\[ \int \frac {\left (b x+c x^2\right )^{5/2}}{(d+e x)^{7/2}} \, dx \]
Optimal antiderivative \[ \frac {2 \left (6 c e x -5 b e +16 c d \right ) \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{15 e^{3} \left (e x +d \right )^{\frac {3}{2}}}-\frac {2 \left (c \,x^{2}+b x \right )^{\frac {5}{2}}}{5 e \left (e x +d \right )^{\frac {5}{2}}}+\frac {4 \left (23 b^{2} e^{2}-128 b c d e +128 c^{2} d^{2}\right ) \EllipticE \left (\frac {\sqrt {c}\, \sqrt {x}}{\sqrt {-b}}, \sqrt {\frac {b e}{c d}}\right ) \sqrt {-b}\, \sqrt {c}\, \sqrt {x}\, \sqrt {1+\frac {c x}{b}}\, \sqrt {e x +d}}{15 e^{6} \sqrt {1+\frac {e x}{d}}\, \sqrt {c \,x^{2}+b x}}-\frac {2 \left (-b e +2 c d \right ) \left (15 b^{2} e^{2}-128 b c d e +128 c^{2} d^{2}\right ) \EllipticF \left (\frac {\sqrt {c}\, \sqrt {x}}{\sqrt {-b}}, \sqrt {\frac {b e}{c d}}\right ) \sqrt {-b}\, \sqrt {x}\, \sqrt {1+\frac {c x}{b}}\, \sqrt {1+\frac {e x}{d}}}{15 e^{6} \sqrt {c}\, \sqrt {e x +d}\, \sqrt {c \,x^{2}+b x}}-\frac {2 \left (128 c^{2} d^{2}-112 b c d e +15 b^{2} e^{2}+16 c e \left (-b e +2 c d \right ) x \right ) \sqrt {c \,x^{2}+b x}}{15 e^{5} \sqrt {e x +d}} \]
command
integrate((c*x^2+b*x)^(5/2)/(e*x+d)^(7/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {2 \, {\left ({\left (256 \, c^{3} d^{6} + b^{3} x^{3} e^{6} + 3 \, {\left (42 \, b^{2} c d x^{3} + b^{3} d x^{2}\right )} e^{5} - 3 \, {\left (128 \, b c^{2} d^{2} x^{3} - 126 \, b^{2} c d^{2} x^{2} - b^{3} d^{2} x\right )} e^{4} + {\left (256 \, c^{3} d^{3} x^{3} - 1152 \, b c^{2} d^{3} x^{2} + 378 \, b^{2} c d^{3} x + b^{3} d^{3}\right )} e^{3} + 6 \, {\left (128 \, c^{3} d^{4} x^{2} - 192 \, b c^{2} d^{4} x + 21 \, b^{2} c d^{4}\right )} e^{2} + 384 \, {\left (2 \, c^{3} d^{5} x - b c^{2} 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^{3} d^{5} e + 23 \, b^{2} c x^{3} e^{6} - {\left (128 \, b c^{2} d x^{3} - 69 \, b^{2} c d x^{2}\right )} e^{5} + {\left (128 \, c^{3} d^{2} x^{3} - 384 \, b c^{2} d^{2} x^{2} + 69 \, b^{2} c d^{2} x\right )} e^{4} + {\left (384 \, c^{3} d^{3} x^{2} - 384 \, b c^{2} d^{3} x + 23 \, b^{2} c d^{3}\right )} e^{3} + 128 \, {\left (3 \, c^{3} d^{4} x - b c^{2} 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^{3} d^{4} e^{2} - {\left (3 \, c^{3} x^{4} + 11 \, b c^{2} x^{3} - 23 \, b^{2} c x^{2}\right )} e^{6} + {\left (10 \, c^{3} d x^{3} - 161 \, b c^{2} d x^{2} + 35 \, b^{2} c d x\right )} e^{5} + {\left (176 \, c^{3} d^{2} x^{2} - 256 \, b c^{2} d^{2} x + 15 \, b^{2} c d^{2}\right )} e^{4} + 16 \, {\left (18 \, c^{3} d^{3} x - 7 \, b c^{2} d^{3}\right )} e^{3}\right )} \sqrt {c x^{2} + b x} \sqrt {x e + d}\right )}}{45 \, {\left (c x^{3} e^{10} + 3 \, c d x^{2} e^{9} + 3 \, c d^{2} x e^{8} + c d^{3} e^{7}\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (c^{2} x^{4} + 2 \, b c x^{3} + b^{2} x^{2}\right )} \sqrt {c x^{2} + b x} \sqrt {e x + d}}{e^{4} x^{4} + 4 \, d e^{3} x^{3} + 6 \, d^{2} e^{2} x^{2} + 4 \, d^{3} e x + d^{4}}, x\right ) \]