Optimal. Leaf size=451 \[ \frac{\sqrt{2} \cos (e+f x) (a \sin (e+f x)+a)^m \left (c d (3 A+4 B m-B+3 C)+d^2 (-4 A m+A-3 B+3 C)-2 c^2 (2 C m+C)\right ) \sqrt{\frac{c+d \sin (e+f x)}{c-d}} F_1\left (m+\frac{1}{2};\frac{1}{2},\frac{3}{2};m+\frac{3}{2};\frac{1}{2} (\sin (e+f x)+1),-\frac{d (\sin (e+f x)+1)}{c-d}\right )}{3 d f (2 m+1) (c-d)^2 (c+d) \sqrt{1-\sin (e+f x)} \sqrt{c+d \sin (e+f x)}}+\frac{\sqrt{2} \cos (e+f x) (a \sin (e+f x)+a)^{m+1} \left (-d^2 (-2 A m+A+3 C)+B c d (1-2 m)+2 c^2 C (m+1)\right ) \sqrt{\frac{c+d \sin (e+f x)}{c-d}} F_1\left (m+\frac{3}{2};\frac{1}{2},\frac{3}{2};m+\frac{5}{2};\frac{1}{2} (\sin (e+f x)+1),-\frac{d (\sin (e+f x)+1)}{c-d}\right )}{3 a d f (2 m+3) (c-d)^2 (c+d) \sqrt{1-\sin (e+f x)} \sqrt{c+d \sin (e+f x)}}+\frac{2 \cos (e+f x) \left (A d^2-B c d+c^2 C\right ) (a \sin (e+f x)+a)^m}{3 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}} \]
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Rubi [A] time = 1.16287, antiderivative size = 451, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 47, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.128, Rules used = {3043, 2987, 2788, 140, 139, 138} \[ \frac{\sqrt{2} \cos (e+f x) (a \sin (e+f x)+a)^m \left (c d (3 A+4 B m-B+3 C)+d^2 (-4 A m+A-3 B+3 C)-2 c^2 (2 C m+C)\right ) \sqrt{\frac{c+d \sin (e+f x)}{c-d}} F_1\left (m+\frac{1}{2};\frac{1}{2},\frac{3}{2};m+\frac{3}{2};\frac{1}{2} (\sin (e+f x)+1),-\frac{d (\sin (e+f x)+1)}{c-d}\right )}{3 d f (2 m+1) (c-d)^2 (c+d) \sqrt{1-\sin (e+f x)} \sqrt{c+d \sin (e+f x)}}+\frac{\sqrt{2} \cos (e+f x) (a \sin (e+f x)+a)^{m+1} \left (-d^2 (-2 A m+A+3 C)+B c d (1-2 m)+2 c^2 C (m+1)\right ) \sqrt{\frac{c+d \sin (e+f x)}{c-d}} F_1\left (m+\frac{3}{2};\frac{1}{2},\frac{3}{2};m+\frac{5}{2};\frac{1}{2} (\sin (e+f x)+1),-\frac{d (\sin (e+f x)+1)}{c-d}\right )}{3 a d f (2 m+3) (c-d)^2 (c+d) \sqrt{1-\sin (e+f x)} \sqrt{c+d \sin (e+f x)}}+\frac{2 \cos (e+f x) \left (A d^2-B c d+c^2 C\right ) (a \sin (e+f x)+a)^m}{3 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}} \]
Antiderivative was successfully verified.
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Rule 3043
Rule 2987
Rule 2788
Rule 140
Rule 139
Rule 138
Rubi steps
\begin{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 &=\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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) F_1\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 ) F_1\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*}
Mathematica [B] time = 33.6076, size = 20723, normalized size = 45.95 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.73, size = 0, normalized size = 0. \begin{align*} \int{ \left ( a+a\sin \left ( fx+e \right ) \right ) ^{m} \left ( A+B\sin \left ( fx+e \right ) +C \left ( \sin \left ( fx+e \right ) \right ) ^{2} \right ) \left ( c+d\sin \left ( fx+e \right ) \right ) ^{-{\frac{5}{2}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (C \cos \left (f x + e\right )^{2} - B \sin \left (f x + e\right ) - A - C\right )} \sqrt{d \sin \left (f x + e\right ) + c}{\left (a \sin \left (f x + e\right ) + a\right )}^{m}}{3 \, c d^{2} \cos \left (f x + e\right )^{2} - c^{3} - 3 \, c d^{2} +{\left (d^{3} \cos \left (f x + e\right )^{2} - 3 \, c^{2} d - d^{3}\right )} \sin \left (f x + e\right )}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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