\[ \int \frac {\left (a+c x^2\right )^{3/2}}{\sqrt {d+e x}} \, dx \]
Optimal antiderivative \[ \frac {2 \left (c \,x^{2}+a \right )^{\frac {3}{2}} \sqrt {e x +d}}{7 e}+\frac {4 \left (-3 c d e x +5 a \,e^{2}+4 c \,d^{2}\right ) \sqrt {e x +d}\, \sqrt {c \,x^{2}+a}}{35 e^{3}}+\frac {32 d \left (2 a \,e^{2}+c \,d^{2}\right ) \EllipticE \left (\frac {\sqrt {1-\frac {x \sqrt {c}}{\sqrt {-a}}}\, \sqrt {2}}{2}, \sqrt {-\frac {2 a e}{-a e +d \sqrt {-a}\, \sqrt {c}}}\right ) \sqrt {-a}\, \sqrt {c}\, \sqrt {e x +d}\, \sqrt {1+\frac {c \,x^{2}}{a}}}{35 e^{4} \sqrt {c \,x^{2}+a}\, \sqrt {\frac {\left (e x +d \right ) \sqrt {c}}{e \sqrt {-a}+d \sqrt {c}}}}-\frac {8 \left (a \,e^{2}+c \,d^{2}\right ) \left (5 a \,e^{2}+4 c \,d^{2}\right ) \EllipticF \left (\frac {\sqrt {1-\frac {x \sqrt {c}}{\sqrt {-a}}}\, \sqrt {2}}{2}, \sqrt {-\frac {2 a e}{-a e +d \sqrt {-a}\, \sqrt {c}}}\right ) \sqrt {-a}\, \sqrt {1+\frac {c \,x^{2}}{a}}\, \sqrt {\frac {\left (e x +d \right ) \sqrt {c}}{e \sqrt {-a}+d \sqrt {c}}}}{35 e^{4} \sqrt {c}\, \sqrt {e x +d}\, \sqrt {c \,x^{2}+a}} \]
command
integrate((c*x^2+a)^(3/2)/(e*x+d)^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {2 \, {\left (4 \, {\left (4 \, c^{2} d^{4} + 11 \, a c d^{2} e^{2} + 15 \, a^{2} e^{4}\right )} \sqrt {c} e^{\frac {1}{2}} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )} e^{\left (-2\right )}}{3 \, c}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )} e^{\left (-3\right )}}{27 \, c}, \frac {1}{3} \, {\left (3 \, x e + d\right )} e^{\left (-1\right )}\right ) + 48 \, {\left (c^{2} d^{3} e + 2 \, a c d e^{3}\right )} \sqrt {c} e^{\frac {1}{2}} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )} e^{\left (-2\right )}}{3 \, c}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )} e^{\left (-3\right )}}{27 \, c}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )} e^{\left (-2\right )}}{3 \, c}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )} e^{\left (-3\right )}}{27 \, c}, \frac {1}{3} \, {\left (3 \, x e + d\right )} e^{\left (-1\right )}\right )\right ) - 3 \, {\left (6 \, c^{2} d x e^{3} - 8 \, c^{2} d^{2} e^{2} - 5 \, {\left (c^{2} x^{2} + 3 \, a c\right )} e^{4}\right )} \sqrt {c x^{2} + a} \sqrt {x e + d}\right )} e^{\left (-5\right )}}{105 \, c} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (c x^{2} + a\right )}^{\frac {3}{2}}}{\sqrt {e x + d}}, x\right ) \]