Optimal. Leaf size=299 \[ \frac {2 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (3 b \left (25 a^2+7 b^2\right ) \sin (c+d x)+a \left (5 a^2+27 b^2\right )\right )}{315 b d}-\frac {4 a \left (5 a^4+22 a^2 b^2-27 b^4\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} F\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{315 b^2 d \sqrt {a+b \sin (c+d x)}}+\frac {4 \left (5 a^4+102 a^2 b^2+21 b^4\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{315 b^2 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {2 b \cos ^3(c+d x) (a+b \sin (c+d x))^{3/2}}{9 d}-\frac {8 a b \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)}}{21 d} \]
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Rubi [A] time = 0.67, antiderivative size = 299, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {2692, 2862, 2865, 2752, 2663, 2661, 2655, 2653} \[ \frac {2 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (3 b \left (25 a^2+7 b^2\right ) \sin (c+d x)+a \left (5 a^2+27 b^2\right )\right )}{315 b d}-\frac {4 a \left (22 a^2 b^2+5 a^4-27 b^4\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} F\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{315 b^2 d \sqrt {a+b \sin (c+d x)}}+\frac {4 \left (102 a^2 b^2+5 a^4+21 b^4\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{315 b^2 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {2 b \cos ^3(c+d x) (a+b \sin (c+d x))^{3/2}}{9 d}-\frac {8 a b \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)}}{21 d} \]
Antiderivative was successfully verified.
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Rule 2653
Rule 2655
Rule 2661
Rule 2663
Rule 2692
Rule 2752
Rule 2862
Rule 2865
Rubi steps
\begin {align*} \int \cos ^2(c+d x) (a+b \sin (c+d x))^{5/2} \, dx &=-\frac {2 b \cos ^3(c+d x) (a+b \sin (c+d x))^{3/2}}{9 d}+\frac {2}{9} \int \cos ^2(c+d x) \sqrt {a+b \sin (c+d x)} \left (\frac {9 a^2}{2}+\frac {3 b^2}{2}+6 a b \sin (c+d x)\right ) \, dx\\ &=-\frac {8 a b \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)}}{21 d}-\frac {2 b \cos ^3(c+d x) (a+b \sin (c+d x))^{3/2}}{9 d}+\frac {4}{63} \int \frac {\cos ^2(c+d x) \left (\frac {3}{4} a \left (21 a^2+11 b^2\right )+\frac {3}{4} b \left (25 a^2+7 b^2\right ) \sin (c+d x)\right )}{\sqrt {a+b \sin (c+d x)}} \, dx\\ &=-\frac {8 a b \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)}}{21 d}-\frac {2 b \cos ^3(c+d x) (a+b \sin (c+d x))^{3/2}}{9 d}+\frac {2 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (a \left (5 a^2+27 b^2\right )+3 b \left (25 a^2+7 b^2\right ) \sin (c+d x)\right )}{315 b d}+\frac {16 \int \frac {6 a b^2 \left (5 a^2+3 b^2\right )+\frac {3}{8} b \left (5 a^4+102 a^2 b^2+21 b^4\right ) \sin (c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx}{945 b^2}\\ &=-\frac {8 a b \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)}}{21 d}-\frac {2 b \cos ^3(c+d x) (a+b \sin (c+d x))^{3/2}}{9 d}+\frac {2 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (a \left (5 a^2+27 b^2\right )+3 b \left (25 a^2+7 b^2\right ) \sin (c+d x)\right )}{315 b d}-\frac {\left (2 a \left (5 a^4+22 a^2 b^2-27 b^4\right )\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}} \, dx}{315 b^2}+\frac {\left (2 \left (5 a^4+102 a^2 b^2+21 b^4\right )\right ) \int \sqrt {a+b \sin (c+d x)} \, dx}{315 b^2}\\ &=-\frac {8 a b \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)}}{21 d}-\frac {2 b \cos ^3(c+d x) (a+b \sin (c+d x))^{3/2}}{9 d}+\frac {2 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (a \left (5 a^2+27 b^2\right )+3 b \left (25 a^2+7 b^2\right ) \sin (c+d x)\right )}{315 b d}+\frac {\left (2 \left (5 a^4+102 a^2 b^2+21 b^4\right ) \sqrt {a+b \sin (c+d x)}\right ) \int \sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}} \, dx}{315 b^2 \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {\left (2 a \left (5 a^4+22 a^2 b^2-27 b^4\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}\right ) \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}} \, dx}{315 b^2 \sqrt {a+b \sin (c+d x)}}\\ &=-\frac {8 a b \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)}}{21 d}-\frac {2 b \cos ^3(c+d x) (a+b \sin (c+d x))^{3/2}}{9 d}+\frac {4 \left (5 a^4+102 a^2 b^2+21 b^4\right ) E\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )|\frac {2 b}{a+b}\right ) \sqrt {a+b \sin (c+d x)}}{315 b^2 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {4 a \left (5 a^4+22 a^2 b^2-27 b^4\right ) F\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )|\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{315 b^2 d \sqrt {a+b \sin (c+d x)}}+\frac {2 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (a \left (5 a^2+27 b^2\right )+3 b \left (25 a^2+7 b^2\right ) \sin (c+d x)\right )}{315 b d}\\ \end {align*}
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Mathematica [A] time = 1.04, size = 239, normalized size = 0.80 \[ \frac {b (a+b \sin (c+d x)) \left (\left (40 a^3-354 a b^2\right ) \cos (c+d x)+2 b \left (\sin (2 (c+d x)) \left (150 a^2-35 b^2 \cos (2 (c+d x))+7 b^2\right )-95 a b \cos (3 (c+d x))\right )\right )-16 \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \left (16 b \left (5 a^3 b+3 a b^3\right ) F\left (\frac {1}{4} (-2 c-2 d x+\pi )|\frac {2 b}{a+b}\right )+\left (5 a^4+102 a^2 b^2+21 b^4\right ) \left ((a+b) E\left (\frac {1}{4} (-2 c-2 d x+\pi )|\frac {2 b}{a+b}\right )-a F\left (\frac {1}{4} (-2 c-2 d x+\pi )|\frac {2 b}{a+b}\right )\right )\right )}{1260 b^2 d \sqrt {a+b \sin (c+d x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.02, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-{\left (b^{2} \cos \left (d x + c\right )^{4} - 2 \, a b \cos \left (d x + c\right )^{2} \sin \left (d x + c\right ) - {\left (a^{2} + b^{2}\right )} \cos \left (d x + c\right )^{2}\right )} \sqrt {b \sin \left (d x + c\right ) + a}, x\right ) \]
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 \[ \int {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \cos \left (d x + c\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.89, size = 1190, normalized size = 3.98 \[ \text {result too large to display} \]
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 \[ \int {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \cos \left (d x + c\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int {\cos \left (c+d\,x\right )}^2\,{\left (a+b\,\sin \left (c+d\,x\right )\right )}^{5/2} \,d x \]
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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