\[ \int \frac {1}{\left (\frac {c}{(a+b x)^3}\right )^{5/2}} \, dx \]
Optimal antiderivative \[ \frac {2 \left (b x +a \right )^{7}}{17 b \,c^{2} \sqrt {\frac {c}{\left (b x +a \right )^{3}}}} \]
command
integrate(1/(c/(b*x+a)^3)^(5/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {2 \, {\left (109395 \, \sqrt {b c x + a c} a^{8} - \frac {291720 \, {\left (3 \, \sqrt {b c x + a c} a c - {\left (b c x + a c\right )}^{\frac {3}{2}}\right )} a^{7}}{c} + \frac {204204 \, {\left (15 \, \sqrt {b c x + a c} a^{2} c^{2} - 10 \, {\left (b c x + a c\right )}^{\frac {3}{2}} a c + 3 \, {\left (b c x + a c\right )}^{\frac {5}{2}}\right )} a^{6}}{c^{2}} - \frac {175032 \, {\left (35 \, \sqrt {b c x + a c} a^{3} c^{3} - 35 \, {\left (b c x + a c\right )}^{\frac {3}{2}} a^{2} c^{2} + 21 \, {\left (b c x + a c\right )}^{\frac {5}{2}} a c - 5 \, {\left (b c x + a c\right )}^{\frac {7}{2}}\right )} a^{5}}{c^{3}} + \frac {24310 \, {\left (315 \, \sqrt {b c x + a c} a^{4} c^{4} - 420 \, {\left (b c x + a c\right )}^{\frac {3}{2}} a^{3} c^{3} + 378 \, {\left (b c x + a c\right )}^{\frac {5}{2}} a^{2} c^{2} - 180 \, {\left (b c x + a c\right )}^{\frac {7}{2}} a c + 35 \, {\left (b c x + a c\right )}^{\frac {9}{2}}\right )} a^{4}}{c^{4}} - \frac {8840 \, {\left (693 \, \sqrt {b c x + a c} a^{5} c^{5} - 1155 \, {\left (b c x + a c\right )}^{\frac {3}{2}} a^{4} c^{4} + 1386 \, {\left (b c x + a c\right )}^{\frac {5}{2}} a^{3} c^{3} - 990 \, {\left (b c x + a c\right )}^{\frac {7}{2}} a^{2} c^{2} + 385 \, {\left (b c x + a c\right )}^{\frac {9}{2}} a c - 63 \, {\left (b c x + a c\right )}^{\frac {11}{2}}\right )} a^{3}}{c^{5}} + \frac {1020 \, {\left (3003 \, \sqrt {b c x + a c} a^{6} c^{6} - 6006 \, {\left (b c x + a c\right )}^{\frac {3}{2}} a^{5} c^{5} + 9009 \, {\left (b c x + a c\right )}^{\frac {5}{2}} a^{4} c^{4} - 8580 \, {\left (b c x + a c\right )}^{\frac {7}{2}} a^{3} c^{3} + 5005 \, {\left (b c x + a c\right )}^{\frac {9}{2}} a^{2} c^{2} - 1638 \, {\left (b c x + a c\right )}^{\frac {11}{2}} a c + 231 \, {\left (b c x + a c\right )}^{\frac {13}{2}}\right )} a^{2}}{c^{6}} - \frac {136 \, {\left (6435 \, \sqrt {b c x + a c} a^{7} c^{7} - 15015 \, {\left (b c x + a c\right )}^{\frac {3}{2}} a^{6} c^{6} + 27027 \, {\left (b c x + a c\right )}^{\frac {5}{2}} a^{5} c^{5} - 32175 \, {\left (b c x + a c\right )}^{\frac {7}{2}} a^{4} c^{4} + 25025 \, {\left (b c x + a c\right )}^{\frac {9}{2}} a^{3} c^{3} - 12285 \, {\left (b c x + a c\right )}^{\frac {11}{2}} a^{2} c^{2} + 3465 \, {\left (b c x + a c\right )}^{\frac {13}{2}} a c - 429 \, {\left (b c x + a c\right )}^{\frac {15}{2}}\right )} a}{c^{7}} + \frac {109395 \, \sqrt {b c x + a c} a^{8} c^{8} - 291720 \, {\left (b c x + a c\right )}^{\frac {3}{2}} a^{7} c^{7} + 612612 \, {\left (b c x + a c\right )}^{\frac {5}{2}} a^{6} c^{6} - 875160 \, {\left (b c x + a c\right )}^{\frac {7}{2}} a^{5} c^{5} + 850850 \, {\left (b c x + a c\right )}^{\frac {9}{2}} a^{4} c^{4} - 556920 \, {\left (b c x + a c\right )}^{\frac {11}{2}} a^{3} c^{3} + 235620 \, {\left (b c x + a c\right )}^{\frac {13}{2}} a^{2} c^{2} - 58344 \, {\left (b c x + a c\right )}^{\frac {15}{2}} a c + 6435 \, {\left (b c x + a c\right )}^{\frac {17}{2}}}{c^{8}}\right )}}{109395 \, b c^{3} \mathrm {sgn}\left (b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}\right ) \mathrm {sgn}\left (b x + a\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int \frac {1}{\left (\frac {c}{{\left (b x + a\right )}^{3}}\right )^{\frac {5}{2}}}\,{d x} \]________________________________________________________________________________________