Integrand size = 47, antiderivative size = 451 \[ \int \frac {(a+a \sin (e+f x))^m \left (A+B \sin (e+f x)+C \sin ^2(e+f x)\right )}{(c+d \sin (e+f x))^{5/2}} \, dx=\frac {2 \left (c^2 C-B c d+A d^2\right ) \cos (e+f x) (a+a \sin (e+f x))^m}{3 d \left (c^2-d^2\right ) f (c+d \sin (e+f x))^{3/2}}+\frac {\sqrt {2} \left (d^2 (A-3 B+3 C-4 A m)+c d (3 A-B+3 C+4 B m)-2 c^2 (C+2 C m)\right ) \operatorname {AppellF1}\left (\frac {1}{2}+m,\frac {1}{2},\frac {3}{2},\frac {3}{2}+m,\frac {1}{2} (1+\sin (e+f x)),-\frac {d (1+\sin (e+f x))}{c-d}\right ) \cos (e+f x) (a+a \sin (e+f x))^m \sqrt {\frac {c+d \sin (e+f x)}{c-d}}}{3 (c-d)^2 d (c+d) f (1+2 m) \sqrt {1-\sin (e+f x)} \sqrt {c+d \sin (e+f x)}}+\frac {\sqrt {2} \left (B c d (1-2 m)+2 c^2 C (1+m)-d^2 (A+3 C-2 A m)\right ) \operatorname {AppellF1}\left (\frac {3}{2}+m,\frac {1}{2},\frac {3}{2},\frac {5}{2}+m,\frac {1}{2} (1+\sin (e+f x)),-\frac {d (1+\sin (e+f x))}{c-d}\right ) \cos (e+f x) (a+a \sin (e+f x))^{1+m} \sqrt {\frac {c+d \sin (e+f x)}{c-d}}}{3 a (c-d)^2 d (c+d) f (3+2 m) \sqrt {1-\sin (e+f x)} \sqrt {c+d \sin (e+f x)}} \]
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Time = 0.75 (sec) , antiderivative size = 451, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.128, Rules used = {3122, 3066, 2867, 145, 144, 143} \[ \int \frac {(a+a \sin (e+f x))^m \left (A+B \sin (e+f x)+C \sin ^2(e+f x)\right )}{(c+d \sin (e+f x))^{5/2}} \, dx=\frac {\sqrt {2} \cos (e+f x) (a \sin (e+f x)+a)^m \left (c d (3 A+4 B m-B+3 C)+d^2 (-4 A m+A-3 B+3 C)-2 c^2 (2 C m+C)\right ) \sqrt {\frac {c+d \sin (e+f x)}{c-d}} \operatorname {AppellF1}\left (m+\frac {1}{2},\frac {1}{2},\frac {3}{2},m+\frac {3}{2},\frac {1}{2} (\sin (e+f x)+1),-\frac {d (\sin (e+f x)+1)}{c-d}\right )}{3 d f (2 m+1) (c-d)^2 (c+d) \sqrt {1-\sin (e+f x)} \sqrt {c+d \sin (e+f x)}}+\frac {\sqrt {2} \cos (e+f x) (a \sin (e+f x)+a)^{m+1} \left (-d^2 (-2 A m+A+3 C)+B c d (1-2 m)+2 c^2 C (m+1)\right ) \sqrt {\frac {c+d \sin (e+f x)}{c-d}} \operatorname {AppellF1}\left (m+\frac {3}{2},\frac {1}{2},\frac {3}{2},m+\frac {5}{2},\frac {1}{2} (\sin (e+f x)+1),-\frac {d (\sin (e+f x)+1)}{c-d}\right )}{3 a d f (2 m+3) (c-d)^2 (c+d) \sqrt {1-\sin (e+f x)} \sqrt {c+d \sin (e+f x)}}+\frac {2 \cos (e+f x) \left (A d^2-B c d+c^2 C\right ) (a \sin (e+f x)+a)^m}{3 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}} \]
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Rule 143
Rule 144
Rule 145
Rule 2867
Rule 3066
Rule 3122
Rubi steps \begin{align*} \text {integral}& = \frac {2 \left (c^2 C-B c d+A d^2\right ) \cos (e+f x) (a+a \sin (e+f x))^m}{3 d \left (c^2-d^2\right ) f (c+d \sin (e+f x))^{3/2}}-\frac {2 \int \frac {(a+a \sin (e+f x))^m \left (-\frac {1}{2} a \left (2 (c C-B d) \left (\frac {3 d}{2}-c m\right )+2 A d \left (\frac {3 c}{2}-d m\right )\right )+\frac {1}{2} a \left (3 C d^2-d (B c-A d) (1-2 m)-2 c^2 C (1+m)\right ) \sin (e+f x)\right )}{(c+d \sin (e+f x))^{3/2}} \, dx}{3 a d \left (c^2-d^2\right )} \\ & = \frac {2 \left (c^2 C-B c d+A d^2\right ) \cos (e+f x) (a+a \sin (e+f x))^m}{3 d \left (c^2-d^2\right ) f (c+d \sin (e+f x))^{3/2}}+\frac {\left (B c d (1-2 m)+2 c^2 C (1+m)-d^2 (A+3 C-2 A m)\right ) \int \frac {(a+a \sin (e+f x))^{1+m}}{(c+d \sin (e+f x))^{3/2}} \, dx}{3 a d \left (c^2-d^2\right )}+\frac {\left (d^2 (A-3 B+3 C-4 A m)+c d (3 A-B+3 C+4 B m)-2 c^2 (C+2 C m)\right ) \int \frac {(a+a \sin (e+f x))^m}{(c+d \sin (e+f x))^{3/2}} \, dx}{3 d \left (c^2-d^2\right )} \\ & = \frac {2 \left (c^2 C-B c d+A d^2\right ) \cos (e+f x) (a+a \sin (e+f x))^m}{3 d \left (c^2-d^2\right ) f (c+d \sin (e+f x))^{3/2}}+\frac {\left (a \left (B c d (1-2 m)+2 c^2 C (1+m)-d^2 (A+3 C-2 A m)\right ) \cos (e+f x)\right ) \text {Subst}\left (\int \frac {(a+a x)^{\frac {1}{2}+m}}{\sqrt {a-a x} (c+d x)^{3/2}} \, dx,x,\sin (e+f x)\right )}{3 d \left (c^2-d^2\right ) f \sqrt {a-a \sin (e+f x)} \sqrt {a+a \sin (e+f x)}}+\frac {\left (a^2 \left (d^2 (A-3 B+3 C-4 A m)+c d (3 A-B+3 C+4 B m)-2 c^2 (C+2 C m)\right ) \cos (e+f x)\right ) \text {Subst}\left (\int \frac {(a+a x)^{-\frac {1}{2}+m}}{\sqrt {a-a x} (c+d x)^{3/2}} \, dx,x,\sin (e+f x)\right )}{3 d \left (c^2-d^2\right ) f \sqrt {a-a \sin (e+f x)} \sqrt {a+a \sin (e+f x)}} \\ & = \frac {2 \left (c^2 C-B c d+A d^2\right ) \cos (e+f x) (a+a \sin (e+f x))^m}{3 d \left (c^2-d^2\right ) f (c+d \sin (e+f x))^{3/2}}+\frac {\left (a \left (B c d (1-2 m)+2 c^2 C (1+m)-d^2 (A+3 C-2 A m)\right ) \cos (e+f x) \sqrt {\frac {a-a \sin (e+f x)}{a}}\right ) \text {Subst}\left (\int \frac {(a+a x)^{\frac {1}{2}+m}}{\sqrt {\frac {1}{2}-\frac {x}{2}} (c+d x)^{3/2}} \, dx,x,\sin (e+f x)\right )}{3 \sqrt {2} d \left (c^2-d^2\right ) f (a-a \sin (e+f x)) \sqrt {a+a \sin (e+f x)}}+\frac {\left (a^2 \left (d^2 (A-3 B+3 C-4 A m)+c d (3 A-B+3 C+4 B m)-2 c^2 (C+2 C m)\right ) \cos (e+f x) \sqrt {\frac {a-a \sin (e+f x)}{a}}\right ) \text {Subst}\left (\int \frac {(a+a x)^{-\frac {1}{2}+m}}{\sqrt {\frac {1}{2}-\frac {x}{2}} (c+d x)^{3/2}} \, dx,x,\sin (e+f x)\right )}{3 \sqrt {2} d \left (c^2-d^2\right ) f (a-a \sin (e+f x)) \sqrt {a+a \sin (e+f x)}} \\ & = \frac {2 \left (c^2 C-B c d+A d^2\right ) \cos (e+f x) (a+a \sin (e+f x))^m}{3 d \left (c^2-d^2\right ) f (c+d \sin (e+f x))^{3/2}}+\frac {\left (a^2 \left (B c d (1-2 m)+2 c^2 C (1+m)-d^2 (A+3 C-2 A m)\right ) \cos (e+f x) \sqrt {\frac {a-a \sin (e+f x)}{a}} \sqrt {\frac {a (c+d \sin (e+f x))}{a c-a d}}\right ) \text {Subst}\left (\int \frac {(a+a x)^{\frac {1}{2}+m}}{\sqrt {\frac {1}{2}-\frac {x}{2}} \left (\frac {a c}{a c-a d}+\frac {a d x}{a c-a d}\right )^{3/2}} \, dx,x,\sin (e+f x)\right )}{3 \sqrt {2} d (a c-a d) \left (c^2-d^2\right ) f (a-a \sin (e+f x)) \sqrt {a+a \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}+\frac {\left (a^3 \left (d^2 (A-3 B+3 C-4 A m)+c d (3 A-B+3 C+4 B m)-2 c^2 (C+2 C m)\right ) \cos (e+f x) \sqrt {\frac {a-a \sin (e+f x)}{a}} \sqrt {\frac {a (c+d \sin (e+f x))}{a c-a d}}\right ) \text {Subst}\left (\int \frac {(a+a x)^{-\frac {1}{2}+m}}{\sqrt {\frac {1}{2}-\frac {x}{2}} \left (\frac {a c}{a c-a d}+\frac {a d x}{a c-a d}\right )^{3/2}} \, dx,x,\sin (e+f x)\right )}{3 \sqrt {2} d (a c-a d) \left (c^2-d^2\right ) f (a-a \sin (e+f x)) \sqrt {a+a \sin (e+f x)} \sqrt {c+d \sin (e+f x)}} \\ & = \frac {2 \left (c^2 C-B c d+A d^2\right ) \cos (e+f x) (a+a \sin (e+f x))^m}{3 d \left (c^2-d^2\right ) f (c+d \sin (e+f x))^{3/2}}+\frac {\sqrt {2} \left (d^2 (A-3 B+3 C-4 A m)+c d (3 A-B+3 C+4 B m)-2 c^2 (C+2 C m)\right ) \operatorname {AppellF1}\left (\frac {1}{2}+m,\frac {1}{2},\frac {3}{2},\frac {3}{2}+m,\frac {1}{2} (1+\sin (e+f x)),-\frac {d (1+\sin (e+f x))}{c-d}\right ) \cos (e+f x) (a+a \sin (e+f x))^m \sqrt {\frac {c+d \sin (e+f x)}{c-d}}}{3 (c-d)^2 d (c+d) f (1+2 m) \sqrt {1-\sin (e+f x)} \sqrt {c+d \sin (e+f x)}}+\frac {\sqrt {2} \left (B c d (1-2 m)+2 c^2 C (1+m)-d^2 (A+3 C-2 A m)\right ) \operatorname {AppellF1}\left (\frac {3}{2}+m,\frac {1}{2},\frac {3}{2},\frac {5}{2}+m,\frac {1}{2} (1+\sin (e+f x)),-\frac {d (1+\sin (e+f x))}{c-d}\right ) \cos (e+f x) \sqrt {1-\sin (e+f x)} (a+a \sin (e+f x))^{1+m} \sqrt {\frac {c+d \sin (e+f x)}{c-d}}}{3 (c-d)^2 d (c+d) f (3+2 m) (a-a \sin (e+f x)) \sqrt {c+d \sin (e+f x)}} \\ \end{align*}
\[ \int \frac {(a+a \sin (e+f x))^m \left (A+B \sin (e+f x)+C \sin ^2(e+f x)\right )}{(c+d \sin (e+f x))^{5/2}} \, dx=\int \frac {(a+a \sin (e+f x))^m \left (A+B \sin (e+f x)+C \sin ^2(e+f x)\right )}{(c+d \sin (e+f x))^{5/2}} \, dx \]
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\[\int \frac {\left (a +a \sin \left (f x +e \right )\right )^{m} \left (A +B \sin \left (f x +e \right )+C \left (\sin ^{2}\left (f x +e \right )\right )\right )}{\left (c +d \sin \left (f x +e \right )\right )^{\frac {5}{2}}}d x\]
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\[ \int \frac {(a+a \sin (e+f x))^m \left (A+B \sin (e+f x)+C \sin ^2(e+f x)\right )}{(c+d \sin (e+f x))^{5/2}} \, dx=\int { \frac {{\left (C \sin \left (f x + e\right )^{2} + B \sin \left (f x + e\right ) + A\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{m}}{{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {(a+a \sin (e+f x))^m \left (A+B \sin (e+f x)+C \sin ^2(e+f x)\right )}{(c+d \sin (e+f x))^{5/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {(a+a \sin (e+f x))^m \left (A+B \sin (e+f x)+C \sin ^2(e+f x)\right )}{(c+d \sin (e+f x))^{5/2}} \, dx=\int { \frac {{\left (C \sin \left (f x + e\right )^{2} + B \sin \left (f x + e\right ) + A\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{m}}{{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}}} \,d x } \]
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\[ \int \frac {(a+a \sin (e+f x))^m \left (A+B \sin (e+f x)+C \sin ^2(e+f x)\right )}{(c+d \sin (e+f x))^{5/2}} \, dx=\int { \frac {{\left (C \sin \left (f x + e\right )^{2} + B \sin \left (f x + e\right ) + A\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{m}}{{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {(a+a \sin (e+f x))^m \left (A+B \sin (e+f x)+C \sin ^2(e+f x)\right )}{(c+d \sin (e+f x))^{5/2}} \, dx=\int \frac {{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m\,\left (C\,{\sin \left (e+f\,x\right )}^2+B\,\sin \left (e+f\,x\right )+A\right )}{{\left (c+d\,\sin \left (e+f\,x\right )\right )}^{5/2}} \,d x \]
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