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 = 0.10, 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} \[ -\frac {a 2^{m+\frac {7}{4}} (e \cos (c+d x))^{3/2} (\sin (c+d x)+1)^{\frac {1}{4}-m} (a \sin (c+d x)+a)^{m-1} \, _2F_1\left (\frac {3}{4},\frac {1}{4}-m;\frac {7}{4};\frac {1}{2} (1-\sin (c+d x))\right )}{3 d e} \]
Antiderivative was successfully verified.
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Rule 69
Rule 70
Rule 2689
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*}
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Mathematica [A] time = 0.08, 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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fricas [F] time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {e \cos \left (d x + c\right )} {\left (a \sin \left (d x + c\right ) + a\right )}^{m}, x\right ) \]
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 \[ \int \sqrt {e \cos \left (d x + c\right )} {\left (a \sin \left (d x + c\right ) + a\right )}^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.22, size = 0, normalized size = 0.00 \[ \int \sqrt {e \cos \left (d x +c \right )}\, \left (a +a \sin \left (d x +c \right )\right )^{m}\, dx \]
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 \[ \int \sqrt {e \cos \left (d x + c\right )} {\left (a \sin \left (d x + c\right ) + a\right )}^{m}\,{d x} \]
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 \[ \int \sqrt {e\,\cos \left (c+d\,x\right )}\,{\left (a+a\,\sin \left (c+d\,x\right )\right )}^m \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a \left (\sin {\left (c + d x \right )} + 1\right )\right )^{m} \sqrt {e \cos {\left (c + d x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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