Optimal. Leaf size=88 \[ -\frac{a 2^{m+\frac{9}{4}} (g \cos (e+f x))^{5/2} (\sin (e+f x)+1)^{-m-\frac{1}{4}} (a \sin (e+f x)+a)^{m-1} \, _2F_1\left (\frac{5}{4},-m-\frac{1}{4};\frac{9}{4};\frac{1}{2} (1-\sin (e+f x))\right )}{5 f g} \]
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Rubi [A] time = 0.0855542, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {2689, 70, 69} \[ -\frac{a 2^{m+\frac{9}{4}} (g \cos (e+f x))^{5/2} (\sin (e+f x)+1)^{-m-\frac{1}{4}} (a \sin (e+f x)+a)^{m-1} \, _2F_1\left (\frac{5}{4},-m-\frac{1}{4};\frac{9}{4};\frac{1}{2} (1-\sin (e+f x))\right )}{5 f g} \]
Antiderivative was successfully verified.
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Rule 2689
Rule 70
Rule 69
Rubi steps
\begin{align*} \int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^m \, dx &=\frac{\left (a^2 (g \cos (e+f x))^{5/2}\right ) \operatorname{Subst}\left (\int \sqrt [4]{a-a x} (a+a x)^{\frac{1}{4}+m} \, dx,x,\sin (e+f x)\right )}{f g (a-a \sin (e+f x))^{5/4} (a+a \sin (e+f x))^{5/4}}\\ &=\frac{\left (2^{\frac{1}{4}+m} a^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{-1+m} \left (\frac{a+a \sin (e+f x)}{a}\right )^{-\frac{1}{4}-m}\right ) \operatorname{Subst}\left (\int \left (\frac{1}{2}+\frac{x}{2}\right )^{\frac{1}{4}+m} \sqrt [4]{a-a x} \, dx,x,\sin (e+f x)\right )}{f g (a-a \sin (e+f x))^{5/4}}\\ &=-\frac{2^{\frac{9}{4}+m} a (g \cos (e+f x))^{5/2} \, _2F_1\left (\frac{5}{4},-\frac{1}{4}-m;\frac{9}{4};\frac{1}{2} (1-\sin (e+f x))\right ) (1+\sin (e+f x))^{-\frac{1}{4}-m} (a+a \sin (e+f x))^{-1+m}}{5 f g}\\ \end{align*}
Mathematica [A] time = 0.155589, size = 85, normalized size = 0.97 \[ -\frac{2^{m+\frac{9}{4}} (g \cos (e+f x))^{5/2} (\sin (e+f x)+1)^{-m-\frac{5}{4}} (a (\sin (e+f x)+1))^m \, _2F_1\left (\frac{5}{4},-m-\frac{1}{4};\frac{9}{4};\frac{1}{2} (1-\sin (e+f x))\right )}{5 f g} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.125, size = 0, normalized size = 0. \begin{align*} \int \left ( g\cos \left ( fx+e \right ) \right ) ^{{\frac{3}{2}}} \left ( a+a\sin \left ( fx+e \right ) \right ) ^{m}\, 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 \left (g \cos \left (f x + e\right )\right )^{\frac{3}{2}}{\left (a \sin \left (f x + e\right ) + a\right )}^{m}\,{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 (\sqrt{g \cos \left (f x + e\right )}{\left (a \sin \left (f x + e\right ) + a\right )}^{m} g \cos \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(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: AttributeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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