\[ \int \frac {\left (b x+c x^2\right )^{5/2}}{(d+e x)^{11/2}} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (d \left (-2 b^{2} e^{2}-11 b c d e +16 c^{2} d^{2}\right )+e \left (3 b^{2} e^{2}-26 b c d e +26 c^{2} d^{2}\right ) x \right ) \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{63 d \,e^{3} \left (-b e +c d \right ) \left (e x +d \right )^{\frac {7}{2}}}-\frac {2 \left (c \,x^{2}+b x \right )^{\frac {5}{2}}}{9 e \left (e x +d \right )^{\frac {9}{2}}}+\frac {4 \left (-b^{4} e^{4}-7 b^{3} c d \,e^{3}+135 b^{2} c^{2} d^{2} e^{2}-256 b \,c^{3} d^{3} e +128 c^{4} d^{4}\right ) \EllipticE \left (\frac {\sqrt {c}\, \sqrt {x}}{\sqrt {-b}}, \sqrt {\frac {b e}{c d}}\right ) \sqrt {-b}\, \sqrt {c}\, \sqrt {x}\, \sqrt {1+\frac {c x}{b}}\, \sqrt {e x +d}}{63 d^{2} e^{6} \left (-b e +c d \right )^{2} \sqrt {1+\frac {e x}{d}}\, \sqrt {c \,x^{2}+b x}}-\frac {2 \left (-b e +2 c d \right ) \left (-b^{2} e^{2}-128 b c d e +128 c^{2} d^{2}\right ) \EllipticF \left (\frac {\sqrt {c}\, \sqrt {x}}{\sqrt {-b}}, \sqrt {\frac {b e}{c d}}\right ) \sqrt {-b}\, \sqrt {c}\, \sqrt {x}\, \sqrt {1+\frac {c x}{b}}\, \sqrt {1+\frac {e x}{d}}}{63 d \,e^{6} \left (-b e +c d \right ) \sqrt {e x +d}\, \sqrt {c \,x^{2}+b x}}-\frac {2 \left (c \,d^{2} \left (-b^{3} e^{3}+111 b^{2} c d \,e^{2}-240 b \,c^{2} d^{2} e +128 c^{3} d^{3}\right )+e \left (-2 b^{4} e^{4}-11 b^{3} c d \,e^{3}+171 b^{2} c^{2} d^{2} e^{2}-320 b \,c^{3} d^{3} e +160 c^{4} d^{4}\right ) x \right ) \sqrt {c \,x^{2}+b x}}{63 d^{2} e^{5} \left (-b e +c d \right )^{2} \left (e x +d \right )^{\frac {3}{2}}} \]
command
integrate((c*x^2+b*x)^(5/2)/(e*x+d)^(11/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {2 \, {\left ({\left (256 \, c^{5} d^{10} - 2 \, b^{5} x^{5} e^{10} - {\left (13 \, b^{4} c d x^{5} + 10 \, b^{5} d x^{4}\right )} e^{9} - {\left (77 \, b^{3} c^{2} d^{2} x^{5} + 65 \, b^{4} c d^{2} x^{4} + 20 \, b^{5} d^{2} x^{3}\right )} e^{8} + {\left (478 \, b^{2} c^{3} d^{3} x^{5} - 385 \, b^{3} c^{2} d^{3} x^{4} - 130 \, b^{4} c d^{3} x^{3} - 20 \, b^{5} d^{3} x^{2}\right )} e^{7} - 10 \, {\left (64 \, b c^{4} d^{4} x^{5} - 239 \, b^{2} c^{3} d^{4} x^{4} + 77 \, b^{3} c^{2} d^{4} x^{3} + 13 \, b^{4} c d^{4} x^{2} + b^{5} d^{4} x\right )} e^{6} + {\left (256 \, c^{5} d^{5} x^{5} - 3200 \, b c^{4} d^{5} x^{4} + 4780 \, b^{2} c^{3} d^{5} x^{3} - 770 \, b^{3} c^{2} d^{5} x^{2} - 65 \, b^{4} c d^{5} x - 2 \, b^{5} d^{5}\right )} e^{5} + {\left (1280 \, c^{5} d^{6} x^{4} - 6400 \, b c^{4} d^{6} x^{3} + 4780 \, b^{2} c^{3} d^{6} x^{2} - 385 \, b^{3} c^{2} d^{6} x - 13 \, b^{4} c d^{6}\right )} e^{4} + {\left (2560 \, c^{5} d^{7} x^{3} - 6400 \, b c^{4} d^{7} x^{2} + 2390 \, b^{2} c^{3} d^{7} x - 77 \, b^{3} c^{2} d^{7}\right )} e^{3} + 2 \, {\left (1280 \, c^{5} d^{8} x^{2} - 1600 \, b c^{4} d^{8} x + 239 \, b^{2} c^{3} d^{8}\right )} e^{2} + 640 \, {\left (2 \, c^{5} d^{9} x - b c^{4} d^{9}\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 (128 \, c^{5} d^{9} e - b^{4} c x^{5} e^{10} - {\left (7 \, b^{3} c^{2} d x^{5} + 5 \, b^{4} c d x^{4}\right )} e^{9} + 5 \, {\left (27 \, b^{2} c^{3} d^{2} x^{5} - 7 \, b^{3} c^{2} d^{2} x^{4} - 2 \, b^{4} c d^{2} x^{3}\right )} e^{8} - {\left (256 \, b c^{4} d^{3} x^{5} - 675 \, b^{2} c^{3} d^{3} x^{4} + 70 \, b^{3} c^{2} d^{3} x^{3} + 10 \, b^{4} c d^{3} x^{2}\right )} e^{7} + {\left (128 \, c^{5} d^{4} x^{5} - 1280 \, b c^{4} d^{4} x^{4} + 1350 \, b^{2} c^{3} d^{4} x^{3} - 70 \, b^{3} c^{2} d^{4} x^{2} - 5 \, b^{4} c d^{4} x\right )} e^{6} + {\left (640 \, c^{5} d^{5} x^{4} - 2560 \, b c^{4} d^{5} x^{3} + 1350 \, b^{2} c^{3} d^{5} x^{2} - 35 \, b^{3} c^{2} d^{5} x - b^{4} c d^{5}\right )} e^{5} + {\left (1280 \, c^{5} d^{6} x^{3} - 2560 \, b c^{4} d^{6} x^{2} + 675 \, b^{2} c^{3} d^{6} x - 7 \, b^{3} c^{2} d^{6}\right )} e^{4} + 5 \, {\left (256 \, c^{5} d^{7} x^{2} - 256 \, b c^{4} d^{7} x + 27 \, b^{2} c^{3} d^{7}\right )} e^{3} + 128 \, {\left (5 \, c^{5} d^{8} x - 2 \, b c^{4} d^{8}\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^{5} d^{8} e^{2} - 2 \, b^{4} c x^{4} e^{10} - {\left (14 \, b^{3} c^{2} d x^{4} + 9 \, b^{4} c d x^{3}\right )} e^{9} + {\left (207 \, b^{2} c^{3} d^{2} x^{4} + 8 \, b^{3} c^{2} d^{2} x^{3}\right )} e^{8} - 2 \, {\left (193 \, b c^{4} d^{3} x^{4} - 291 \, b^{2} c^{3} d^{3} x^{3} + 5 \, b^{3} c^{2} d^{3} x^{2}\right )} e^{7} + {\left (193 \, c^{5} d^{4} x^{4} - 1239 \, b c^{4} d^{4} x^{3} + 783 \, b^{2} c^{3} d^{4} x^{2} - 5 \, b^{3} c^{2} d^{4} x\right )} e^{6} + {\left (650 \, c^{5} d^{5} x^{3} - 1665 \, b c^{4} d^{5} x^{2} + 477 \, b^{2} c^{3} d^{5} x - b^{3} c^{2} d^{5}\right )} e^{5} + {\left (880 \, c^{5} d^{6} x^{2} - 1024 \, b c^{4} d^{6} x + 111 \, b^{2} c^{3} d^{6}\right )} e^{4} + 16 \, {\left (34 \, c^{5} d^{7} x - 15 \, b c^{4} d^{7}\right )} e^{3}\right )} \sqrt {c x^{2} + b x} \sqrt {x e + d}\right )}}{189 \, {\left (c^{3} d^{9} e^{7} + b^{2} c d^{2} x^{5} e^{14} - {\left (2 \, b c^{2} d^{3} x^{5} - 5 \, b^{2} c d^{3} x^{4}\right )} e^{13} + {\left (c^{3} d^{4} x^{5} - 10 \, b c^{2} d^{4} x^{4} + 10 \, b^{2} c d^{4} x^{3}\right )} e^{12} + 5 \, {\left (c^{3} d^{5} x^{4} - 4 \, b c^{2} d^{5} x^{3} + 2 \, b^{2} c d^{5} x^{2}\right )} e^{11} + 5 \, {\left (2 \, c^{3} d^{6} x^{3} - 4 \, b c^{2} d^{6} x^{2} + b^{2} c d^{6} x\right )} e^{10} + {\left (10 \, c^{3} d^{7} x^{2} - 10 \, b c^{2} d^{7} x + b^{2} c d^{7}\right )} e^{9} + {\left (5 \, c^{3} d^{8} x - 2 \, b c^{2} d^{8}\right )} e^{8}\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (c^{2} x^{4} + 2 \, b c x^{3} + b^{2} x^{2}\right )} \sqrt {c x^{2} + b x} \sqrt {e x + d}}{e^{6} x^{6} + 6 \, d e^{5} x^{5} + 15 \, d^{2} e^{4} x^{4} + 20 \, d^{3} e^{3} x^{3} + 15 \, d^{4} e^{2} x^{2} + 6 \, d^{5} e x + d^{6}}, x\right ) \]