\[ \int \frac {\left (b x+c x^2\right )^{5/2}}{(d+e x)^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (c \,x^{2}+b x \right )^{\frac {5}{2}}}{e \sqrt {e x +d}}-\frac {10 \left (-14 c e x -15 b e +16 c d \right ) \left (c \,x^{2}+b x \right )^{\frac {3}{2}} \sqrt {e x +d}}{63 e^{3}}+\frac {4 \left (-b^{4} e^{4}-7 b^{3} c d \,e^{3}+135 b^{2} c^{2} d^{2} e^{2}-256 b \,c^{3} d^{3} e +128 c^{4} d^{4}\right ) \EllipticE \left (\frac {\sqrt {c}\, \sqrt {x}}{\sqrt {-b}}, \sqrt {\frac {b e}{c d}}\right ) \sqrt {-b}\, \sqrt {x}\, \sqrt {1+\frac {c x}{b}}\, \sqrt {e x +d}}{63 c^{\frac {3}{2}} e^{6} \sqrt {1+\frac {e x}{d}}\, \sqrt {c \,x^{2}+b x}}-\frac {2 d \left (-b e +c d \right ) \left (-b e +2 c d \right ) \left (-b^{2} e^{2}-128 b c d e +128 c^{2} d^{2}\right ) \EllipticF \left (\frac {\sqrt {c}\, \sqrt {x}}{\sqrt {-b}}, \sqrt {\frac {b e}{c d}}\right ) \sqrt {-b}\, \sqrt {x}\, \sqrt {1+\frac {c x}{b}}\, \sqrt {1+\frac {e x}{d}}}{63 c^{\frac {3}{2}} e^{6} \sqrt {e x +d}\, \sqrt {c \,x^{2}+b x}}-\frac {2 \left (128 c^{3} d^{3}-240 b \,c^{2} d^{2} e +111 b^{2} c d \,e^{2}-b^{3} e^{3}-3 c e \left (b^{2} e^{2}-32 b c d e +32 c^{2} d^{2}\right ) x \right ) \sqrt {e x +d}\, \sqrt {c \,x^{2}+b x}}{63 c \,e^{5}} \]
command
integrate((c*x^2+b*x)^(5/2)/(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 (256 \, c^{5} d^{6} - 2 \, b^{5} x e^{6} - {\left (13 \, b^{4} c d x + 2 \, b^{5} d\right )} e^{5} - {\left (77 \, b^{3} c^{2} d^{2} x + 13 \, b^{4} c d^{2}\right )} e^{4} + {\left (478 \, b^{2} c^{3} d^{3} x - 77 \, b^{3} c^{2} d^{3}\right )} e^{3} - 2 \, {\left (320 \, b c^{4} d^{4} x - 239 \, b^{2} c^{3} d^{4}\right )} e^{2} + 128 \, {\left (2 \, c^{5} d^{5} x - 5 \, b c^{4} 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 ) + 6 \, {\left (128 \, c^{5} d^{5} e - b^{4} c x e^{6} - {\left (7 \, b^{3} c^{2} d x + b^{4} c d\right )} e^{5} + {\left (135 \, b^{2} c^{3} d^{2} x - 7 \, b^{3} c^{2} d^{2}\right )} e^{4} - {\left (256 \, b c^{4} d^{3} x - 135 \, b^{2} c^{3} d^{3}\right )} e^{3} + 128 \, {\left (c^{5} d^{4} x - 2 \, b c^{4} 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^{5} d^{4} e^{2} - {\left (7 \, c^{5} x^{4} + 19 \, b c^{4} x^{3} + 15 \, b^{2} c^{3} x^{2} + b^{3} c^{2} x\right )} e^{6} + {\left (10 \, c^{5} d x^{3} + 31 \, b c^{4} d x^{2} + 33 \, b^{2} c^{3} d x - b^{3} c^{2} d\right )} e^{5} - {\left (16 \, c^{5} d^{2} x^{2} + 64 \, b c^{4} d^{2} x - 111 \, b^{2} c^{3} d^{2}\right )} e^{4} + 16 \, {\left (2 \, c^{5} d^{3} x - 15 \, b c^{4} d^{3}\right )} e^{3}\right )} \sqrt {c x^{2} + b x} \sqrt {x e + d}\right )}}{189 \, {\left (c^{3} x e^{8} + c^{3} d e^{7}\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (c^{2} x^{4} + 2 \, b c x^{3} + b^{2} x^{2}\right )} \sqrt {c x^{2} + b x} \sqrt {e x + d}}{e^{2} x^{2} + 2 \, d e x + d^{2}}, x\right ) \]