\[ \int \frac {(A+B x) \left (a+c x^2\right )^{3/2}}{(d+e x)^{5/2}} \, dx \]
Optimal antiderivative \[ \frac {2 \left (3 B e x -5 A e +8 B d \right ) \left (c \,x^{2}+a \right )^{\frac {3}{2}}}{15 e^{2} \left (e x +d \right )^{\frac {3}{2}}}-\frac {4 \left (9 a B \,e^{2}+4 c d \left (-5 A e +8 B d \right )+c e \left (-5 A e +8 B d \right ) x \right ) \sqrt {c \,x^{2}+a}}{15 e^{4} \sqrt {e x +d}}-\frac {8 \left (9 a B \,e^{2}+4 c d \left (-5 A e +8 B d \right )\right ) \EllipticE \left (\frac {\sqrt {1-\frac {x \sqrt {c}}{\sqrt {-a}}}\, \sqrt {2}}{2}, \sqrt {-\frac {2 a e}{-a e +d \sqrt {-a}\, \sqrt {c}}}\right ) \sqrt {-a}\, \sqrt {c}\, \sqrt {e x +d}\, \sqrt {1+\frac {c \,x^{2}}{a}}}{15 e^{5} \sqrt {c \,x^{2}+a}\, \sqrt {\frac {\left (e x +d \right ) \sqrt {c}}{e \sqrt {-a}+d \sqrt {c}}}}+\frac {8 \left (-5 a A \,e^{3}-20 A c \,d^{2} e +17 a B d \,e^{2}+32 B c \,d^{3}\right ) \EllipticF \left (\frac {\sqrt {1-\frac {x \sqrt {c}}{\sqrt {-a}}}\, \sqrt {2}}{2}, \sqrt {-\frac {2 a e}{-a e +d \sqrt {-a}\, \sqrt {c}}}\right ) \sqrt {-a}\, \sqrt {c}\, \sqrt {1+\frac {c \,x^{2}}{a}}\, \sqrt {\frac {\left (e x +d \right ) \sqrt {c}}{e \sqrt {-a}+d \sqrt {c}}}}{15 e^{5} \sqrt {e x +d}\, \sqrt {c \,x^{2}+a}} \]
command
integrate((B*x+A)*(c*x^2+a)^(3/2)/(e*x+d)^(5/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {2 \, {\left (4 \, {\left (32 \, B c d^{5} - 15 \, A a x^{2} e^{5} + 3 \, {\left (11 \, B a d x^{2} - 10 \, A a d x\right )} e^{4} - {\left (20 \, A c d^{2} x^{2} - 66 \, B a d^{2} x + 15 \, A a d^{2}\right )} e^{3} + {\left (32 \, B c d^{3} x^{2} - 40 \, A c d^{3} x + 33 \, B a d^{3}\right )} e^{2} + 4 \, {\left (16 \, B c d^{4} x - 5 \, A c d^{4}\right )} e\right )} \sqrt {c} e^{\frac {1}{2}} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )} e^{\left (-2\right )}}{3 \, c}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )} e^{\left (-3\right )}}{27 \, c}, \frac {1}{3} \, {\left (3 \, x e + d\right )} e^{\left (-1\right )}\right ) + 12 \, {\left (32 \, B c d^{4} e + 9 \, B a x^{2} e^{5} - 2 \, {\left (10 \, A c d x^{2} - 9 \, B a d x\right )} e^{4} + {\left (32 \, B c d^{2} x^{2} - 40 \, A c d^{2} x + 9 \, B a d^{2}\right )} e^{3} + 4 \, {\left (16 \, B c d^{3} x - 5 \, A c d^{3}\right )} e^{2}\right )} \sqrt {c} e^{\frac {1}{2}} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )} e^{\left (-2\right )}}{3 \, c}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )} e^{\left (-3\right )}}{27 \, c}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )} e^{\left (-2\right )}}{3 \, c}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )} e^{\left (-3\right )}}{27 \, c}, \frac {1}{3} \, {\left (3 \, x e + d\right )} e^{\left (-1\right )}\right )\right ) + 3 \, {\left (64 \, B c d^{3} e^{2} - {\left (3 \, B c x^{3} + 5 \, A c x^{2} - 15 \, B a x - 5 \, A a\right )} e^{5} + 2 \, {\left (4 \, B c d x^{2} - 25 \, A c d x + 5 \, B a d\right )} e^{4} + 40 \, {\left (2 \, B c d^{2} x - A c d^{2}\right )} e^{3}\right )} \sqrt {c x^{2} + a} \sqrt {x e + d}\right )}}{45 \, {\left (x^{2} e^{8} + 2 \, d x e^{7} + d^{2} e^{6}\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (B c x^{3} + A c x^{2} + B a x + A a\right )} \sqrt {c x^{2} + a} \sqrt {e x + d}}{e^{3} x^{3} + 3 \, d e^{2} x^{2} + 3 \, d^{2} e x + d^{3}}, x\right ) \]