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