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