Optimal. Leaf size=134 \[ \frac{e (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^{m+1} F_1\left (m+1;-\frac{3}{4},-\frac{3}{4};m+2;\frac{a+b \sin (c+d x)}{a-b},\frac{a+b \sin (c+d x)}{a+b}\right )}{b d (m+1) \left (1-\frac{a+b \sin (c+d x)}{a-b}\right )^{3/4} \left (1-\frac{a+b \sin (c+d x)}{a+b}\right )^{3/4}} \]
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Rubi [A] time = 0.100995, antiderivative size = 134, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08, Rules used = {2704, 138} \[ \frac{e (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^{m+1} F_1\left (m+1;-\frac{3}{4},-\frac{3}{4};m+2;\frac{a+b \sin (c+d x)}{a-b},\frac{a+b \sin (c+d x)}{a+b}\right )}{b d (m+1) \left (1-\frac{a+b \sin (c+d x)}{a-b}\right )^{3/4} \left (1-\frac{a+b \sin (c+d x)}{a+b}\right )^{3/4}} \]
Antiderivative was successfully verified.
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Rule 2704
Rule 138
Rubi steps
\begin{align*} \int (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^m \, dx &=\frac{\left (e (e \cos (c+d x))^{3/2}\right ) \operatorname{Subst}\left (\int (a+b x)^m \left (-\frac{b}{a-b}-\frac{b x}{a-b}\right )^{3/4} \left (\frac{b}{a+b}-\frac{b x}{a+b}\right )^{3/4} \, dx,x,\sin (c+d x)\right )}{d \left (1-\frac{a+b \sin (c+d x)}{a-b}\right )^{3/4} \left (1-\frac{a+b \sin (c+d x)}{a+b}\right )^{3/4}}\\ &=\frac{e F_1\left (1+m;-\frac{3}{4},-\frac{3}{4};2+m;\frac{a+b \sin (c+d x)}{a-b},\frac{a+b \sin (c+d x)}{a+b}\right ) (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^{1+m}}{b d (1+m) \left (1-\frac{a+b \sin (c+d x)}{a-b}\right )^{3/4} \left (1-\frac{a+b \sin (c+d x)}{a+b}\right )^{3/4}}\\ \end{align*}
Mathematica [F] time = 56.6606, size = 0, normalized size = 0. \[ \int (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^m \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 0.121, size = 0, normalized size = 0. \begin{align*} \int \left ( e\cos \left ( dx+c \right ) \right ) ^{{\frac{5}{2}}} \left ( a+b\sin \left ( dx+c \right ) \right ) ^{m}\, 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 \left (e \cos \left (d x + c\right )\right )^{\frac{5}{2}}{\left (b \sin \left (d x + c\right ) + a\right )}^{m}\,{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 (\sqrt{e \cos \left (d x + c\right )}{\left (b \sin \left (d x + c\right ) + a\right )}^{m} e^{2} \cos \left (d x + c\right )^{2}, 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 \left (e \cos \left (d x + c\right )\right )^{\frac{5}{2}}{\left (b \sin \left (d x + c\right ) + a\right )}^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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