\[ \int \frac {A+B x}{(d+e x)^{3/2} \sqrt {a+c x^2}} \, dx \]
Optimal antiderivative \[ \frac {2 \left (-A e +B d \right ) \sqrt {c \,x^{2}+a}}{\left (a \,e^{2}+c \,d^{2}\right ) \sqrt {e x +d}}+\frac {2 \left (-A e +B d \right ) \EllipticE \left (\frac {\sqrt {1-\frac {x \sqrt {c}}{\sqrt {-a}}}\, \sqrt {2}}{2}, \sqrt {-\frac {2 a e}{-a e +d \sqrt {-a}\, \sqrt {c}}}\right ) \sqrt {-a}\, \sqrt {c}\, \sqrt {e x +d}\, \sqrt {1+\frac {c \,x^{2}}{a}}}{e \left (a \,e^{2}+c \,d^{2}\right ) \sqrt {c \,x^{2}+a}\, \sqrt {\frac {\left (e x +d \right ) \sqrt {c}}{e \sqrt {-a}+d \sqrt {c}}}}-\frac {2 B \EllipticF \left (\frac {\sqrt {1-\frac {x \sqrt {c}}{\sqrt {-a}}}\, \sqrt {2}}{2}, \sqrt {-\frac {2 a e}{-a e +d \sqrt {-a}\, \sqrt {c}}}\right ) \sqrt {-a}\, \sqrt {1+\frac {c \,x^{2}}{a}}\, \sqrt {\frac {\left (e x +d \right ) \sqrt {c}}{e \sqrt {-a}+d \sqrt {c}}}}{e \sqrt {c}\, \sqrt {e x +d}\, \sqrt {c \,x^{2}+a}} \]
command
integrate((B*x+A)/(e*x+d)^(3/2)/(c*x^2+a)^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {2 \, {\left ({\left (B c d^{3} + 3 \, B a x e^{3} + {\left (2 \, A c d x + 3 \, B a d\right )} e^{2} + {\left (B c d^{2} x + 2 \, A c d^{2}\right )} e\right )} \sqrt {c} e^{\frac {1}{2}} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )} e^{\left (-2\right )}}{3 \, c}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )} e^{\left (-3\right )}}{27 \, c}, \frac {1}{3} \, {\left (3 \, x e + d\right )} e^{\left (-1\right )}\right ) + 3 \, {\left (B c d^{2} e - A c x e^{3} + {\left (B c d x - A c d\right )} e^{2}\right )} \sqrt {c} e^{\frac {1}{2}} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )} e^{\left (-2\right )}}{3 \, c}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )} e^{\left (-3\right )}}{27 \, c}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )} e^{\left (-2\right )}}{3 \, c}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )} e^{\left (-3\right )}}{27 \, c}, \frac {1}{3} \, {\left (3 \, x e + d\right )} e^{\left (-1\right )}\right )\right ) + 3 \, {\left (B c d e^{2} - A c e^{3}\right )} \sqrt {c x^{2} + a} \sqrt {x e + d}\right )}}{3 \, {\left (c^{2} d^{2} x e^{3} + c^{2} d^{3} e^{2} + a c x e^{5} + a c d e^{4}\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {c x^{2} + a} {\left (B x + A\right )} \sqrt {e x + d}}{c e^{2} x^{4} + 2 \, c d e x^{3} + 2 \, a d e x + a d^{2} + {\left (c d^{2} + a e^{2}\right )} x^{2}}, x\right ) \]