Optimal. Leaf size=24 \[ \frac {2 (a+b \sin (c+d x))^{7/2}}{7 b d} \]
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Rubi [A] time = 0.04, 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 = {2668, 32} \[ \frac {2 (a+b \sin (c+d x))^{7/2}}{7 b d} \]
Antiderivative was successfully verified.
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Rule 32
Rule 2668
Rubi steps
\begin {align*} \int \cos (c+d x) (a+b \sin (c+d x))^{5/2} \, dx &=\frac {\operatorname {Subst}\left (\int (a+x)^{5/2} \, dx,x,b \sin (c+d x)\right )}{b d}\\ &=\frac {2 (a+b \sin (c+d x))^{7/2}}{7 b d}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 24, normalized size = 1.00 \[ \frac {2 (a+b \sin (c+d x))^{7/2}}{7 b d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.76, size = 77, normalized size = 3.21 \[ -\frac {2 \, {\left (3 \, a b^{2} \cos \left (d x + c\right )^{2} - a^{3} - 3 \, a b^{2} + {\left (b^{3} \cos \left (d x + c\right )^{2} - 3 \, a^{2} b - b^{3}\right )} \sin \left (d x + c\right )\right )} \sqrt {b \sin \left (d x + c\right ) + a}}{7 \, b d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \cos \left (d x + c\right )\,{d x} \]
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 \[ \frac {2 \left (a +b \sin \left (d x +c \right )\right )^{\frac {7}{2}}}{7 b d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 20, normalized size = 0.83 \[ \frac {2 \, {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {7}{2}}}{7 \, b d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.57, size = 20, normalized size = 0.83 \[ \frac {2\,{\left (a+b\,\sin \left (c+d\,x\right )\right )}^{7/2}}{7\,b\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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