\[ \int \frac {(A+B x) (d+e x)^{3/2}}{\left (a+c x^2\right )^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {\left (a \left (A e +B d \right )-\left (A c d -a B e \right ) x \right ) \sqrt {e x +d}}{a c \sqrt {c \,x^{2}+a}}-\frac {\left (A c d -3 a B e \right ) \EllipticE \left (\frac {\sqrt {1-\frac {x \sqrt {c}}{\sqrt {-a}}}\, \sqrt {2}}{2}, \sqrt {-\frac {2 a e}{-a e +d \sqrt {-a}\, \sqrt {c}}}\right ) \sqrt {e x +d}\, \sqrt {1+\frac {c \,x^{2}}{a}}}{c^{\frac {3}{2}} \sqrt {-a}\, \sqrt {c \,x^{2}+a}\, \sqrt {\frac {\left (e x +d \right ) \sqrt {c}}{e \sqrt {-a}+d \sqrt {c}}}}+\frac {A \left (a \,e^{2}+c \,d^{2}\right ) \EllipticF \left (\frac {\sqrt {1-\frac {x \sqrt {c}}{\sqrt {-a}}}\, \sqrt {2}}{2}, \sqrt {-\frac {2 a e}{-a e +d \sqrt {-a}\, \sqrt {c}}}\right ) \sqrt {1+\frac {c \,x^{2}}{a}}\, \sqrt {\frac {\left (e x +d \right ) \sqrt {c}}{e \sqrt {-a}+d \sqrt {c}}}}{c^{\frac {3}{2}} \sqrt {-a}\, \sqrt {e x +d}\, \sqrt {c \,x^{2}+a}} \]
command
integrate((B*x+A)*(e*x+d)^(3/2)/(c*x^2+a)^(3/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {{\left ({\left (A c^{2} d^{2} x^{2} + A a c d^{2} + 3 \, {\left (A a c x^{2} + A a^{2}\right )} e^{2} + 6 \, {\left (B a c d x^{2} + B a^{2} d\right )} e\right )} \sqrt {c} e^{\frac {1}{2}} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )} e^{\left (-2\right )}}{3 \, c}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )} e^{\left (-3\right )}}{27 \, c}, \frac {1}{3} \, {\left (3 \, x e + d\right )} e^{\left (-1\right )}\right ) - 3 \, {\left (3 \, {\left (B a c x^{2} + B a^{2}\right )} e^{2} - {\left (A c^{2} d x^{2} + A a c d\right )} e\right )} \sqrt {c} e^{\frac {1}{2}} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )} e^{\left (-2\right )}}{3 \, c}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )} e^{\left (-3\right )}}{27 \, c}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )} e^{\left (-2\right )}}{3 \, c}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )} e^{\left (-3\right )}}{27 \, c}, \frac {1}{3} \, {\left (3 \, x e + d\right )} e^{\left (-1\right )}\right )\right ) - 3 \, \sqrt {c x^{2} + a} {\left ({\left (B a c x + A a c\right )} e^{2} - {\left (A c^{2} d x - B a c d\right )} e\right )} \sqrt {x e + d}\right )} e^{\left (-1\right )}}{3 \, {\left (a c^{3} x^{2} + a^{2} c^{2}\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (B e x^{2} + A d + {\left (B d + A e\right )} x\right )} \sqrt {c x^{2} + a} \sqrt {e x + d}}{c^{2} x^{4} + 2 \, a c x^{2} + a^{2}}, x\right ) \]