Optimal. Leaf size=92 \[ \frac{\cos (e+f x) (a \sin (e+f x)+a)^{5/2} \sqrt{c-c \sin (e+f x)}}{4 a f}+\frac{c \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{6 a f \sqrt{c-c \sin (e+f x)}} \]
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Rubi [A] time = 0.390201, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.079, Rules used = {2841, 2740, 2738} \[ \frac{\cos (e+f x) (a \sin (e+f x)+a)^{5/2} \sqrt{c-c \sin (e+f x)}}{4 a f}+\frac{c \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{6 a f \sqrt{c-c \sin (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 2841
Rule 2740
Rule 2738
Rubi steps
\begin{align*} \int \cos ^2(e+f x) (a+a \sin (e+f x))^{3/2} \sqrt{c-c \sin (e+f x)} \, dx &=\frac{\int (a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2} \, dx}{a c}\\ &=\frac{\cos (e+f x) (a+a \sin (e+f x))^{5/2} \sqrt{c-c \sin (e+f x)}}{4 a f}+\frac{\int (a+a \sin (e+f x))^{5/2} \sqrt{c-c \sin (e+f x)} \, dx}{2 a}\\ &=\frac{c \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{6 a f \sqrt{c-c \sin (e+f x)}}+\frac{\cos (e+f x) (a+a \sin (e+f x))^{5/2} \sqrt{c-c \sin (e+f x)}}{4 a f}\\ \end{align*}
Mathematica [A] time = 0.398599, size = 83, normalized size = 0.9 \[ \frac{a \sec (e+f x) \sqrt{a (\sin (e+f x)+1)} \sqrt{c-c \sin (e+f x)} (8 (9 \sin (e+f x)+\sin (3 (e+f x)))-12 \cos (2 (e+f x))-3 \cos (4 (e+f x)))}{96 f} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.216, size = 90, normalized size = 1. \begin{align*} -{\frac{\sin \left ( fx+e \right ) \left ( -3\, \left ( \cos \left ( fx+e \right ) \right ) ^{4}+ \left ( \cos \left ( fx+e \right ) \right ) ^{2}\sin \left ( fx+e \right ) -4\, \left ( \cos \left ( fx+e \right ) \right ) ^{2}+5\,\sin \left ( fx+e \right ) -5 \right ) }{12\,f \left ( \cos \left ( fx+e \right ) \right ) ^{3}}\sqrt{-c \left ( -1+\sin \left ( fx+e \right ) \right ) } \left ( a \left ( 1+\sin \left ( fx+e \right ) \right ) \right ) ^{{\frac{3}{2}}}} \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 (a \sin \left (f x + e\right ) + a\right )}^{\frac{3}{2}} \sqrt{-c \sin \left (f x + e\right ) + c} \cos \left (f x + e\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.6993, size = 194, normalized size = 2.11 \begin{align*} -\frac{{\left (3 \, a \cos \left (f x + e\right )^{4} - 4 \,{\left (a \cos \left (f x + e\right )^{2} + 2 \, a\right )} \sin \left (f x + e\right ) - 3 \, a\right )} \sqrt{a \sin \left (f x + e\right ) + a} \sqrt{-c \sin \left (f x + e\right ) + c}}{12 \, f \cos \left (f x + e\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [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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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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