Optimal. Leaf size=247 \[ -\frac{7 a^2 \sqrt{a \sin (c+d x)+a} \sqrt{e \cos (c+d x)}}{4 d e}+\frac{21 a^2 \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \tan ^{-1}\left (\frac{\sqrt{e} \sin (c+d x)}{\sqrt{\cos (c+d x)+1} \sqrt{e \cos (c+d x)}}\right )}{4 d \sqrt{e} (\sin (c+d x)+\cos (c+d x)+1)}-\frac{21 a^2 \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \sinh ^{-1}\left (\frac{\sqrt{e \cos (c+d x)}}{\sqrt{e}}\right )}{4 d \sqrt{e} (\sin (c+d x)+\cos (c+d x)+1)}-\frac{a (a \sin (c+d x)+a)^{3/2} \sqrt{e \cos (c+d x)}}{2 d e} \]
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Rubi [A] time = 0.358691, antiderivative size = 247, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.259, Rules used = {2678, 2677, 2775, 203, 2833, 63, 215} \[ -\frac{7 a^2 \sqrt{a \sin (c+d x)+a} \sqrt{e \cos (c+d x)}}{4 d e}+\frac{21 a^2 \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \tan ^{-1}\left (\frac{\sqrt{e} \sin (c+d x)}{\sqrt{\cos (c+d x)+1} \sqrt{e \cos (c+d x)}}\right )}{4 d \sqrt{e} (\sin (c+d x)+\cos (c+d x)+1)}-\frac{21 a^2 \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \sinh ^{-1}\left (\frac{\sqrt{e \cos (c+d x)}}{\sqrt{e}}\right )}{4 d \sqrt{e} (\sin (c+d x)+\cos (c+d x)+1)}-\frac{a (a \sin (c+d x)+a)^{3/2} \sqrt{e \cos (c+d x)}}{2 d e} \]
Antiderivative was successfully verified.
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Rule 2678
Rule 2677
Rule 2775
Rule 203
Rule 2833
Rule 63
Rule 215
Rubi steps
\begin{align*} \int \frac{(a+a \sin (c+d x))^{5/2}}{\sqrt{e \cos (c+d x)}} \, dx &=-\frac{a \sqrt{e \cos (c+d x)} (a+a \sin (c+d x))^{3/2}}{2 d e}+\frac{1}{4} (7 a) \int \frac{(a+a \sin (c+d x))^{3/2}}{\sqrt{e \cos (c+d x)}} \, dx\\ &=-\frac{7 a^2 \sqrt{e \cos (c+d x)} \sqrt{a+a \sin (c+d x)}}{4 d e}-\frac{a \sqrt{e \cos (c+d x)} (a+a \sin (c+d x))^{3/2}}{2 d e}+\frac{1}{8} \left (21 a^2\right ) \int \frac{\sqrt{a+a \sin (c+d x)}}{\sqrt{e \cos (c+d x)}} \, dx\\ &=-\frac{7 a^2 \sqrt{e \cos (c+d x)} \sqrt{a+a \sin (c+d x)}}{4 d e}-\frac{a \sqrt{e \cos (c+d x)} (a+a \sin (c+d x))^{3/2}}{2 d e}+\frac{\left (21 a^3 \sqrt{1+\cos (c+d x)} \sqrt{a+a \sin (c+d x)}\right ) \int \frac{\sqrt{1+\cos (c+d x)}}{\sqrt{e \cos (c+d x)}} \, dx}{8 (a+a \cos (c+d x)+a \sin (c+d x))}+\frac{\left (21 a^3 \sqrt{1+\cos (c+d x)} \sqrt{a+a \sin (c+d x)}\right ) \int \frac{\sin (c+d x)}{\sqrt{e \cos (c+d x)} \sqrt{1+\cos (c+d x)}} \, dx}{8 (a+a \cos (c+d x)+a \sin (c+d x))}\\ &=-\frac{7 a^2 \sqrt{e \cos (c+d x)} \sqrt{a+a \sin (c+d x)}}{4 d e}-\frac{a \sqrt{e \cos (c+d x)} (a+a \sin (c+d x))^{3/2}}{2 d e}-\frac{\left (21 a^3 \sqrt{1+\cos (c+d x)} \sqrt{a+a \sin (c+d x)}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{e x} \sqrt{1+x}} \, dx,x,\cos (c+d x)\right )}{8 d (a+a \cos (c+d x)+a \sin (c+d x))}-\frac{\left (21 a^3 \sqrt{1+\cos (c+d x)} \sqrt{a+a \sin (c+d x)}\right ) \operatorname{Subst}\left (\int \frac{1}{1+e x^2} \, dx,x,-\frac{\sin (c+d x)}{\sqrt{e \cos (c+d x)} \sqrt{1+\cos (c+d x)}}\right )}{4 d (a+a \cos (c+d x)+a \sin (c+d x))}\\ &=-\frac{7 a^2 \sqrt{e \cos (c+d x)} \sqrt{a+a \sin (c+d x)}}{4 d e}-\frac{a \sqrt{e \cos (c+d x)} (a+a \sin (c+d x))^{3/2}}{2 d e}+\frac{21 a^3 \tan ^{-1}\left (\frac{\sqrt{e} \sin (c+d x)}{\sqrt{e \cos (c+d x)} \sqrt{1+\cos (c+d x)}}\right ) \sqrt{1+\cos (c+d x)} \sqrt{a+a \sin (c+d x)}}{4 d \sqrt{e} (a+a \cos (c+d x)+a \sin (c+d x))}-\frac{\left (21 a^3 \sqrt{1+\cos (c+d x)} \sqrt{a+a \sin (c+d x)}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{e}}} \, dx,x,\sqrt{e \cos (c+d x)}\right )}{4 d e (a+a \cos (c+d x)+a \sin (c+d x))}\\ &=-\frac{7 a^2 \sqrt{e \cos (c+d x)} \sqrt{a+a \sin (c+d x)}}{4 d e}-\frac{a \sqrt{e \cos (c+d x)} (a+a \sin (c+d x))^{3/2}}{2 d e}-\frac{21 a^3 \sinh ^{-1}\left (\frac{\sqrt{e \cos (c+d x)}}{\sqrt{e}}\right ) \sqrt{1+\cos (c+d x)} \sqrt{a+a \sin (c+d x)}}{4 d \sqrt{e} (a+a \cos (c+d x)+a \sin (c+d x))}+\frac{21 a^3 \tan ^{-1}\left (\frac{\sqrt{e} \sin (c+d x)}{\sqrt{e \cos (c+d x)} \sqrt{1+\cos (c+d x)}}\right ) \sqrt{1+\cos (c+d x)} \sqrt{a+a \sin (c+d x)}}{4 d \sqrt{e} (a+a \cos (c+d x)+a \sin (c+d x))}\\ \end{align*}
Mathematica [C] time = 0.103092, size = 76, normalized size = 0.31 \[ -\frac{8\ 2^{3/4} a (a (\sin (c+d x)+1))^{3/2} \sqrt{e \cos (c+d x)} \, _2F_1\left (-\frac{7}{4},\frac{1}{4};\frac{5}{4};\frac{1}{2} (1-\sin (c+d x))\right )}{d e (\sin (c+d x)+1)^{7/4}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.169, size = 284, normalized size = 1.2 \begin{align*} -{\frac{1}{8\,d \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{2}\sin \left ( dx+c \right ) - \left ( \cos \left ( dx+c \right ) \right ) ^{3}+2\,\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) +3\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}-4\,\sin \left ( dx+c \right ) +2\,\cos \left ( dx+c \right ) -4 \right ) } \left ( a \left ( 1+\sin \left ( dx+c \right ) \right ) \right ) ^{{\frac{5}{2}}} \left ( 21\,\sqrt{-2\,{\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }}}\arctan \left ( 1/2\,\sqrt{2}\sqrt{-2\,{\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }}} \right ) \sqrt{2}\sin \left ( dx+c \right ) -21\,\sqrt{-2\,{\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }}}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{2}\sin \left ( dx+c \right ) }{\cos \left ( dx+c \right ) }\sqrt{-2\,{\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }}}} \right ) \sqrt{2}\sin \left ( dx+c \right ) +4\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}\sin \left ( dx+c \right ) +4\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}-22\,\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) +18\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}-22\,\cos \left ( dx+c \right ) \right ){\frac{1}{\sqrt{e\cos \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 \frac{{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{5}{2}}}{\sqrt{e \cos \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(-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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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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{5}{2}}}{\sqrt{e \cos \left (d x + c\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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