}} \, dx \]
Optimal antiderivative \[ \frac {2 \left (3 B e x -5 A e +8 B d \right ) \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{15 e^{2} \left (e x +d \right )^{\frac {3}{2}}}-\frac {2 \left (40 A c e \left (-b e +2 c d \right )-B \left (3 b^{2} e^{2}-88 b c d e +128 c^{2} d^{2}\right )\right ) \EllipticE \left (\frac {\sqrt {c}\, \sqrt {x}}{\sqrt {-b}}, \sqrt {\frac {b e}{c d}}\right ) \sqrt {-b}\, \sqrt {x}\, \sqrt {1+\frac {c x}{b}}\, \sqrt {e x +d}}{15 e^{5} \sqrt {c}\, \sqrt {1+\frac {e x}{d}}\, \sqrt {c \,x^{2}+b x}}+\frac {2 \left (5 A e \left (3 b^{2} e^{2}-16 b c d e +16 c^{2} d^{2}\right )-B d \left (39 b^{2} e^{2}-152 b c d e +128 c^{2} d^{2}\right )\right ) \EllipticF \left (\frac {\sqrt {c}\, \sqrt {x}}{\sqrt {-b}}, \sqrt {\frac {b e}{c d}}\right ) \sqrt {-b}\, \sqrt {x}\, \sqrt {1+\frac {c x}{b}}\, \sqrt {1+\frac {e x}{d}}}{15 e^{5} \sqrt {c}\, \sqrt {e x +d}\, \sqrt {c \,x^{2}+b x}}-\frac {2 \left (4 B d \left (-9 b e +16 c d \right )-5 A e \left (-3 b e +8 c d \right )+e \left (-10 A c e -3 b B e +16 B c d \right ) x \right ) \sqrt {c \,x^{2}+b x}}{15 e^{4} \sqrt {e x +d}} \]
command
integrate((B*x+A)*(c*x^2+b*x)^(3/2)/(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 (128 \, B c^{3} d^{5} + {\left (3 \, B b^{3} - 5 \, A b^{2} c\right )} x^{2} e^{5} + {\left ({\left (23 \, B b^{2} c + 80 \, A b c^{2}\right )} d x^{2} + 2 \, {\left (3 \, B b^{3} - 5 \, A b^{2} c\right )} d x\right )} e^{4} - {\left (8 \, {\left (19 \, B b c^{2} + 10 \, A c^{3}\right )} d^{2} x^{2} - 2 \, {\left (23 \, B b^{2} c + 80 \, A b c^{2}\right )} d^{2} x - {\left (3 \, B b^{3} - 5 \, A b^{2} c\right )} d^{2}\right )} e^{3} + {\left (128 \, B c^{3} d^{3} x^{2} - 16 \, {\left (19 \, B b c^{2} + 10 \, A c^{3}\right )} d^{3} x + {\left (23 \, B b^{2} c + 80 \, A b c^{2}\right )} d^{3}\right )} e^{2} + 8 \, {\left (32 \, B c^{3} d^{4} x - {\left (19 \, B b c^{2} + 10 \, A c^{3}\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 (128 \, B c^{3} d^{4} e + {\left (3 \, B b^{2} c + 40 \, A b c^{2}\right )} x^{2} e^{5} - 2 \, {\left (4 \, {\left (11 \, B b c^{2} + 10 \, A c^{3}\right )} d x^{2} - {\left (3 \, B b^{2} c + 40 \, A b c^{2}\right )} d x\right )} e^{4} + {\left (128 \, B c^{3} d^{2} x^{2} - 16 \, {\left (11 \, B b c^{2} + 10 \, A c^{3}\right )} d^{2} x + {\left (3 \, B b^{2} c + 40 \, A b c^{2}\right )} d^{2}\right )} e^{3} + 8 \, {\left (32 \, B c^{3} d^{3} x - {\left (11 \, B b c^{2} + 10 \, A c^{3}\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 (64 \, B c^{3} d^{3} e^{2} - {\left (3 \, B c^{3} x^{3} - 20 \, A b c^{2} x + {\left (6 \, B b c^{2} + 5 \, A c^{3}\right )} x^{2}\right )} e^{5} + {\left (8 \, B c^{3} d x^{2} + 15 \, A b c^{2} d - {\left (47 \, B b c^{2} + 50 \, A c^{3}\right )} d x\right )} e^{4} + 4 \, {\left (20 \, B c^{3} d^{2} x - {\left (9 \, B b c^{2} + 10 \, A c^{3}\right )} d^{2}\right )} e^{3}\right )} \sqrt {c x^{2} + b x} \sqrt {x e + d}\right )}}{45 \, {\left (c^{2} x^{2} e^{8} + 2 \, c^{2} d x e^{7} + c^{2} d^{2} e^{6}\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (B c x^{3} + A b x + {\left (B b + A c\right )} x^{2}\right )} \sqrt {c x^{2} + b x} \sqrt {e x + d}}{e^{3} x^{3} + 3 \, d e^{2} x^{2} + 3 \, d^{2} e x + d^{3}}, x\right ) \]