\[ \int \frac {(d \sin (e+f x))^{5/2}}{(g \cos (e+f x))^{3/2} (a+b \sin (e+f x))} \, dx \]
Optimal antiderivative \[ \frac {a^{2} d^{\frac {5}{2}} \arctan \! \left (-1+\frac {\sqrt {2}\, \sqrt {d}\, \sqrt {g \cos \left (f x +e \right )}}{\sqrt {g}\, \sqrt {d \sin \left (f x +e \right )}}\right ) \sqrt {2}}{2 b \left (a^{2}-b^{2}\right ) f \,g^{\frac {3}{2}}}-\frac {b \,d^{\frac {5}{2}} \arctan \! \left (-1+\frac {\sqrt {2}\, \sqrt {d}\, \sqrt {g \cos \left (f x +e \right )}}{\sqrt {g}\, \sqrt {d \sin \left (f x +e \right )}}\right ) \sqrt {2}}{2 \left (a^{2}-b^{2}\right ) f \,g^{\frac {3}{2}}}+\frac {a^{2} d^{\frac {5}{2}} \arctan \! \left (1+\frac {\sqrt {2}\, \sqrt {d}\, \sqrt {g \cos \left (f x +e \right )}}{\sqrt {g}\, \sqrt {d \sin \left (f x +e \right )}}\right ) \sqrt {2}}{2 b \left (a^{2}-b^{2}\right ) f \,g^{\frac {3}{2}}}-\frac {b \,d^{\frac {5}{2}} \arctan \! \left (1+\frac {\sqrt {2}\, \sqrt {d}\, \sqrt {g \cos \left (f x +e \right )}}{\sqrt {g}\, \sqrt {d \sin \left (f x +e \right )}}\right ) \sqrt {2}}{2 \left (a^{2}-b^{2}\right ) f \,g^{\frac {3}{2}}}+\frac {a^{2} d^{\frac {5}{2}} \ln \! \left (\sqrt {g}+\cot \! \left (f x +e \right ) \sqrt {g}-\frac {\sqrt {2}\, \sqrt {d}\, \sqrt {g \cos \left (f x +e \right )}}{\sqrt {d \sin \left (f x +e \right )}}\right ) \sqrt {2}}{4 b \left (a^{2}-b^{2}\right ) f \,g^{\frac {3}{2}}}-\frac {b \,d^{\frac {5}{2}} \ln \! \left (\sqrt {g}+\cot \! \left (f x +e \right ) \sqrt {g}-\frac {\sqrt {2}\, \sqrt {d}\, \sqrt {g \cos \left (f x +e \right )}}{\sqrt {d \sin \left (f x +e \right )}}\right ) \sqrt {2}}{4 \left (a^{2}-b^{2}\right ) f \,g^{\frac {3}{2}}}-\frac {a^{2} d^{\frac {5}{2}} \ln \! \left (\sqrt {g}+\cot \! \left (f x +e \right ) \sqrt {g}+\frac {\sqrt {2}\, \sqrt {d}\, \sqrt {g \cos \left (f x +e \right )}}{\sqrt {d \sin \left (f x +e \right )}}\right ) \sqrt {2}}{4 b \left (a^{2}-b^{2}\right ) f \,g^{\frac {3}{2}}}+\frac {b \,d^{\frac {5}{2}} \ln \! \left (\sqrt {g}+\cot \! \left (f x +e \right ) \sqrt {g}+\frac {\sqrt {2}\, \sqrt {d}\, \sqrt {g \cos \left (f x +e \right )}}{\sqrt {d \sin \left (f x +e \right )}}\right ) \sqrt {2}}{4 \left (a^{2}-b^{2}\right ) f \,g^{\frac {3}{2}}}+\frac {2 a d \left (d \sin \! \left (f x +e \right )\right )^{\frac {3}{2}}}{\left (a^{2}-b^{2}\right ) f g \sqrt {g \cos \! \left (f x +e \right )}}-\frac {2 a^{3} d^{3} \EllipticPi \! \left (\frac {\sqrt {g \cos \! \left (f x +e \right )}}{\sqrt {g}\, \sqrt {1+\sin \! \left (f x +e \right )}}, -\frac {\sqrt {-a +b}}{\sqrt {a +b}}, \mathrm {I}\right ) \sqrt {2}\, \left (\sqrt {\sin }\left (f x +e \right )\right )}{b \left (-a +b \right )^{\frac {3}{2}} \left (a +b \right )^{\frac {3}{2}} f \,g^{\frac {3}{2}} \sqrt {d \sin \! \left (f x +e \right )}}+\frac {2 a^{3} d^{3} \EllipticPi \! \left (\frac {\sqrt {g \cos \! \left (f x +e \right )}}{\sqrt {g}\, \sqrt {1+\sin \! \left (f x +e \right )}}, \frac {\sqrt {-a +b}}{\sqrt {a +b}}, \mathrm {I}\right ) \sqrt {2}\, \left (\sqrt {\sin }\left (f x +e \right )\right )}{b \left (-a +b \right )^{\frac {3}{2}} \left (a +b \right )^{\frac {3}{2}} f \,g^{\frac {3}{2}} \sqrt {d \sin \! \left (f x +e \right )}}-\frac {2 b \,d^{2} \sqrt {d \sin \! \left (f x +e \right )}}{\left (a^{2}-b^{2}\right ) f g \sqrt {g \cos \! \left (f x +e \right )}}+\frac {2 a \,d^{2} \sqrt {\frac {1}{2}+\frac {\sin \left (2 f x +2 e \right )}{2}}\, \EllipticE \! \left (\cos \! \left (e +\frac {\pi }{4}+f x \right ), \sqrt {2}\right ) \sqrt {g \cos \! \left (f x +e \right )}\, \sqrt {d \sin \! \left (f x +e \right )}}{\sin \! \left (e +\frac {\pi }{4}+f x \right ) \left (a^{2}-b^{2}\right ) f \,g^{2} \sqrt {\sin \! \left (2 f x +2 e \right )}} \]
command
Integrate[(d*Sin[e + f*x])^(5/2)/((g*Cos[e + f*x])^(3/2)*(a + b*Sin[e + f*x])),x]
Mathematica 13.1 output
\[ \text {\$Aborted} \]
Mathematica 12.3 output
\[ \frac {2 \cot (e+f x) \csc (e+f x) (d \sin (e+f x))^{5/2} (a \sin (e+f x)-b)}{\left (a^2-b^2\right ) f (g \cos (e+f x))^{3/2}}-\frac {\cos ^{\frac {3}{2}}(e+f x) (d \sin (e+f x))^{5/2} \left (-\frac {2 \left (3 a^2-b^2\right ) \left (a F_1\left (\frac {3}{4};\frac {1}{4},1;\frac {7}{4};\cos ^2(e+f x),\frac {b^2 \cos ^2(e+f x)}{b^2-a^2}\right )-b F_1\left (\frac {3}{4};-\frac {1}{4},1;\frac {7}{4};\cos ^2(e+f x),\frac {b^2 \cos ^2(e+f x)}{b^2-a^2}\right )\right ) \cos ^{\frac {3}{2}}(e+f x) \left (a+b \sqrt {1-\cos ^2(e+f x)}\right ) \sin ^{\frac {3}{2}}(e+f x)}{3 \left (a^2-b^2\right ) \left (1-\cos ^2(e+f x)\right )^{3/4} (a+b \sin (e+f x))}-\frac {\cos (2 (e+f x)) \sqrt {\tan (e+f x)} \left (\sqrt {\tan ^2(e+f x)+1} a+b \tan (e+f x)\right ) \left (24 b \left (b^2-a^2\right ) F_1\left (\frac {7}{4};\frac {1}{2},1;\frac {11}{4};-\tan ^2(e+f x),\left (\frac {b^2}{a^2}-1\right ) \tan ^2(e+f x)\right ) \tan ^{\frac {7}{2}}(e+f x)+56 b \left (b^2-3 a^2\right ) F_1\left (\frac {3}{4};\frac {1}{2},1;\frac {7}{4};-\tan ^2(e+f x),\left (\frac {b^2}{a^2}-1\right ) \tan ^2(e+f x)\right ) \tan ^{\frac {3}{2}}(e+f x)+21 a^{3/2} \left (-\frac {4 \sqrt {2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{a^2-b^2} \sqrt {\tan (e+f x)}}{\sqrt {a}}\right ) a^2}{\sqrt [4]{a^2-b^2}}+\frac {4 \sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a^2-b^2} \sqrt {\tan (e+f x)}}{\sqrt {a}}+1\right ) a^2}{\sqrt [4]{a^2-b^2}}-\frac {2 \sqrt {2} \log \left (-a+\sqrt {2} \sqrt [4]{a^2-b^2} \sqrt {\tan (e+f x)} \sqrt {a}-\sqrt {a^2-b^2} \tan (e+f x)\right ) a^2}{\sqrt [4]{a^2-b^2}}+\frac {2 \sqrt {2} \log \left (a+\sqrt {2} \sqrt [4]{a^2-b^2} \sqrt {\tan (e+f x)} \sqrt {a}+\sqrt {a^2-b^2} \tan (e+f x)\right ) a^2}{\sqrt [4]{a^2-b^2}}+4 \sqrt {2} \tan ^{-1}\left (1-\sqrt {2} \sqrt {\tan (e+f x)}\right ) a^{3/2}-4 \sqrt {2} \tan ^{-1}\left (\sqrt {2} \sqrt {\tan (e+f x)}+1\right ) a^{3/2}+2 \sqrt {2} \log \left (\tan (e+f x)-\sqrt {2} \sqrt {\tan (e+f x)}+1\right ) a^{3/2}-2 \sqrt {2} \log \left (\tan (e+f x)+\sqrt {2} \sqrt {\tan (e+f x)}+1\right ) a^{3/2}+\frac {8 b \tan ^{\frac {3}{2}}(e+f x) \sqrt {a}}{\sqrt {\tan ^2(e+f x)+1}}+\frac {2 \sqrt {2} b^2 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{a^2-b^2} \sqrt {\tan (e+f x)}}{\sqrt {a}}\right )}{\sqrt [4]{a^2-b^2}}-\frac {2 \sqrt {2} b^2 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a^2-b^2} \sqrt {\tan (e+f x)}}{\sqrt {a}}+1\right )}{\sqrt [4]{a^2-b^2}}+\frac {\sqrt {2} b^2 \log \left (-a+\sqrt {2} \sqrt [4]{a^2-b^2} \sqrt {\tan (e+f x)} \sqrt {a}-\sqrt {a^2-b^2} \tan (e+f x)\right )}{\sqrt [4]{a^2-b^2}}-\frac {\sqrt {2} b^2 \log \left (a+\sqrt {2} \sqrt [4]{a^2-b^2} \sqrt {\tan (e+f x)} \sqrt {a}+\sqrt {a^2-b^2} \tan (e+f x)\right )}{\sqrt [4]{a^2-b^2}}\right )\right )}{84 a b \cos ^{\frac {3}{2}}(e+f x) (a+b \sin (e+f x)) \left (\tan ^2(e+f x)-1\right ) \sqrt {\tan ^2(e+f x)+1} \sqrt {\sin (e+f x)}}\right )}{(a-b) (a+b) f (g \cos (e+f x))^{3/2} \sin ^{\frac {5}{2}}(e+f x)} \]