Optimal. Leaf size=264 \[ \frac{2 \left (9 a^2 A+14 a b B+7 A b^2\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 d}+\frac{2 \left (9 a^2 A+14 a b B+7 A b^2\right ) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{45 d}+\frac{10 \left (11 a (a B+2 A b)+9 b^2 B\right ) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{231 d}+\frac{2 \left (11 a (a B+2 A b)+9 b^2 B\right ) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{77 d}+\frac{10 \left (11 a (a B+2 A b)+9 b^2 B\right ) \sin (c+d x) \sqrt{\cos (c+d x)}}{231 d}+\frac{2 b (13 a B+11 A b) \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{99 d}+\frac{2 b B \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x) (a+b \cos (c+d x))}{11 d} \]
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Rubi [A] time = 0.381274, antiderivative size = 264, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {2990, 3023, 2748, 2635, 2639, 2641} \[ \frac{2 \left (9 a^2 A+14 a b B+7 A b^2\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 d}+\frac{2 \left (9 a^2 A+14 a b B+7 A b^2\right ) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{45 d}+\frac{10 \left (11 a (a B+2 A b)+9 b^2 B\right ) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{231 d}+\frac{2 \left (11 a (a B+2 A b)+9 b^2 B\right ) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{77 d}+\frac{10 \left (11 a (a B+2 A b)+9 b^2 B\right ) \sin (c+d x) \sqrt{\cos (c+d x)}}{231 d}+\frac{2 b (13 a B+11 A b) \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{99 d}+\frac{2 b B \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x) (a+b \cos (c+d x))}{11 d} \]
Antiderivative was successfully verified.
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Rule 2990
Rule 3023
Rule 2748
Rule 2635
Rule 2639
Rule 2641
Rubi steps
\begin{align*} \int \cos ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))^2 (A+B \cos (c+d x)) \, dx &=\frac{2 b B \cos ^{\frac{7}{2}}(c+d x) (a+b \cos (c+d x)) \sin (c+d x)}{11 d}+\frac{2}{11} \int \cos ^{\frac{5}{2}}(c+d x) \left (\frac{1}{2} a (11 a A+7 b B)+\frac{1}{2} \left (9 b^2 B+11 a (2 A b+a B)\right ) \cos (c+d x)+\frac{1}{2} b (11 A b+13 a B) \cos ^2(c+d x)\right ) \, dx\\ &=\frac{2 b (11 A b+13 a B) \cos ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{99 d}+\frac{2 b B \cos ^{\frac{7}{2}}(c+d x) (a+b \cos (c+d x)) \sin (c+d x)}{11 d}+\frac{4}{99} \int \cos ^{\frac{5}{2}}(c+d x) \left (\frac{11}{4} \left (9 a^2 A+7 A b^2+14 a b B\right )+\frac{9}{4} \left (9 b^2 B+11 a (2 A b+a B)\right ) \cos (c+d x)\right ) \, dx\\ &=\frac{2 b (11 A b+13 a B) \cos ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{99 d}+\frac{2 b B \cos ^{\frac{7}{2}}(c+d x) (a+b \cos (c+d x)) \sin (c+d x)}{11 d}+\frac{1}{9} \left (9 a^2 A+7 A b^2+14 a b B\right ) \int \cos ^{\frac{5}{2}}(c+d x) \, dx+\frac{1}{11} \left (9 b^2 B+11 a (2 A b+a B)\right ) \int \cos ^{\frac{7}{2}}(c+d x) \, dx\\ &=\frac{2 \left (9 a^2 A+7 A b^2+14 a b B\right ) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{45 d}+\frac{2 \left (9 b^2 B+11 a (2 A b+a B)\right ) \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{77 d}+\frac{2 b (11 A b+13 a B) \cos ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{99 d}+\frac{2 b B \cos ^{\frac{7}{2}}(c+d x) (a+b \cos (c+d x)) \sin (c+d x)}{11 d}+\frac{1}{15} \left (9 a^2 A+7 A b^2+14 a b B\right ) \int \sqrt{\cos (c+d x)} \, dx+\frac{1}{77} \left (5 \left (9 b^2 B+11 a (2 A b+a B)\right )\right ) \int \cos ^{\frac{3}{2}}(c+d x) \, dx\\ &=\frac{2 \left (9 a^2 A+7 A b^2+14 a b B\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 d}+\frac{10 \left (9 b^2 B+11 a (2 A b+a B)\right ) \sqrt{\cos (c+d x)} \sin (c+d x)}{231 d}+\frac{2 \left (9 a^2 A+7 A b^2+14 a b B\right ) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{45 d}+\frac{2 \left (9 b^2 B+11 a (2 A b+a B)\right ) \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{77 d}+\frac{2 b (11 A b+13 a B) \cos ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{99 d}+\frac{2 b B \cos ^{\frac{7}{2}}(c+d x) (a+b \cos (c+d x)) \sin (c+d x)}{11 d}+\frac{1}{231} \left (5 \left (9 b^2 B+11 a (2 A b+a B)\right )\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx\\ &=\frac{2 \left (9 a^2 A+7 A b^2+14 a b B\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 d}+\frac{10 \left (9 b^2 B+11 a (2 A b+a B)\right ) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{231 d}+\frac{10 \left (9 b^2 B+11 a (2 A b+a B)\right ) \sqrt{\cos (c+d x)} \sin (c+d x)}{231 d}+\frac{2 \left (9 a^2 A+7 A b^2+14 a b B\right ) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{45 d}+\frac{2 \left (9 b^2 B+11 a (2 A b+a B)\right ) \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{77 d}+\frac{2 b (11 A b+13 a B) \cos ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{99 d}+\frac{2 b B \cos ^{\frac{7}{2}}(c+d x) (a+b \cos (c+d x)) \sin (c+d x)}{11 d}\\ \end{align*}
Mathematica [A] time = 1.71103, size = 196, normalized size = 0.74 \[ \frac{1200 \left (11 a^2 B+22 a A b+9 b^2 B\right ) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )+3696 \left (9 a^2 A+14 a b B+7 A b^2\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )+2 \sin (c+d x) \sqrt{\cos (c+d x)} \left (154 \left (36 a^2 A+86 a b B+43 A b^2\right ) \cos (c+d x)+180 \left (11 a^2 B+22 a A b+16 b^2 B\right ) \cos (2 (c+d x))+15 \left (572 a^2 B+1144 a A b+21 b^2 B \cos (4 (c+d x))+531 b^2 B\right )+770 b (2 a B+A b) \cos (3 (c+d x))\right )}{27720 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 3.161, size = 666, normalized size = 2.5 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (B \cos \left (d x + c\right ) + A\right )}{\left (b \cos \left (d x + c\right ) + a\right )}^{2} \cos \left (d x + c\right )^{\frac{5}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (B b^{2} \cos \left (d x + c\right )^{5} + A a^{2} \cos \left (d x + c\right )^{2} +{\left (2 \, B a b + A b^{2}\right )} \cos \left (d x + c\right )^{4} +{\left (B a^{2} + 2 \, A a b\right )} \cos \left (d x + c\right )^{3}\right )} \sqrt{\cos \left (d x + c\right )}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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