\[ \int \frac {(d+e x)^{9/2}}{\left (b x+c x^2\right )^{5/2}} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (e x +d \right )^{\frac {7}{2}} \left (b d +\left (-b e +2 c d \right ) x \right )}{3 b^{2} \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}+\frac {2 \left (e x +d \right )^{\frac {3}{2}} \left (b c \,d^{2} \left (-11 b e +8 c d \right )+\left (-b e +2 c d \right ) \left (-3 b^{2} e^{2}-8 b c d e +8 c^{2} d^{2}\right ) x \right )}{3 b^{4} c \sqrt {c \,x^{2}+b x}}-\frac {2 \left (-8 b^{4} e^{4}+7 b^{3} c d \,e^{3}+9 b^{2} c^{2} d^{2} e^{2}-32 b \,c^{3} d^{3} e +16 c^{4} d^{4}\right ) \EllipticE \left (\frac {\sqrt {c}\, \sqrt {x}}{\sqrt {-b}}, \sqrt {\frac {b e}{c d}}\right ) \sqrt {x}\, \sqrt {1+\frac {c x}{b}}\, \sqrt {e x +d}}{3 \left (-b \right )^{\frac {7}{2}} c^{\frac {5}{2}} \sqrt {1+\frac {e x}{d}}\, \sqrt {c \,x^{2}+b x}}+\frac {8 d \left (-b e +c d \right ) \left (-b e +2 c d \right ) \left (-b^{2} e^{2}-2 b c d e +2 c^{2} d^{2}\right ) \EllipticF \left (\frac {\sqrt {c}\, \sqrt {x}}{\sqrt {-b}}, \sqrt {\frac {b e}{c d}}\right ) \sqrt {x}\, \sqrt {1+\frac {c x}{b}}\, \sqrt {1+\frac {e x}{d}}}{3 \left (-b \right )^{\frac {7}{2}} c^{\frac {5}{2}} \sqrt {e x +d}\, \sqrt {c \,x^{2}+b x}}-\frac {8 e \left (b^{3} e^{3}-6 b \,c^{2} d^{2} e +4 c^{3} d^{3}\right ) \sqrt {e x +d}\, \sqrt {c \,x^{2}+b x}}{3 b^{4} c^{2}} \]
command
integrate((e*x+d)^(9/2)/(c*x^2+b*x)^(5/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {2 \, {\left ({\left (16 \, c^{7} d^{5} x^{4} + 32 \, b c^{6} d^{5} x^{3} + 16 \, b^{2} c^{5} d^{5} x^{2} - 8 \, {\left (b^{5} c^{2} x^{4} + 2 \, b^{6} c x^{3} + b^{7} x^{2}\right )} e^{5} + 11 \, {\left (b^{4} c^{3} d x^{4} + 2 \, b^{5} c^{2} d x^{3} + b^{6} c d x^{2}\right )} e^{4} + 7 \, {\left (b^{3} c^{4} d^{2} x^{4} + 2 \, b^{4} c^{3} d^{2} x^{3} + b^{5} c^{2} d^{2} x^{2}\right )} e^{3} + 22 \, {\left (b^{2} c^{5} d^{3} x^{4} + 2 \, b^{3} c^{4} d^{3} x^{3} + b^{4} c^{3} d^{3} x^{2}\right )} e^{2} - 40 \, {\left (b c^{6} d^{4} x^{4} + 2 \, b^{2} c^{5} d^{4} x^{3} + b^{3} c^{4} d^{4} x^{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 (8 \, {\left (b^{4} c^{3} x^{4} + 2 \, b^{5} c^{2} x^{3} + b^{6} c x^{2}\right )} e^{5} - 7 \, {\left (b^{3} c^{4} d x^{4} + 2 \, b^{4} c^{3} d x^{3} + b^{5} c^{2} d x^{2}\right )} e^{4} - 9 \, {\left (b^{2} c^{5} d^{2} x^{4} + 2 \, b^{3} c^{4} d^{2} x^{3} + b^{4} c^{3} d^{2} x^{2}\right )} e^{3} + 32 \, {\left (b c^{6} d^{3} x^{4} + 2 \, b^{2} c^{5} d^{3} x^{3} + b^{3} c^{4} d^{3} x^{2}\right )} e^{2} - 16 \, {\left (c^{7} d^{4} x^{4} + 2 \, b c^{6} d^{4} x^{3} + b^{2} c^{5} d^{4} x^{2}\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 (5 \, b^{4} c^{3} x^{3} + 4 \, b^{5} c^{2} x^{2}\right )} e^{5} - {\left (7 \, b^{3} c^{4} d x^{3} + 3 \, b^{4} c^{3} d x^{2}\right )} e^{4} - 3 \, {\left (3 \, b^{2} c^{5} d^{2} x^{3} + 5 \, b^{3} c^{4} d^{2} x^{2}\right )} e^{3} + {\left (32 \, b c^{6} d^{3} x^{3} + 49 \, b^{2} c^{5} d^{3} x^{2} + 13 \, b^{3} c^{4} d^{3} x\right )} e^{2} - {\left (16 \, c^{7} d^{4} x^{3} + 24 \, b c^{6} d^{4} x^{2} + 6 \, b^{2} c^{5} d^{4} x - b^{3} c^{4} d^{4}\right )} e\right )} \sqrt {x e + d}\right )} e^{\left (-1\right )}}{9 \, {\left (b^{4} c^{6} x^{4} + 2 \, b^{5} c^{5} x^{3} + b^{6} c^{4} x^{2}\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (e^{4} x^{4} + 4 \, d e^{3} x^{3} + 6 \, d^{2} e^{2} x^{2} + 4 \, d^{3} e x + d^{4}\right )} \sqrt {c x^{2} + b x} \sqrt {e x + d}}{c^{3} x^{6} + 3 \, b c^{2} x^{5} + 3 \, b^{2} c x^{4} + b^{3} x^{3}}, x\right ) \]