\[ \int \frac {\left (b x+c x^2\right )^{3/2}}{(d+e x)^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{e \sqrt {e x +d}}+\frac {2 \left (b^{2} e^{2}-16 b c d e +16 c^{2} d^{2}\right ) \EllipticE \left (\frac {\sqrt {c}\, \sqrt {x}}{\sqrt {-b}}, \sqrt {\frac {b e}{c d}}\right ) \sqrt {-b}\, \sqrt {x}\, \sqrt {1+\frac {c x}{b}}\, \sqrt {e x +d}}{5 e^{4} \sqrt {c}\, \sqrt {1+\frac {e x}{d}}\, \sqrt {c \,x^{2}+b x}}-\frac {16 d \left (-b e +c d \right ) \left (-b e +2 c d \right ) \EllipticF \left (\frac {\sqrt {c}\, \sqrt {x}}{\sqrt {-b}}, \sqrt {\frac {b e}{c d}}\right ) \sqrt {-b}\, \sqrt {x}\, \sqrt {1+\frac {c x}{b}}\, \sqrt {1+\frac {e x}{d}}}{5 e^{4} \sqrt {c}\, \sqrt {e x +d}\, \sqrt {c \,x^{2}+b x}}-\frac {2 \left (-6 c e x -7 b e +8 c d \right ) \sqrt {e x +d}\, \sqrt {c \,x^{2}+b x}}{5 e^{3}} \]
command
integrate((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
\[ -\frac {2 \, {\left ({\left (16 \, c^{3} d^{4} + b^{3} x e^{4} + {\left (6 \, b^{2} c d x + b^{3} d\right )} e^{3} - 6 \, {\left (4 \, b c^{2} d^{2} x - b^{2} c d^{2}\right )} e^{2} + 8 \, {\left (2 \, c^{3} d^{3} x - 3 \, b c^{2} d^{3}\right )} e\right )} \sqrt {c} e^{\frac {1}{2}} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )} e^{\left (-2\right )}}{3 \, c^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )} e^{\left (-3\right )}}{27 \, c^{3}}, \frac {{\left (c d + {\left (3 \, c x + b\right )} e\right )} e^{\left (-1\right )}}{3 \, c}\right ) + 3 \, {\left (16 \, c^{3} d^{3} e + b^{2} c x e^{4} - {\left (16 \, b c^{2} d x - b^{2} c d\right )} e^{3} + 16 \, {\left (c^{3} d^{2} x - b c^{2} d^{2}\right )} e^{2}\right )} \sqrt {c} e^{\frac {1}{2}} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )} e^{\left (-2\right )}}{3 \, c^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )} e^{\left (-3\right )}}{27 \, c^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )} e^{\left (-2\right )}}{3 \, c^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )} e^{\left (-3\right )}}{27 \, c^{3}}, \frac {{\left (c d + {\left (3 \, c x + b\right )} e\right )} e^{\left (-1\right )}}{3 \, c}\right )\right ) + 3 \, {\left (8 \, c^{3} d^{2} e^{2} - {\left (c^{3} x^{2} + 2 \, b c^{2} x\right )} e^{4} + {\left (2 \, c^{3} d x - 7 \, b c^{2} d\right )} e^{3}\right )} \sqrt {c x^{2} + b x} \sqrt {x e + d}\right )}}{15 \, {\left (c^{2} x e^{6} + c^{2} d e^{5}\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (c x^{2} + b x\right )}^{\frac {3}{2}} \sqrt {e x + d}}{e^{2} x^{2} + 2 \, d e x + d^{2}}, x\right ) \]