\[ \int \frac {1}{(d+e x)^{3/2} \left (b x+c x^2\right )^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (b \left (-b e +c d \right )+c \left (-b e +2 c d \right ) x \right )}{b^{2} d \left (-b e +c d \right ) \sqrt {e x +d}\, \sqrt {c \,x^{2}+b x}}+\frac {4 \left (b^{2} e^{2}-b c d e +c^{2} d^{2}\right ) \EllipticE \left (\frac {\sqrt {c}\, \sqrt {x}}{\sqrt {-b}}, \sqrt {\frac {b e}{c d}}\right ) \sqrt {c}\, \sqrt {x}\, \sqrt {1+\frac {c x}{b}}\, \sqrt {e x +d}}{\left (-b \right )^{\frac {3}{2}} d^{2} \left (-b e +c d \right )^{2} \sqrt {1+\frac {e x}{d}}\, \sqrt {c \,x^{2}+b x}}-\frac {2 \left (-b e +2 c d \right ) \EllipticF \left (\frac {\sqrt {c}\, \sqrt {x}}{\sqrt {-b}}, \sqrt {\frac {b e}{c d}}\right ) \sqrt {c}\, \sqrt {x}\, \sqrt {1+\frac {c x}{b}}\, \sqrt {1+\frac {e x}{d}}}{\left (-b \right )^{\frac {3}{2}} d \left (-b e +c d \right ) \sqrt {e x +d}\, \sqrt {c \,x^{2}+b x}}-\frac {4 e \left (b^{2} e^{2}-b c d e +c^{2} d^{2}\right ) \sqrt {c \,x^{2}+b x}}{b^{2} d^{2} \left (-b e +c d \right )^{2} \sqrt {e x +d}} \]
command
integrate(1/(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 (2 \, c^{4} d^{4} x^{2} + 2 \, b c^{3} d^{4} x + 2 \, {\left (b^{3} c x^{3} + b^{4} x^{2}\right )} e^{4} - {\left (3 \, b^{2} c^{2} d x^{3} + b^{3} c d x^{2} - 2 \, b^{4} d x\right )} e^{3} - 3 \, {\left (b c^{3} d^{2} x^{3} + 2 \, b^{2} c^{2} d^{2} x^{2} + b^{3} c d^{2} x\right )} e^{2} + {\left (2 \, c^{4} d^{3} x^{3} - b c^{3} d^{3} x^{2} - 3 \, b^{2} c^{2} d^{3} 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 ) + 6 \, {\left ({\left (b^{2} c^{2} x^{3} + b^{3} c x^{2}\right )} e^{4} - {\left (b c^{3} d x^{3} - b^{3} c d x\right )} e^{3} + {\left (c^{4} d^{2} x^{3} - b^{2} c^{2} d^{2} x\right )} e^{2} + {\left (c^{4} d^{3} x^{2} + b c^{3} d^{3} 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 (2 \, {\left (b^{2} c^{2} x^{2} + b^{3} c x\right )} e^{4} - {\left (2 \, b c^{3} d x^{2} + b^{2} c^{2} d x - b^{3} c d\right )} e^{3} + {\left (2 \, c^{4} d^{2} x^{2} - b c^{3} d^{2} x - 2 \, b^{2} c^{2} d^{2}\right )} e^{2} + {\left (2 \, c^{4} d^{3} x + b c^{3} d^{3}\right )} e\right )} \sqrt {x e + d}\right )}}{3 \, {\left ({\left (b^{4} c^{2} d^{2} x^{3} + b^{5} c d^{2} x^{2}\right )} e^{4} - {\left (2 \, b^{3} c^{3} d^{3} x^{3} + b^{4} c^{2} d^{3} x^{2} - b^{5} c d^{3} x\right )} e^{3} + {\left (b^{2} c^{4} d^{4} x^{3} - b^{3} c^{3} d^{4} x^{2} - 2 \, b^{4} c^{2} d^{4} x\right )} e^{2} + {\left (b^{2} c^{4} d^{5} x^{2} + b^{3} c^{3} d^{5} x\right )} e\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {c x^{2} + b x} \sqrt {e x + d}}{c^{2} e^{2} x^{6} + b^{2} d^{2} x^{2} + 2 \, {\left (c^{2} d e + b c e^{2}\right )} x^{5} + {\left (c^{2} d^{2} + 4 \, b c d e + b^{2} e^{2}\right )} x^{4} + 2 \, {\left (b c d^{2} + b^{2} d e\right )} x^{3}}, x\right ) \]