\[ \int \frac {\left (b x+c x^2\right )^{5/2}}{(d+e x)^{5/2}} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (c \,x^{2}+b x \right )^{\frac {5}{2}}}{3 e \left (e x +d \right )^{\frac {3}{2}}}+\frac {10 \left (2 c e x -7 b e +16 c d \right ) \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{21 e^{3} \sqrt {e x +d}}-\frac {2 \left (-b e +2 c d \right ) \left (3 b^{2} e^{2}-128 b c d e +128 c^{2} d^{2}\right ) \EllipticE \left (\frac {\sqrt {c}\, \sqrt {x}}{\sqrt {-b}}, \sqrt {\frac {b e}{c d}}\right ) \sqrt {-b}\, \sqrt {x}\, \sqrt {1+\frac {c x}{b}}\, \sqrt {e x +d}}{21 e^{6} \sqrt {c}\, \sqrt {1+\frac {e x}{d}}\, \sqrt {c \,x^{2}+b x}}+\frac {4 d \left (-b e +c d \right ) \left (27 b^{2} e^{2}-128 b c d e +128 c^{2} d^{2}\right ) \EllipticF \left (\frac {\sqrt {c}\, \sqrt {x}}{\sqrt {-b}}, \sqrt {\frac {b e}{c d}}\right ) \sqrt {-b}\, \sqrt {x}\, \sqrt {1+\frac {c x}{b}}\, \sqrt {1+\frac {e x}{d}}}{21 e^{6} \sqrt {c}\, \sqrt {e x +d}\, \sqrt {c \,x^{2}+b x}}+\frac {2 \left (128 c^{2} d^{2}-176 b c d e +51 b^{2} e^{2}-48 c e \left (-b e +2 c d \right ) x \right ) \sqrt {e x +d}\, \sqrt {c \,x^{2}+b x}}{21 e^{5}} \]
command
integrate((c*x^2+b*x)^(5/2)/(e*x+d)^(5/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {2 \, {\left ({\left (256 \, c^{4} d^{6} - 3 \, b^{4} x^{2} e^{6} - 2 \, {\left (11 \, b^{3} c d x^{2} + 3 \, b^{4} d x\right )} e^{5} + {\left (278 \, b^{2} c^{2} d^{2} x^{2} - 44 \, b^{3} c d^{2} x - 3 \, b^{4} d^{2}\right )} e^{4} - 2 \, {\left (256 \, b c^{3} d^{3} x^{2} - 278 \, b^{2} c^{2} d^{3} x + 11 \, b^{3} c d^{3}\right )} e^{3} + 2 \, {\left (128 \, c^{4} d^{4} x^{2} - 512 \, b c^{3} d^{4} x + 139 \, b^{2} c^{2} d^{4}\right )} e^{2} + 512 \, {\left (c^{4} d^{5} x - b c^{3} d^{5}\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 (256 \, c^{4} d^{5} e - 3 \, b^{3} c x^{2} e^{6} + 2 \, {\left (67 \, b^{2} c^{2} d x^{2} - 3 \, b^{3} c d x\right )} e^{5} - {\left (384 \, b c^{3} d^{2} x^{2} - 268 \, b^{2} c^{2} d^{2} x + 3 \, b^{3} c d^{2}\right )} e^{4} + 2 \, {\left (128 \, c^{4} d^{3} x^{2} - 384 \, b c^{3} d^{3} x + 67 \, b^{2} c^{2} d^{3}\right )} e^{3} + 128 \, {\left (4 \, c^{4} d^{4} x - 3 \, b c^{3} d^{4}\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 (128 \, c^{4} d^{4} e^{2} + 3 \, {\left (c^{4} x^{4} + 3 \, b c^{3} x^{3} + 3 \, b^{2} c^{2} x^{2}\right )} e^{6} - {\left (6 \, c^{4} d x^{3} + 25 \, b c^{3} d x^{2} - 67 \, b^{2} c^{2} d x\right )} e^{5} + {\left (16 \, c^{4} d^{2} x^{2} - 224 \, b c^{3} d^{2} x + 51 \, b^{2} c^{2} d^{2}\right )} e^{4} + 16 \, {\left (10 \, c^{4} d^{3} x - 11 \, b c^{3} d^{3}\right )} e^{3}\right )} \sqrt {c x^{2} + b x} \sqrt {x e + d}\right )}}{63 \, {\left (c^{2} x^{2} e^{9} + 2 \, c^{2} d x e^{8} + c^{2} d^{2} e^{7}\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (c^{2} x^{4} + 2 \, b c x^{3} + b^{2} x^{2}\right )} \sqrt {c x^{2} + b x} \sqrt {e x + d}}{e^{3} x^{3} + 3 \, d e^{2} x^{2} + 3 \, d^{2} e x + d^{3}}, x\right ) \]