Optimal. Leaf size=114 \[ \frac{2 \left (5 a^2+2 b^2\right ) E\left (\left .\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )\right |2\right ) \sqrt{e \sin (c+d x)}}{5 d \sqrt{\sin (c+d x)}}+\frac{14 a b (e \sin (c+d x))^{3/2}}{15 d e}+\frac{2 b (e \sin (c+d x))^{3/2} (a+b \cos (c+d x))}{5 d e} \]
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Rubi [A] time = 0.131006, antiderivative size = 114, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used = {2692, 2669, 2640, 2639} \[ \frac{2 \left (5 a^2+2 b^2\right ) E\left (\left .\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )\right |2\right ) \sqrt{e \sin (c+d x)}}{5 d \sqrt{\sin (c+d x)}}+\frac{14 a b (e \sin (c+d x))^{3/2}}{15 d e}+\frac{2 b (e \sin (c+d x))^{3/2} (a+b \cos (c+d x))}{5 d e} \]
Antiderivative was successfully verified.
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Rule 2692
Rule 2669
Rule 2640
Rule 2639
Rubi steps
\begin{align*} \int (a+b \cos (c+d x))^2 \sqrt{e \sin (c+d x)} \, dx &=\frac{2 b (a+b \cos (c+d x)) (e \sin (c+d x))^{3/2}}{5 d e}+\frac{2}{5} \int \left (\frac{5 a^2}{2}+b^2+\frac{7}{2} a b \cos (c+d x)\right ) \sqrt{e \sin (c+d x)} \, dx\\ &=\frac{14 a b (e \sin (c+d x))^{3/2}}{15 d e}+\frac{2 b (a+b \cos (c+d x)) (e \sin (c+d x))^{3/2}}{5 d e}+\frac{1}{5} \left (5 a^2+2 b^2\right ) \int \sqrt{e \sin (c+d x)} \, dx\\ &=\frac{14 a b (e \sin (c+d x))^{3/2}}{15 d e}+\frac{2 b (a+b \cos (c+d x)) (e \sin (c+d x))^{3/2}}{5 d e}+\frac{\left (\left (5 a^2+2 b^2\right ) \sqrt{e \sin (c+d x)}\right ) \int \sqrt{\sin (c+d x)} \, dx}{5 \sqrt{\sin (c+d x)}}\\ &=\frac{2 \left (5 a^2+2 b^2\right ) E\left (\left .\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )\right |2\right ) \sqrt{e \sin (c+d x)}}{5 d \sqrt{\sin (c+d x)}}+\frac{14 a b (e \sin (c+d x))^{3/2}}{15 d e}+\frac{2 b (a+b \cos (c+d x)) (e \sin (c+d x))^{3/2}}{5 d e}\\ \end{align*}
Mathematica [A] time = 0.270983, size = 83, normalized size = 0.73 \[ \frac{2 \sqrt{e \sin (c+d x)} \left (b \sin ^{\frac{3}{2}}(c+d x) (10 a+3 b \cos (c+d x))-3 \left (5 a^2+2 b^2\right ) E\left (\left .\frac{1}{4} (-2 c-2 d x+\pi )\right |2\right )\right )}{15 d \sqrt{\sin (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 1.878, size = 294, normalized size = 2.6 \begin{align*} -{\frac{e}{15\,d\cos \left ( dx+c \right ) } \left ( 30\,\sqrt{1-\sin \left ( dx+c \right ) }\sqrt{2+2\,\sin \left ( dx+c \right ) }\sqrt{\sin \left ( dx+c \right ) }{\it EllipticE} \left ( \sqrt{1-\sin \left ( dx+c \right ) },1/2\,\sqrt{2} \right ){a}^{2}+12\,\sqrt{1-\sin \left ( dx+c \right ) }\sqrt{2+2\,\sin \left ( dx+c \right ) }\sqrt{\sin \left ( dx+c \right ) }{\it EllipticE} \left ( \sqrt{1-\sin \left ( dx+c \right ) },1/2\,\sqrt{2} \right ){b}^{2}-15\,\sqrt{1-\sin \left ( dx+c \right ) }\sqrt{2+2\,\sin \left ( dx+c \right ) }\sqrt{\sin \left ( dx+c \right ) }{\it EllipticF} \left ( \sqrt{1-\sin \left ( dx+c \right ) },1/2\,\sqrt{2} \right ){a}^{2}-6\,\sqrt{1-\sin \left ( dx+c \right ) }\sqrt{2+2\,\sin \left ( dx+c \right ) }\sqrt{\sin \left ( dx+c \right ) }{\it EllipticF} \left ( \sqrt{1-\sin \left ( dx+c \right ) },1/2\,\sqrt{2} \right ){b}^{2}+6\,{b}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{4}+20\,ab \left ( \cos \left ( dx+c \right ) \right ) ^{3}-6\,{b}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{2}-20\,ab\cos \left ( dx+c \right ) \right ){\frac{1}{\sqrt{e\sin \left ( dx+c \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 )}^{2} \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^{2} \cos \left (d x + c\right )^{2} + 2 \, a b \cos \left (d x + c\right ) + a^{2}\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 )^{2}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [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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