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 = 0.09, antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, 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 69
Rule 70
Rule 2689
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*}
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Mathematica [A] time = 0.15, 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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fricas [F] time = 0.49, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {g \cos \left (f x + e\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{m} g \cos \left (f x + e\right ), x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (g \cos \left (f x + e\right )\right )^{\frac {3}{2}} {\left (a \sin \left (f x + e\right ) + a\right )}^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.31, size = 0, normalized size = 0.00 \[ \int \left (g \cos \left (f x +e \right )\right )^{\frac {3}{2}} \left (a +a \sin \left (f x +e \right )\right )^{m}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (g \cos \left (f x + e\right )\right )^{\frac {3}{2}} {\left (a \sin \left (f x + e\right ) + a\right )}^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (g\,\cos \left (e+f\,x\right )\right )}^{3/2}\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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