\[ \int \frac {(A+B x) \sqrt {b x+c x^2}}{(d+e x)^{7/2}} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (2 A e \left (b^{2} e^{2}-b c d e +c^{2} d^{2}\right )+B d \left (3 b^{2} e^{2}-13 b c d e +8 c^{2} d^{2}\right )\right ) \EllipticE \left (\frac {\sqrt {c}\, \sqrt {x}}{\sqrt {-b}}, \sqrt {\frac {b e}{c d}}\right ) \sqrt {-b}\, \sqrt {c}\, \sqrt {x}\, \sqrt {1+\frac {c x}{b}}\, \sqrt {e x +d}}{15 d^{2} e^{3} \left (-b e +c d \right )^{2} \sqrt {1+\frac {e x}{d}}\, \sqrt {c \,x^{2}+b x}}+\frac {2 \left (B d \left (-9 b e +8 c d \right )+A e \left (-b e +2 c d \right )\right ) \EllipticF \left (\frac {\sqrt {c}\, \sqrt {x}}{\sqrt {-b}}, \sqrt {\frac {b e}{c d}}\right ) \sqrt {-b}\, \sqrt {c}\, \sqrt {x}\, \sqrt {1+\frac {c x}{b}}\, \sqrt {1+\frac {e x}{d}}}{15 d \,e^{3} \left (-b e +c d \right ) \sqrt {e x +d}\, \sqrt {c \,x^{2}+b x}}-\frac {2 \left (d \left (B d \left (-3 b e +4 c d \right )+A e \left (-2 b e +c d \right )\right )+e \left (B d \left (-6 b e +7 c d \right )-A e \left (-b e +2 c d \right )\right ) x \right ) \sqrt {c \,x^{2}+b x}}{15 d \,e^{2} \left (-b e +c d \right ) \left (e x +d \right )^{\frac {5}{2}}}+\frac {2 \left (2 A e \left (b^{2} e^{2}-b c d e +c^{2} d^{2}\right )+B d \left (3 b^{2} e^{2}-13 b c d e +8 c^{2} d^{2}\right )\right ) \sqrt {c \,x^{2}+b x}}{15 d^{2} e^{2} \left (-b e +c d \right )^{2} \sqrt {e x +d}} \]
command
integrate((B*x+A)*(c*x^2+b*x)^(1/2)/(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 (8 \, B c^{3} d^{7} + 2 \, A b^{3} x^{3} e^{7} + 3 \, {\left (2 \, A b^{3} d x^{2} + {\left (B b^{3} - A b^{2} c\right )} d x^{3}\right )} e^{6} + {\left (6 \, A b^{3} d^{2} x + {\left (8 \, B b^{2} c - 3 \, A b c^{2}\right )} d^{2} x^{3} + 9 \, {\left (B b^{3} - A b^{2} c\right )} d^{2} x^{2}\right )} e^{5} + {\left (2 \, A b^{3} d^{3} - {\left (17 \, B b c^{2} - 2 \, A c^{3}\right )} d^{3} x^{3} + 3 \, {\left (8 \, B b^{2} c - 3 \, A b c^{2}\right )} d^{3} x^{2} + 9 \, {\left (B b^{3} - A b^{2} c\right )} d^{3} x\right )} e^{4} + {\left (8 \, B c^{3} d^{4} x^{3} - 3 \, {\left (17 \, B b c^{2} - 2 \, A c^{3}\right )} d^{4} x^{2} + 3 \, {\left (8 \, B b^{2} c - 3 \, A b c^{2}\right )} d^{4} x + 3 \, {\left (B b^{3} - A b^{2} c\right )} d^{4}\right )} e^{3} + {\left (24 \, B c^{3} d^{5} x^{2} - 3 \, {\left (17 \, B b c^{2} - 2 \, A c^{3}\right )} d^{5} x + {\left (8 \, B b^{2} c - 3 \, A b c^{2}\right )} d^{5}\right )} e^{2} + {\left (24 \, B c^{3} d^{6} x - {\left (17 \, B b c^{2} - 2 \, A c^{3}\right )} d^{6}\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 (8 \, B c^{3} d^{6} e + 2 \, A b^{2} c x^{3} e^{7} + {\left (6 \, A b^{2} c d x^{2} + {\left (3 \, B b^{2} c - 2 \, A b c^{2}\right )} d x^{3}\right )} e^{6} + {\left (6 \, A b^{2} c d^{2} x - {\left (13 \, B b c^{2} - 2 \, A c^{3}\right )} d^{2} x^{3} + 3 \, {\left (3 \, B b^{2} c - 2 \, A b c^{2}\right )} d^{2} x^{2}\right )} e^{5} + {\left (8 \, B c^{3} d^{3} x^{3} + 2 \, A b^{2} c d^{3} - 3 \, {\left (13 \, B b c^{2} - 2 \, A c^{3}\right )} d^{3} x^{2} + 3 \, {\left (3 \, B b^{2} c - 2 \, A b c^{2}\right )} d^{3} x\right )} e^{4} + {\left (24 \, B c^{3} d^{4} x^{2} - 3 \, {\left (13 \, B b c^{2} - 2 \, A c^{3}\right )} d^{4} x + {\left (3 \, B b^{2} c - 2 \, A b c^{2}\right )} d^{4}\right )} e^{3} + {\left (24 \, B c^{3} d^{5} x - {\left (13 \, B b c^{2} - 2 \, A c^{3}\right )} d^{5}\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 (4 \, B c^{3} d^{5} e^{2} + 2 \, A b^{2} c x^{2} e^{7} + {\left (5 \, A b^{2} c d x + {\left (3 \, B b^{2} c - 2 \, A b c^{2}\right )} d x^{2}\right )} e^{6} - {\left (7 \, A b c^{2} d^{2} x + {\left (13 \, B b c^{2} - 2 \, A c^{3}\right )} d^{2} x^{2}\right )} e^{5} + {\left (8 \, B c^{3} d^{3} x^{2} + A b c^{2} d^{3} - {\left (13 \, B b c^{2} - 6 \, A c^{3}\right )} d^{3} x\right )} e^{4} + {\left (9 \, B c^{3} d^{4} x - {\left (6 \, B b c^{2} - A c^{3}\right )} d^{4}\right )} e^{3}\right )} \sqrt {c x^{2} + b x} \sqrt {x e + d}\right )}}{45 \, {\left (c^{3} d^{7} e^{4} + b^{2} c d^{2} x^{3} e^{9} - {\left (2 \, b c^{2} d^{3} x^{3} - 3 \, b^{2} c d^{3} x^{2}\right )} e^{8} + {\left (c^{3} d^{4} x^{3} - 6 \, b c^{2} d^{4} x^{2} + 3 \, b^{2} c d^{4} x\right )} e^{7} + {\left (3 \, c^{3} d^{5} x^{2} - 6 \, b c^{2} d^{5} x + b^{2} c d^{5}\right )} e^{6} + {\left (3 \, c^{3} d^{6} x - 2 \, b c^{2} d^{6}\right )} e^{5}\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {c x^{2} + b x} {\left (B x + A\right )} \sqrt {e x + d}}{e^{4} x^{4} + 4 \, d e^{3} x^{3} + 6 \, d^{2} e^{2} x^{2} + 4 \, d^{3} e x + d^{4}}, x\right ) \]