Optimal. Leaf size=68 \[ \frac{2 a E\left (\left .\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )\right |2\right ) \sqrt{e \sin (c+d x)}}{d \sqrt{\sin (c+d x)}}+\frac{2 b (e \sin (c+d x))^{3/2}}{3 d e} \]
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Rubi [A] time = 0.0489842, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {2669, 2640, 2639} \[ \frac{2 a E\left (\left .\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )\right |2\right ) \sqrt{e \sin (c+d x)}}{d \sqrt{\sin (c+d x)}}+\frac{2 b (e \sin (c+d x))^{3/2}}{3 d e} \]
Antiderivative was successfully verified.
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Rule 2669
Rule 2640
Rule 2639
Rubi steps
\begin{align*} \int (a+b \cos (c+d x)) \sqrt{e \sin (c+d x)} \, dx &=\frac{2 b (e \sin (c+d x))^{3/2}}{3 d e}+a \int \sqrt{e \sin (c+d x)} \, dx\\ &=\frac{2 b (e \sin (c+d x))^{3/2}}{3 d e}+\frac{\left (a \sqrt{e \sin (c+d x)}\right ) \int \sqrt{\sin (c+d x)} \, dx}{\sqrt{\sin (c+d x)}}\\ &=\frac{2 a E\left (\left .\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )\right |2\right ) \sqrt{e \sin (c+d x)}}{d \sqrt{\sin (c+d x)}}+\frac{2 b (e \sin (c+d x))^{3/2}}{3 d e}\\ \end{align*}
Mathematica [A] time = 0.097279, size = 60, normalized size = 0.88 \[ \frac{2 \sqrt{e \sin (c+d x)} \left (b \sin ^{\frac{3}{2}}(c+d x)-3 a E\left (\left .\frac{1}{4} (-2 c-2 d x+\pi )\right |2\right )\right )}{3 d \sqrt{\sin (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 1.609, size = 117, normalized size = 1.7 \begin{align*}{\frac{1}{d} \left ({\frac{2\,b}{3\,e} \left ( e\sin \left ( dx+c \right ) \right ) ^{{\frac{3}{2}}}}-{\frac{ae}{\cos \left ( dx+c \right ) }\sqrt{1-\sin \left ( dx+c \right ) }\sqrt{2+2\,\sin \left ( dx+c \right ) }\sqrt{\sin \left ( dx+c \right ) } \left ( 2\,{\it EllipticE} \left ( \sqrt{1-\sin \left ( dx+c \right ) },1/2\,\sqrt{2} \right ) -{\it EllipticF} \left ( \sqrt{1-\sin \left ( dx+c \right ) },{\frac{\sqrt{2}}{2}} \right ) \right ){\frac{1}{\sqrt{e\sin \left ( dx+c \right ) }}}} \right ) } \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 (b \cos \left (d x + c\right ) + a\right )} \sqrt{e \sin \left (d x + c\right )}\,{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 ({\left (b \cos \left (d x + c\right ) + a\right )} \sqrt{e \sin \left (d x + c\right )}, 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 \sqrt{e \sin{\left (c + d x \right )}} \left (a + b \cos{\left (c + d x \right )}\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{\left (b \cos \left (d x + c\right ) + a\right )} \sqrt{e \sin \left (d x + c\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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