Optimal. Leaf size=100 \[ \frac{2 a e^2 \sqrt{\sin (c+d x)} F\left (\left .\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )\right |2\right )}{3 d \sqrt{e \sin (c+d x)}}-\frac{2 a e \cos (c+d x) \sqrt{e \sin (c+d x)}}{3 d}+\frac{2 b (e \sin (c+d x))^{5/2}}{5 d e} \]
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Rubi [A] time = 0.0713243, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {2669, 2635, 2642, 2641} \[ \frac{2 a e^2 \sqrt{\sin (c+d x)} F\left (\left .\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )\right |2\right )}{3 d \sqrt{e \sin (c+d x)}}-\frac{2 a e \cos (c+d x) \sqrt{e \sin (c+d x)}}{3 d}+\frac{2 b (e \sin (c+d x))^{5/2}}{5 d e} \]
Antiderivative was successfully verified.
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Rule 2669
Rule 2635
Rule 2642
Rule 2641
Rubi steps
\begin{align*} \int (a+b \cos (c+d x)) (e \sin (c+d x))^{3/2} \, dx &=\frac{2 b (e \sin (c+d x))^{5/2}}{5 d e}+a \int (e \sin (c+d x))^{3/2} \, dx\\ &=-\frac{2 a e \cos (c+d x) \sqrt{e \sin (c+d x)}}{3 d}+\frac{2 b (e \sin (c+d x))^{5/2}}{5 d e}+\frac{1}{3} \left (a e^2\right ) \int \frac{1}{\sqrt{e \sin (c+d x)}} \, dx\\ &=-\frac{2 a e \cos (c+d x) \sqrt{e \sin (c+d x)}}{3 d}+\frac{2 b (e \sin (c+d x))^{5/2}}{5 d e}+\frac{\left (a e^2 \sqrt{\sin (c+d x)}\right ) \int \frac{1}{\sqrt{\sin (c+d x)}} \, dx}{3 \sqrt{e \sin (c+d x)}}\\ &=\frac{2 a e^2 F\left (\left .\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )\right |2\right ) \sqrt{\sin (c+d x)}}{3 d \sqrt{e \sin (c+d x)}}-\frac{2 a e \cos (c+d x) \sqrt{e \sin (c+d x)}}{3 d}+\frac{2 b (e \sin (c+d x))^{5/2}}{5 d e}\\ \end{align*}
Mathematica [A] time = 0.458702, size = 80, normalized size = 0.8 \[ \frac{2 (e \sin (c+d x))^{3/2} \left (\sqrt{\sin (c+d x)} \left (3 b \sin ^2(c+d x)-5 a \cos (c+d x)\right )-5 a F\left (\left .\frac{1}{4} (-2 c-2 d x+\pi )\right |2\right )\right )}{15 d \sin ^{\frac{3}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 1.426, size = 116, normalized size = 1.2 \begin{align*}{\frac{1}{d} \left ({\frac{2\,b}{5\,e} \left ( e\sin \left ( dx+c \right ) \right ) ^{{\frac{5}{2}}}}-{\frac{{e}^{2}a}{3\,\cos \left ( dx+c \right ) } \left ( \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 ) },{\frac{\sqrt{2}}{2}} \right ) -2\, \left ( \sin \left ( dx+c \right ) \right ) ^{3}+2\,\sin \left ( dx+c \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 )} \left (e \sin \left (d x + c\right )\right )^{\frac{3}{2}}\,{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 e \cos \left (d x + c\right ) + a e\right )} \sqrt{e \sin \left (d x + c\right )} \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 \left (e \sin{\left (c + d x \right )}\right )^{\frac{3}{2}} \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 )} \left (e \sin \left (d x + c\right )\right )^{\frac{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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