\[ \int \frac {A+B x}{(d+e x)^{3/2} \sqrt {b x+c x^2}} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (-A e +B d \right ) \EllipticE \left (\frac {\sqrt {c}\, \sqrt {x}}{\sqrt {-b}}, \sqrt {\frac {b e}{c d}}\right ) \sqrt {-b}\, \sqrt {c}\, \sqrt {x}\, \sqrt {1+\frac {c x}{b}}\, \sqrt {e x +d}}{d e \left (-b e +c d \right ) \sqrt {1+\frac {e x}{d}}\, \sqrt {c \,x^{2}+b x}}+\frac {2 B \EllipticF \left (\frac {\sqrt {c}\, \sqrt {x}}{\sqrt {-b}}, \sqrt {\frac {b e}{c d}}\right ) \sqrt {-b}\, \sqrt {x}\, \sqrt {1+\frac {c x}{b}}\, \sqrt {1+\frac {e x}{d}}}{e \sqrt {c}\, \sqrt {e x +d}\, \sqrt {c \,x^{2}+b x}}+\frac {2 \left (-A e +B d \right ) \sqrt {c \,x^{2}+b x}}{d \left (-b e +c d \right ) \sqrt {e x +d}} \]
command
integrate((B*x+A)/(e*x+d)^(3/2)/(c*x^2+b*x)^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {2 \, {\left ({\left (B c d^{3} - A b x e^{3} - {\left (A b d + 2 \, {\left (B b - A c\right )} d x\right )} e^{2} + {\left (B c d^{2} x - 2 \, {\left (B b - A c\right )} d^{2}\right )} e\right )} \sqrt {c} e^{\frac {1}{2}} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )} e^{\left (-2\right )}}{3 \, c^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )} e^{\left (-3\right )}}{27 \, c^{3}}, \frac {{\left (c d + {\left (3 \, c x + b\right )} e\right )} e^{\left (-1\right )}}{3 \, c}\right ) + 3 \, {\left (B c d^{2} e - A c x e^{3} + {\left (B c d x - A c d\right )} e^{2}\right )} \sqrt {c} e^{\frac {1}{2}} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )} e^{\left (-2\right )}}{3 \, c^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )} e^{\left (-3\right )}}{27 \, c^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )} e^{\left (-2\right )}}{3 \, c^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )} e^{\left (-3\right )}}{27 \, c^{3}}, \frac {{\left (c d + {\left (3 \, c x + b\right )} e\right )} e^{\left (-1\right )}}{3 \, c}\right )\right ) + 3 \, {\left (B c d e^{2} - A c e^{3}\right )} \sqrt {c x^{2} + b x} \sqrt {x e + d}\right )}}{3 \, {\left (c^{2} d^{3} e^{2} - b c d x e^{4} + {\left (c^{2} d^{2} x - b c d^{2}\right )} e^{3}\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {c x^{2} + b x} {\left (B x + A\right )} \sqrt {e x + d}}{c e^{2} x^{4} + b d^{2} x + {\left (2 \, c d e + b e^{2}\right )} x^{3} + {\left (c d^{2} + 2 \, b d e\right )} x^{2}}, x\right ) \]