Optimal. Leaf size=154 \[ \frac {4 \left (3 a^2-b^2\right ) (a+b \sin (c+d x))^{7/2}}{7 b^5 d}-\frac {8 a \left (a^2-b^2\right ) (a+b \sin (c+d x))^{5/2}}{5 b^5 d}+\frac {2 \left (a^2-b^2\right )^2 (a+b \sin (c+d x))^{3/2}}{3 b^5 d}+\frac {2 (a+b \sin (c+d x))^{11/2}}{11 b^5 d}-\frac {8 a (a+b \sin (c+d x))^{9/2}}{9 b^5 d} \]
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Rubi [A] time = 0.12, antiderivative size = 154, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {2668, 697} \[ \frac {4 \left (3 a^2-b^2\right ) (a+b \sin (c+d x))^{7/2}}{7 b^5 d}-\frac {8 a \left (a^2-b^2\right ) (a+b \sin (c+d x))^{5/2}}{5 b^5 d}+\frac {2 \left (a^2-b^2\right )^2 (a+b \sin (c+d x))^{3/2}}{3 b^5 d}+\frac {2 (a+b \sin (c+d x))^{11/2}}{11 b^5 d}-\frac {8 a (a+b \sin (c+d x))^{9/2}}{9 b^5 d} \]
Antiderivative was successfully verified.
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Rule 697
Rule 2668
Rubi steps
\begin {align*} \int \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)} \, dx &=\frac {\operatorname {Subst}\left (\int \sqrt {a+x} \left (b^2-x^2\right )^2 \, dx,x,b \sin (c+d x)\right )}{b^5 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (\left (a^2-b^2\right )^2 \sqrt {a+x}-4 \left (a^3-a b^2\right ) (a+x)^{3/2}+2 \left (3 a^2-b^2\right ) (a+x)^{5/2}-4 a (a+x)^{7/2}+(a+x)^{9/2}\right ) \, dx,x,b \sin (c+d x)\right )}{b^5 d}\\ &=\frac {2 \left (a^2-b^2\right )^2 (a+b \sin (c+d x))^{3/2}}{3 b^5 d}-\frac {8 a \left (a^2-b^2\right ) (a+b \sin (c+d x))^{5/2}}{5 b^5 d}+\frac {4 \left (3 a^2-b^2\right ) (a+b \sin (c+d x))^{7/2}}{7 b^5 d}-\frac {8 a (a+b \sin (c+d x))^{9/2}}{9 b^5 d}+\frac {2 (a+b \sin (c+d x))^{11/2}}{11 b^5 d}\\ \end {align*}
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Mathematica [A] time = 0.34, size = 117, normalized size = 0.76 \[ \frac {2 (a+b \sin (c+d x))^{3/2} \left (8 \left (16 a^4+\left (99 a b^3-24 a^3 b\right ) \sin (c+d x)+15 b^2 \left (2 a^2-3 b^2\right ) \sin ^2(c+d x)-66 a^2 b^2-35 a b^3 \sin ^3(c+d x)+105 b^4\right )+315 b^4 \cos ^4(c+d x)\right )}{3465 b^5 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.78, size = 142, normalized size = 0.92 \[ \frac {2 \, {\left (35 \, a b^{4} \cos \left (d x + c\right )^{4} + 128 \, a^{5} - 480 \, a^{3} b^{2} + 992 \, a b^{4} - 16 \, {\left (3 \, a^{3} b^{2} - 8 \, a b^{4}\right )} \cos \left (d x + c\right )^{2} + {\left (315 \, b^{5} \cos \left (d x + c\right )^{4} - 64 \, a^{4} b + 224 \, a^{2} b^{3} + 480 \, b^{5} + 40 \, {\left (a^{2} b^{3} + 9 \, b^{5}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )\right )} \sqrt {b \sin \left (d x + c\right ) + a}}{3465 \, b^{5} d} \]
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 \[ \int \sqrt {b \sin \left (d x + c\right ) + a} \cos \left (d x + c\right )^{5}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.60, size = 126, normalized size = 0.82 \[ \frac {2 \left (a +b \sin \left (d x +c \right )\right )^{\frac {3}{2}} \left (315 b^{4} \left (\cos ^{4}\left (d x +c \right )\right )+280 a \,b^{3} \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )-240 a^{2} b^{2} \left (\cos ^{2}\left (d x +c \right )\right )+360 b^{4} \left (\cos ^{2}\left (d x +c \right )\right )-192 a^{3} b \sin \left (d x +c \right )+512 a \,b^{3} \sin \left (d x +c \right )+128 a^{4}-288 a^{2} b^{2}+480 b^{4}\right )}{3465 b^{5} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 116, normalized size = 0.75 \[ \frac {2 \, {\left (315 \, {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {11}{2}} - 1540 \, {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {9}{2}} a + 990 \, {\left (3 \, a^{2} - b^{2}\right )} {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {7}{2}} - 2772 \, {\left (a^{3} - a b^{2}\right )} {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {5}{2}} + 1155 \, {\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}}\right )}}{3465 \, b^{5} d} \]
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 \[ \int {\cos \left (c+d\,x\right )}^5\,\sqrt {a+b\,\sin \left (c+d\,x\right )} \,d x \]
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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