Optimal. Leaf size=203 \[ -\frac{2 a^2 \sin ^4(c+d x) \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{11 d}-\frac{46 a^3 \sin ^4(c+d x) \cos (c+d x)}{99 d \sqrt{a \sin (c+d x)+a}}-\frac{710 a^3 \sin ^3(c+d x) \cos (c+d x)}{693 d \sqrt{a \sin (c+d x)+a}}+\frac{568 a^2 \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{693 d}-\frac{284 a^3 \cos (c+d x)}{99 d \sqrt{a \sin (c+d x)+a}}-\frac{284 a \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{231 d} \]
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Rubi [A] time = 0.351971, antiderivative size = 203, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261, Rules used = {2763, 2981, 2770, 2759, 2751, 2646} \[ -\frac{2 a^2 \sin ^4(c+d x) \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{11 d}-\frac{46 a^3 \sin ^4(c+d x) \cos (c+d x)}{99 d \sqrt{a \sin (c+d x)+a}}-\frac{710 a^3 \sin ^3(c+d x) \cos (c+d x)}{693 d \sqrt{a \sin (c+d x)+a}}+\frac{568 a^2 \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{693 d}-\frac{284 a^3 \cos (c+d x)}{99 d \sqrt{a \sin (c+d x)+a}}-\frac{284 a \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{231 d} \]
Antiderivative was successfully verified.
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Rule 2763
Rule 2981
Rule 2770
Rule 2759
Rule 2751
Rule 2646
Rubi steps
\begin{align*} \int \sin ^3(c+d x) (a+a \sin (c+d x))^{5/2} \, dx &=-\frac{2 a^2 \cos (c+d x) \sin ^4(c+d x) \sqrt{a+a \sin (c+d x)}}{11 d}+\frac{2}{11} \int \sin ^3(c+d x) \sqrt{a+a \sin (c+d x)} \left (\frac{19 a^2}{2}+\frac{23}{2} a^2 \sin (c+d x)\right ) \, dx\\ &=-\frac{46 a^3 \cos (c+d x) \sin ^4(c+d x)}{99 d \sqrt{a+a \sin (c+d x)}}-\frac{2 a^2 \cos (c+d x) \sin ^4(c+d x) \sqrt{a+a \sin (c+d x)}}{11 d}+\frac{1}{99} \left (355 a^2\right ) \int \sin ^3(c+d x) \sqrt{a+a \sin (c+d x)} \, dx\\ &=-\frac{710 a^3 \cos (c+d x) \sin ^3(c+d x)}{693 d \sqrt{a+a \sin (c+d x)}}-\frac{46 a^3 \cos (c+d x) \sin ^4(c+d x)}{99 d \sqrt{a+a \sin (c+d x)}}-\frac{2 a^2 \cos (c+d x) \sin ^4(c+d x) \sqrt{a+a \sin (c+d x)}}{11 d}+\frac{1}{231} \left (710 a^2\right ) \int \sin ^2(c+d x) \sqrt{a+a \sin (c+d x)} \, dx\\ &=-\frac{710 a^3 \cos (c+d x) \sin ^3(c+d x)}{693 d \sqrt{a+a \sin (c+d x)}}-\frac{46 a^3 \cos (c+d x) \sin ^4(c+d x)}{99 d \sqrt{a+a \sin (c+d x)}}-\frac{2 a^2 \cos (c+d x) \sin ^4(c+d x) \sqrt{a+a \sin (c+d x)}}{11 d}-\frac{284 a \cos (c+d x) (a+a \sin (c+d x))^{3/2}}{231 d}+\frac{1}{231} (284 a) \int \left (\frac{3 a}{2}-a \sin (c+d x)\right ) \sqrt{a+a \sin (c+d x)} \, dx\\ &=-\frac{710 a^3 \cos (c+d x) \sin ^3(c+d x)}{693 d \sqrt{a+a \sin (c+d x)}}-\frac{46 a^3 \cos (c+d x) \sin ^4(c+d x)}{99 d \sqrt{a+a \sin (c+d x)}}+\frac{568 a^2 \cos (c+d x) \sqrt{a+a \sin (c+d x)}}{693 d}-\frac{2 a^2 \cos (c+d x) \sin ^4(c+d x) \sqrt{a+a \sin (c+d x)}}{11 d}-\frac{284 a \cos (c+d x) (a+a \sin (c+d x))^{3/2}}{231 d}+\frac{1}{99} \left (142 a^2\right ) \int \sqrt{a+a \sin (c+d x)} \, dx\\ &=-\frac{284 a^3 \cos (c+d x)}{99 d \sqrt{a+a \sin (c+d x)}}-\frac{710 a^3 \cos (c+d x) \sin ^3(c+d x)}{693 d \sqrt{a+a \sin (c+d x)}}-\frac{46 a^3 \cos (c+d x) \sin ^4(c+d x)}{99 d \sqrt{a+a \sin (c+d x)}}+\frac{568 a^2 \cos (c+d x) \sqrt{a+a \sin (c+d x)}}{693 d}-\frac{2 a^2 \cos (c+d x) \sin ^4(c+d x) \sqrt{a+a \sin (c+d x)}}{11 d}-\frac{284 a \cos (c+d x) (a+a \sin (c+d x))^{3/2}}{231 d}\\ \end{align*}
Mathematica [A] time = 1.26886, size = 189, normalized size = 0.93 \[ -\frac{(a (\sin (c+d x)+1))^{5/2} \left (-31878 \sin \left (\frac{1}{2} (c+d x)\right )+8778 \sin \left (\frac{3}{2} (c+d x)\right )+3465 \sin \left (\frac{5}{2} (c+d x)\right )-1287 \sin \left (\frac{7}{2} (c+d x)\right )-385 \sin \left (\frac{9}{2} (c+d x)\right )+63 \sin \left (\frac{11}{2} (c+d x)\right )+31878 \cos \left (\frac{1}{2} (c+d x)\right )+8778 \cos \left (\frac{3}{2} (c+d x)\right )-3465 \cos \left (\frac{5}{2} (c+d x)\right )-1287 \cos \left (\frac{7}{2} (c+d x)\right )+385 \cos \left (\frac{9}{2} (c+d x)\right )+63 \cos \left (\frac{11}{2} (c+d x)\right )\right )}{11088 d \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.628, size = 95, normalized size = 0.5 \begin{align*}{\frac{ \left ( 2+2\,\sin \left ( dx+c \right ) \right ){a}^{3} \left ( \sin \left ( dx+c \right ) -1 \right ) \left ( 63\, \left ( \sin \left ( dx+c \right ) \right ) ^{5}+224\, \left ( \sin \left ( dx+c \right ) \right ) ^{4}+355\, \left ( \sin \left ( dx+c \right ) \right ) ^{3}+426\, \left ( \sin \left ( dx+c \right ) \right ) ^{2}+568\,\sin \left ( dx+c \right ) +1136 \right ) }{693\,d\cos \left ( dx+c \right ) }{\frac{1}{\sqrt{a+a\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 (a \sin \left (d x + c\right ) + a\right )}^{\frac{5}{2}} \sin \left (d x + c\right )^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.53847, size = 508, normalized size = 2.5 \begin{align*} -\frac{2 \,{\left (63 \, a^{2} \cos \left (d x + c\right )^{6} + 224 \, a^{2} \cos \left (d x + c\right )^{5} - 320 \, a^{2} \cos \left (d x + c\right )^{4} - 874 \, a^{2} \cos \left (d x + c\right )^{3} + 593 \, a^{2} \cos \left (d x + c\right )^{2} + 1786 \, a^{2} \cos \left (d x + c\right ) + 800 \, a^{2} +{\left (63 \, a^{2} \cos \left (d x + c\right )^{5} - 161 \, a^{2} \cos \left (d x + c\right )^{4} - 481 \, a^{2} \cos \left (d x + c\right )^{3} + 393 \, a^{2} \cos \left (d x + c\right )^{2} + 986 \, a^{2} \cos \left (d x + c\right ) - 800 \, a^{2}\right )} \sin \left (d x + c\right )\right )} \sqrt{a \sin \left (d x + c\right ) + a}}{693 \,{\left (d \cos \left (d x + c\right ) + d \sin \left (d x + c\right ) + d\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] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{5}{2}} \sin \left (d x + c\right )^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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