Optimal. Leaf size=24 \[ \frac {2 (a \sin (c+d x)+a)^{5/2}}{5 a d} \]
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Rubi [A] time = 0.03, antiderivative size = 24, normalized size of antiderivative = 1.00, 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 = {2667, 32} \[ \frac {2 (a \sin (c+d x)+a)^{5/2}}{5 a d} \]
Antiderivative was successfully verified.
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Rule 32
Rule 2667
Rubi steps
\begin {align*} \int \cos (c+d x) (a+a \sin (c+d x))^{3/2} \, dx &=\frac {\operatorname {Subst}\left (\int (a+x)^{3/2} \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac {2 (a+a \sin (c+d x))^{5/2}}{5 a d}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 24, normalized size = 1.00 \[ \frac {2 (a \sin (c+d x)+a)^{5/2}}{5 a d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.72, size = 40, normalized size = 1.67 \[ -\frac {2 \, {\left (a \cos \left (d x + c\right )^{2} - 2 \, a \sin \left (d x + c\right ) - 2 \, a\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{5 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.39, size = 133, normalized size = 5.54 \[ -\frac {1}{30} \, \sqrt {2} {\left (\frac {5 \, a \cos \left (\frac {1}{4} \, \pi + \frac {3}{2} \, d x + \frac {3}{2} \, c\right ) \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}{d} + \frac {3 \, a \cos \left (-\frac {1}{4} \, \pi + \frac {5}{2} \, d x + \frac {5}{2} \, c\right ) \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}{d} - \frac {30 \, a \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{d} - \frac {10 \, a \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {3}{2} \, d x + \frac {3}{2} \, c\right )}{d}\right )} \sqrt {a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 21, normalized size = 0.88 \[ \frac {2 \left (a +a \sin \left (d x +c \right )\right )^{\frac {5}{2}}}{5 d a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.51, size = 20, normalized size = 0.83 \[ \frac {2 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {5}{2}}}{5 \, a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.62, size = 20, normalized size = 0.83 \[ \frac {2\,{\left (a\,\left (\sin \left (c+d\,x\right )+1\right )\right )}^{5/2}}{5\,a\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 29.54, size = 90, normalized size = 3.75 \[ \begin {cases} \frac {2 a \sqrt {a \sin {\left (c + d x \right )} + a} \sin ^{2}{\left (c + d x \right )}}{5 d} + \frac {4 a \sqrt {a \sin {\left (c + d x \right )} + a} \sin {\left (c + d x \right )}}{5 d} + \frac {2 a \sqrt {a \sin {\left (c + d x \right )} + a}}{5 d} & \text {for}\: d \neq 0 \\x \left (a \sin {\relax (c )} + a\right )^{\frac {3}{2}} \cos {\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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