\[ \int \frac {\left (b x+c x^2\right )^{5/2}}{\sqrt {d+e x}} \, dx \]
Optimal antiderivative \[ \frac {10 \left (16 c^{2} d^{2}-23 b c d e +3 b^{2} e^{2}-7 c e \left (-b e +2 c d \right ) x \right ) \left (c \,x^{2}+b x \right )^{\frac {3}{2}} \sqrt {e x +d}}{693 c \,e^{3}}+\frac {2 \left (c \,x^{2}+b x \right )^{\frac {5}{2}} \sqrt {e x +d}}{11 e}-\frac {2 \left (-b e +2 c d \right ) \left (8 b^{4} e^{4}+29 b^{3} c d \,e^{3}+99 b^{2} c^{2} d^{2} e^{2}-256 b \,c^{3} d^{3} e +128 c^{4} d^{4}\right ) \EllipticE \left (\frac {\sqrt {c}\, \sqrt {x}}{\sqrt {-b}}, \sqrt {\frac {b e}{c d}}\right ) \sqrt {-b}\, \sqrt {x}\, \sqrt {1+\frac {c x}{b}}\, \sqrt {e x +d}}{693 c^{\frac {5}{2}} e^{6} \sqrt {1+\frac {e x}{d}}\, \sqrt {c \,x^{2}+b x}}+\frac {4 d \left (-b e +c d \right ) \left (2 b^{4} e^{4}+5 b^{3} c d \,e^{3}+123 b^{2} c^{2} d^{2} e^{2}-256 b \,c^{3} d^{3} e +128 c^{4} d^{4}\right ) \EllipticF \left (\frac {\sqrt {c}\, \sqrt {x}}{\sqrt {-b}}, \sqrt {\frac {b e}{c d}}\right ) \sqrt {-b}\, \sqrt {x}\, \sqrt {1+\frac {c x}{b}}\, \sqrt {1+\frac {e x}{d}}}{693 c^{\frac {5}{2}} e^{6} \sqrt {e x +d}\, \sqrt {c \,x^{2}+b x}}+\frac {2 \left (128 c^{4} d^{4}-304 b \,c^{3} d^{3} e +195 b^{2} c^{2} d^{2} e^{2}-7 b^{3} c d \,e^{3}-4 b^{4} e^{4}-12 c e \left (-b e +2 c d \right ) \left (-b^{2} e^{2}-4 b c d e +4 c^{2} d^{2}\right ) x \right ) \sqrt {e x +d}\, \sqrt {c \,x^{2}+b x}}{693 c^{2} e^{5}} \]
command
integrate((c*x^2+b*x)^(5/2)/(e*x+d)^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {2 \, {\left ({\left (256 \, c^{6} d^{6} - 768 \, b c^{5} d^{5} e + 726 \, b^{2} c^{4} d^{4} e^{2} - 172 \, b^{3} c^{3} d^{3} e^{3} - 33 \, b^{4} c^{2} d^{2} e^{4} - 9 \, b^{5} c d e^{5} - 8 \, b^{6} e^{6}\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^{6} d^{5} e - 640 \, b c^{5} d^{4} e^{2} + 454 \, b^{2} c^{4} d^{3} e^{3} - 41 \, b^{3} c^{3} d^{2} e^{4} - 13 \, b^{4} c^{2} d e^{5} - 8 \, b^{5} c e^{6}\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^{6} d^{4} e^{2} + {\left (63 \, c^{6} x^{4} + 161 \, b c^{5} x^{3} + 113 \, b^{2} c^{4} x^{2} + 3 \, b^{3} c^{3} x - 4 \, b^{4} c^{2}\right )} e^{6} - {\left (70 \, c^{6} d x^{3} + 185 \, b c^{5} d x^{2} + 139 \, b^{2} c^{4} d x + 7 \, b^{3} c^{3} d\right )} e^{5} + {\left (80 \, c^{6} d^{2} x^{2} + 224 \, b c^{5} d^{2} x + 195 \, b^{2} c^{4} d^{2}\right )} e^{4} - 16 \, {\left (6 \, c^{6} d^{3} x + 19 \, b c^{5} d^{3}\right )} e^{3}\right )} \sqrt {c x^{2} + b x} \sqrt {x e + d}\right )} e^{\left (-7\right )}}{2079 \, c^{4}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (c^{2} x^{4} + 2 \, b c x^{3} + b^{2} x^{2}\right )} \sqrt {c x^{2} + b x}}{\sqrt {e x + d}}, x\right ) \]