Optimal. Leaf size=88 \[ -\frac{a 2^{m+\frac{7}{4}} (e \cos (c+d x))^{3/2} (\sin (c+d x)+1)^{\frac{1}{4}-m} (a \sin (c+d x)+a)^{m-1} \, _2F_1\left (\frac{3}{4},\frac{1}{4}-m;\frac{7}{4};\frac{1}{2} (1-\sin (c+d x))\right )}{3 d e} \]
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Rubi [A] time = 0.0992657, 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} \[ -\frac{a 2^{m+\frac{7}{4}} (e \cos (c+d x))^{3/2} (\sin (c+d x)+1)^{\frac{1}{4}-m} (a \sin (c+d x)+a)^{m-1} \, _2F_1\left (\frac{3}{4},\frac{1}{4}-m;\frac{7}{4};\frac{1}{2} (1-\sin (c+d x))\right )}{3 d e} \]
Antiderivative was successfully verified.
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Rule 2689
Rule 70
Rule 69
Rubi steps
\begin{align*} \int \sqrt{e \cos (c+d x)} (a+a \sin (c+d x))^m \, dx &=\frac{\left (a^2 (e \cos (c+d x))^{3/2}\right ) \operatorname{Subst}\left (\int \frac{(a+a x)^{-\frac{1}{4}+m}}{\sqrt [4]{a-a x}} \, dx,x,\sin (c+d x)\right )}{d e (a-a \sin (c+d x))^{3/4} (a+a \sin (c+d x))^{3/4}}\\ &=\frac{\left (2^{-\frac{1}{4}+m} a^2 (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^{-1+m} \left (\frac{a+a \sin (c+d x)}{a}\right )^{\frac{1}{4}-m}\right ) \operatorname{Subst}\left (\int \frac{\left (\frac{1}{2}+\frac{x}{2}\right )^{-\frac{1}{4}+m}}{\sqrt [4]{a-a x}} \, dx,x,\sin (c+d x)\right )}{d e (a-a \sin (c+d x))^{3/4}}\\ &=-\frac{2^{\frac{7}{4}+m} a (e \cos (c+d x))^{3/2} \, _2F_1\left (\frac{3}{4},\frac{1}{4}-m;\frac{7}{4};\frac{1}{2} (1-\sin (c+d x))\right ) (1+\sin (c+d x))^{\frac{1}{4}-m} (a+a \sin (c+d x))^{-1+m}}{3 d e}\\ \end{align*}
Mathematica [A] time = 0.0831012, size = 85, normalized size = 0.97 \[ -\frac{2^{m+\frac{7}{4}} (e \cos (c+d x))^{3/2} (\sin (c+d x)+1)^{-m-\frac{3}{4}} (a (\sin (c+d x)+1))^m \, _2F_1\left (\frac{3}{4},\frac{1}{4}-m;\frac{7}{4};\frac{1}{2} (1-\sin (c+d x))\right )}{3 d e} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.098, size = 0, normalized size = 0. \begin{align*} \int \sqrt{e\cos \left ( dx+c \right ) } \left ( a+a\sin \left ( dx+c \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 \sqrt{e \cos \left (d x + c\right )}{\left (a \sin \left (d x + c\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{e \cos \left (d x + c\right )}{\left (a \sin \left (d x + c\right ) + a\right )}^{m}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a \left (\sin{\left (c + d x \right )} + 1\right )\right )^{m} \sqrt{e \cos{\left (c + d x \right )}}\, dx \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 \sqrt{e \cos \left (d x + c\right )}{\left (a \sin \left (d x + c\right ) + a\right )}^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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