Optimal. Leaf size=85 \[ \frac{2^{m+\frac{1}{4}} (\sin (c+d x)+1)^{\frac{3}{4}-m} (a \sin (c+d x)+a)^m \, _2F_1\left (-\frac{3}{4},\frac{7}{4}-m;\frac{1}{4};\frac{1}{2} (1-\sin (c+d x))\right )}{3 d e (e \cos (c+d x))^{3/2}} \]
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Rubi [A] time = 0.0941946, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {2689, 70, 69} \[ \frac{2^{m+\frac{1}{4}} (\sin (c+d x)+1)^{\frac{3}{4}-m} (a \sin (c+d x)+a)^m \, _2F_1\left (-\frac{3}{4},\frac{7}{4}-m;\frac{1}{4};\frac{1}{2} (1-\sin (c+d x))\right )}{3 d e (e \cos (c+d x))^{3/2}} \]
Antiderivative was successfully verified.
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Rule 2689
Rule 70
Rule 69
Rubi steps
\begin{align*} \int \frac{(a+a \sin (c+d x))^m}{(e \cos (c+d x))^{5/2}} \, dx &=\frac{\left (a^2 (a-a \sin (c+d x))^{3/4} (a+a \sin (c+d x))^{3/4}\right ) \operatorname{Subst}\left (\int \frac{(a+a x)^{-\frac{7}{4}+m}}{(a-a x)^{7/4}} \, dx,x,\sin (c+d x)\right )}{d e (e \cos (c+d x))^{3/2}}\\ &=\frac{\left (2^{-\frac{7}{4}+m} a (a-a \sin (c+d x))^{3/4} (a+a \sin (c+d x))^m \left (\frac{a+a \sin (c+d x)}{a}\right )^{\frac{3}{4}-m}\right ) \operatorname{Subst}\left (\int \frac{\left (\frac{1}{2}+\frac{x}{2}\right )^{-\frac{7}{4}+m}}{(a-a x)^{7/4}} \, dx,x,\sin (c+d x)\right )}{d e (e \cos (c+d x))^{3/2}}\\ &=\frac{2^{\frac{1}{4}+m} \, _2F_1\left (-\frac{3}{4},\frac{7}{4}-m;\frac{1}{4};\frac{1}{2} (1-\sin (c+d x))\right ) (1+\sin (c+d x))^{\frac{3}{4}-m} (a+a \sin (c+d x))^m}{3 d e (e \cos (c+d x))^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.115325, size = 85, normalized size = 1. \[ \frac{2^{m+\frac{1}{4}} (\sin (c+d x)+1)^{\frac{3}{4}-m} (a (\sin (c+d x)+1))^m \, _2F_1\left (-\frac{3}{4},\frac{7}{4}-m;\frac{1}{4};\frac{1}{2} (1-\sin (c+d x))\right )}{3 d e (e \cos (c+d x))^{3/2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.087, size = 0, normalized size = 0. \begin{align*} \int{ \left ( a+a\sin \left ( dx+c \right ) \right ) ^{m} \left ( e\cos \left ( dx+c \right ) \right ) ^{-{\frac{5}{2}}}}\, dx \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 )}^{m}}{\left (e \cos \left (d x + c\right )\right )^{\frac{5}{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 (\frac{\sqrt{e \cos \left (d x + c\right )}{\left (a \sin \left (d x + c\right ) + a\right )}^{m}}{e^{3} \cos \left (d x + c\right )^{3}}, x\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 \frac{{\left (a \sin \left (d x + c\right ) + a\right )}^{m}}{\left (e \cos \left (d x + c\right )\right )^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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