Optimal. Leaf size=89 \[ -\frac {64 a^3 \cos (c+d x)}{15 d \sqrt {a+a \sin (c+d x)}}-\frac {16 a^2 \cos (c+d x) \sqrt {a+a \sin (c+d x)}}{15 d}-\frac {2 a \cos (c+d x) (a+a \sin (c+d x))^{3/2}}{5 d} \]
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Rubi [A]
time = 0.04, antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2726, 2725}
\begin {gather*} -\frac {64 a^3 \cos (c+d x)}{15 d \sqrt {a \sin (c+d x)+a}}-\frac {16 a^2 \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{15 d}-\frac {2 a \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{5 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 2725
Rule 2726
Rubi steps
\begin {align*} \int (a+a \sin (c+d x))^{5/2} \, dx &=-\frac {2 a \cos (c+d x) (a+a \sin (c+d x))^{3/2}}{5 d}+\frac {1}{5} (8 a) \int (a+a \sin (c+d x))^{3/2} \, dx\\ &=-\frac {16 a^2 \cos (c+d x) \sqrt {a+a \sin (c+d x)}}{15 d}-\frac {2 a \cos (c+d x) (a+a \sin (c+d x))^{3/2}}{5 d}+\frac {1}{15} \left (32 a^2\right ) \int \sqrt {a+a \sin (c+d x)} \, dx\\ &=-\frac {64 a^3 \cos (c+d x)}{15 d \sqrt {a+a \sin (c+d x)}}-\frac {16 a^2 \cos (c+d x) \sqrt {a+a \sin (c+d x)}}{15 d}-\frac {2 a \cos (c+d x) (a+a \sin (c+d x))^{3/2}}{5 d}\\ \end {align*}
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Mathematica [A]
time = 0.21, size = 117, normalized size = 1.31 \begin {gather*} -\frac {(a (1+\sin (c+d x)))^{5/2} \left (150 \cos \left (\frac {1}{2} (c+d x)\right )+25 \cos \left (\frac {3}{2} (c+d x)\right )-3 \cos \left (\frac {5}{2} (c+d x)\right )-150 \sin \left (\frac {1}{2} (c+d x)\right )+25 \sin \left (\frac {3}{2} (c+d x)\right )+3 \sin \left (\frac {5}{2} (c+d x)\right )\right )}{30 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^5} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.88, size = 65, normalized size = 0.73
method | result | size |
default | \(\frac {2 \left (1+\sin \left (d x +c \right )\right ) a^{3} \left (\sin \left (d x +c \right )-1\right ) \left (3 \left (\sin ^{2}\left (d x +c \right )\right )+14 \sin \left (d x +c \right )+43\right )}{15 \cos \left (d x +c \right ) \sqrt {a +a \sin \left (d x +c \right )}\, d}\) | \(65\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.40, size = 115, normalized size = 1.29 \begin {gather*} \frac {2 \, {\left (3 \, a^{2} \cos \left (d x + c\right )^{3} - 11 \, a^{2} \cos \left (d x + c\right )^{2} - 46 \, a^{2} \cos \left (d x + c\right ) - 32 \, a^{2} - {\left (3 \, a^{2} \cos \left (d x + c\right )^{2} + 14 \, a^{2} \cos \left (d x + c\right ) - 32 \, a^{2}\right )} \sin \left (d x + c\right )\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{15 \, {\left (d \cos \left (d x + c\right ) + d \sin \left (d x + c\right ) + d\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a \sin {\left (c + d x \right )} + a\right )^{\frac {5}{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.51, size = 102, normalized size = 1.15 \begin {gather*} \frac {\sqrt {2} {\left (150 \, a^{2} \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 ) + 25 \, a^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {3}{4} \, \pi + \frac {3}{2} \, d x + \frac {3}{2} \, c\right ) + 3 \, a^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {5}{4} \, \pi + \frac {5}{2} \, d x + \frac {5}{2} \, c\right )\right )} \sqrt {a}}{30 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\left (a+a\,\sin \left (c+d\,x\right )\right )}^{5/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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