\[ \int \frac {(c x)^{7/2}}{\sqrt {a+b x^2}} \, dx \]
Optimal antiderivative \[ \frac {2 c \left (c x \right )^{\frac {5}{2}} \sqrt {b \,x^{2}+a}}{7 b}-\frac {10 a \,c^{3} \sqrt {c x}\, \sqrt {b \,x^{2}+a}}{21 b^{2}}+\frac {5 a^{\frac {7}{4}} c^{\frac {7}{2}} \sqrt {\frac {\cos \left (4 \arctan \left (\frac {b^{\frac {1}{4}} \sqrt {c x}}{a^{\frac {1}{4}} \sqrt {c}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (2 \arctan \left (\frac {b^{\frac {1}{4}} \sqrt {c x}}{a^{\frac {1}{4}} \sqrt {c}}\right )\right ), \frac {\sqrt {2}}{2}\right ) \left (\sqrt {a}+x \sqrt {b}\right ) \sqrt {\frac {b \,x^{2}+a}{\left (\sqrt {a}+x \sqrt {b}\right )^{2}}}}{21 \cos \left (2 \arctan \left (\frac {b^{\frac {1}{4}} \sqrt {c x}}{a^{\frac {1}{4}} \sqrt {c}}\right )\right ) b^{\frac {9}{4}} \sqrt {b \,x^{2}+a}} \]
command
integrate((c*x)^(7/2)/(b*x^2+a)^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {2 \, {\left (5 \, \sqrt {b c} a^{2} c^{3} {\rm weierstrassPInverse}\left (-\frac {4 \, a}{b}, 0, x\right ) + {\left (3 \, b^{2} c^{3} x^{2} - 5 \, a b c^{3}\right )} \sqrt {b x^{2} + a} \sqrt {c x}\right )}}{21 \, b^{3}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {c x} c^{3} x^{3}}{\sqrt {b x^{2} + a}}, x\right ) \]