\[ \int \frac {(c x)^{7/2}}{\left (a+b x^2\right )^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {c \left (c x \right )^{\frac {5}{2}}}{b \sqrt {b \,x^{2}+a}}+\frac {5 c^{3} \sqrt {c x}\, \sqrt {b \,x^{2}+a}}{3 b^{2}}-\frac {5 a^{\frac {3}{4}} c^{\frac {7}{2}} \sqrt {\frac {\cos \left (4 \arctan \left (\frac {b^{\frac {1}{4}} \sqrt {c x}}{a^{\frac {1}{4}} \sqrt {c}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (2 \arctan \left (\frac {b^{\frac {1}{4}} \sqrt {c x}}{a^{\frac {1}{4}} \sqrt {c}}\right )\right ), \frac {\sqrt {2}}{2}\right ) \left (\sqrt {a}+x \sqrt {b}\right ) \sqrt {\frac {b \,x^{2}+a}{\left (\sqrt {a}+x \sqrt {b}\right )^{2}}}}{6 \cos \left (2 \arctan \left (\frac {b^{\frac {1}{4}} \sqrt {c x}}{a^{\frac {1}{4}} \sqrt {c}}\right )\right ) b^{\frac {9}{4}} \sqrt {b \,x^{2}+a}} \]
command
integrate((c*x)^(7/2)/(b*x^2+a)^(3/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {5 \, {\left (a b c^{3} x^{2} + a^{2} c^{3}\right )} \sqrt {b c} {\rm weierstrassPInverse}\left (-\frac {4 \, a}{b}, 0, x\right ) - {\left (2 \, b^{2} c^{3} x^{2} + 5 \, a b c^{3}\right )} \sqrt {b x^{2} + a} \sqrt {c x}}{3 \, {\left (b^{4} x^{2} + a b^{3}\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {b x^{2} + a} \sqrt {c x} c^{3} x^{3}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}, x\right ) \]