\[ \int \frac {(A+B x) \left (b x+c x^2\right )^{3/2}}{(d+e x)^{3/2}} \, dx \]
Optimal antiderivative \[ \frac {2 \left (B e x -7 A e +8 B d \right ) \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{7 e^{2} \sqrt {e x +d}}+\frac {2 \left (5 b c e \left (-7 A e +8 B d \right ) \left (-b e +2 c d \right )-\left (-14 A c e -b B e +16 B c d \right ) \left (-2 b^{2} e^{2}-3 b c d e +8 c^{2} d^{2}\right )\right ) \EllipticE \left (\frac {\sqrt {c}\, \sqrt {x}}{\sqrt {-b}}, \sqrt {\frac {b e}{c d}}\right ) \sqrt {-b}\, \sqrt {x}\, \sqrt {1+\frac {c x}{b}}\, \sqrt {e x +d}}{35 c^{\frac {3}{2}} e^{5} \sqrt {1+\frac {e x}{d}}\, \sqrt {c \,x^{2}+b x}}-\frac {2 d \left (-b e +c d \right ) \left (56 A c e \left (-b e +2 c d \right )-B \left (-b^{2} e^{2}-72 b c d e +128 c^{2} d^{2}\right )\right ) \EllipticF \left (\frac {\sqrt {c}\, \sqrt {x}}{\sqrt {-b}}, \sqrt {\frac {b e}{c d}}\right ) \sqrt {-b}\, \sqrt {x}\, \sqrt {1+\frac {c x}{b}}\, \sqrt {1+\frac {e x}{d}}}{35 c^{\frac {3}{2}} e^{5} \sqrt {e x +d}\, \sqrt {c \,x^{2}+b x}}-\frac {2 \left (7 A c e \left (-7 b e +8 c d \right )-B \left (b^{2} e^{2}-60 b c d e +64 c^{2} d^{2}\right )+3 c e \left (-14 A c e -b B e +16 B c d \right ) x \right ) \sqrt {e x +d}\, \sqrt {c \,x^{2}+b x}}{35 c \,e^{4}} \]
command
integrate((B*x+A)*(c*x^2+b*x)^(3/2)/(e*x+d)^(3/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {2 \, {\left ({\left (128 \, B c^{4} d^{5} + {\left (2 \, B b^{4} - 7 \, A b^{3} c\right )} x e^{5} + {\left (2 \, {\left (5 \, B b^{3} c - 21 \, A b^{2} c^{2}\right )} d x + {\left (2 \, B b^{4} - 7 \, A b^{3} c\right )} d\right )} e^{4} + {\left ({\left (55 \, B b^{2} c^{2} + 168 \, A b c^{3}\right )} d^{2} x + 2 \, {\left (5 \, B b^{3} c - 21 \, A b^{2} c^{2}\right )} d^{2}\right )} e^{3} - {\left (8 \, {\left (25 \, B b c^{3} + 14 \, A c^{4}\right )} d^{3} x - {\left (55 \, B b^{2} c^{2} + 168 \, A b c^{3}\right )} d^{3}\right )} e^{2} + 8 \, {\left (16 \, B c^{4} d^{4} x - {\left (25 \, B b c^{3} + 14 \, A c^{4}\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^{4} d^{4} e + {\left (2 \, B b^{3} c - 7 \, A b^{2} c^{2}\right )} x e^{5} + {\left ({\left (11 \, B b^{2} c^{2} + 112 \, A b c^{3}\right )} d x + {\left (2 \, B b^{3} c - 7 \, A b^{2} c^{2}\right )} d\right )} e^{4} - {\left (8 \, {\left (17 \, B b c^{3} + 14 \, A c^{4}\right )} d^{2} x - {\left (11 \, B b^{2} c^{2} + 112 \, A b c^{3}\right )} d^{2}\right )} e^{3} + 8 \, {\left (16 \, B c^{4} d^{3} x - {\left (17 \, B b c^{3} + 14 \, A c^{4}\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^{4} d^{3} e^{2} + {\left (5 \, B c^{4} x^{3} + {\left (8 \, B b c^{3} + 7 \, A c^{4}\right )} x^{2} + {\left (B b^{2} c^{2} + 14 \, A b c^{3}\right )} x\right )} e^{5} - {\left (8 \, B c^{4} d x^{2} + {\left (17 \, B b c^{3} + 14 \, A c^{4}\right )} d x - {\left (B b^{2} c^{2} + 49 \, A b c^{3}\right )} d\right )} e^{4} + 4 \, {\left (4 \, B c^{4} d^{2} x - {\left (15 \, B b c^{3} + 14 \, A c^{4}\right )} d^{2}\right )} e^{3}\right )} \sqrt {c x^{2} + b x} \sqrt {x e + d}\right )}}{105 \, {\left (c^{3} x e^{7} + c^{3} d e^{6}\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (B c x^{3} + A b x + {\left (B b + A c\right )} x^{2}\right )} \sqrt {c x^{2} + b x} \sqrt {e x + d}}{e^{2} x^{2} + 2 \, d e x + d^{2}}, x\right ) \]