Optimal. Leaf size=24 \[ \frac{2 (a+b \sin (c+d x))^{5/2}}{5 b d} \]
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Rubi [A] time = 0.0387544, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {2668, 32} \[ \frac{2 (a+b \sin (c+d x))^{5/2}}{5 b d} \]
Antiderivative was successfully verified.
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Rule 2668
Rule 32
Rubi steps
\begin{align*} \int \cos (c+d x) (a+b \sin (c+d x))^{3/2} \, dx &=\frac{\operatorname{Subst}\left (\int (a+x)^{3/2} \, dx,x,b \sin (c+d x)\right )}{b d}\\ &=\frac{2 (a+b \sin (c+d x))^{5/2}}{5 b d}\\ \end{align*}
Mathematica [A] time = 0.0212864, size = 24, normalized size = 1. \[ \frac{2 (a+b \sin (c+d x))^{5/2}}{5 b d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 21, normalized size = 0.9 \begin{align*}{\frac{2}{5\,bd} \left ( a+b\sin \left ( dx+c \right ) \right ) ^{{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.949259, size = 27, normalized size = 1.12 \begin{align*} \frac{2 \,{\left (b \sin \left (d x + c\right ) + a\right )}^{\frac{5}{2}}}{5 \, b d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.35848, size = 123, normalized size = 5.12 \begin{align*} -\frac{2 \,{\left (b^{2} \cos \left (d x + c\right )^{2} - 2 \, a b \sin \left (d x + c\right ) - a^{2} - b^{2}\right )} \sqrt{b \sin \left (d x + c\right ) + a}}{5 \, b d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 27.4081, size = 116, normalized size = 4.83 \begin{align*} \begin{cases} a^{\frac{3}{2}} x \cos{\left (c \right )} & \text{for}\: b = 0 \wedge d = 0 \\x \left (a + b \sin{\left (c \right )}\right )^{\frac{3}{2}} \cos{\left (c \right )} & \text{for}\: d = 0 \\\frac{a^{\frac{3}{2}} \sin{\left (c + d x \right )}}{d} & \text{for}\: b = 0 \\\frac{2 a^{2} \sqrt{a + b \sin{\left (c + d x \right )}}}{5 b d} + \frac{4 a \sqrt{a + b \sin{\left (c + d x \right )}} \sin{\left (c + d x \right )}}{5 d} + \frac{2 b \sqrt{a + b \sin{\left (c + d x \right )}} \sin ^{2}{\left (c + d x \right )}}{5 d} & \text{otherwise} \end{cases} \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 (b \sin \left (d x + c\right ) + a\right )}^{\frac{3}{2}} \cos \left (d x + c\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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