Optimal. Leaf size=125 \[ \frac{2 \sin (c+d x) (b \cos (c+d x))^{9/2}}{11 b^2 d}+\frac{30 b^2 \sin (c+d x) \sqrt{b \cos (c+d x)}}{77 d}+\frac{30 b^3 \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{77 d \sqrt{b \cos (c+d x)}}+\frac{18 \sin (c+d x) (b \cos (c+d x))^{5/2}}{77 d} \]
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Rubi [A] time = 0.0804959, antiderivative size = 125, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {16, 2635, 2642, 2641} \[ \frac{2 \sin (c+d x) (b \cos (c+d x))^{9/2}}{11 b^2 d}+\frac{30 b^2 \sin (c+d x) \sqrt{b \cos (c+d x)}}{77 d}+\frac{30 b^3 \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{77 d \sqrt{b \cos (c+d x)}}+\frac{18 \sin (c+d x) (b \cos (c+d x))^{5/2}}{77 d} \]
Antiderivative was successfully verified.
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Rule 16
Rule 2635
Rule 2642
Rule 2641
Rubi steps
\begin{align*} \int \cos ^3(c+d x) (b \cos (c+d x))^{5/2} \, dx &=\frac{\int (b \cos (c+d x))^{11/2} \, dx}{b^3}\\ &=\frac{2 (b \cos (c+d x))^{9/2} \sin (c+d x)}{11 b^2 d}+\frac{9 \int (b \cos (c+d x))^{7/2} \, dx}{11 b}\\ &=\frac{18 (b \cos (c+d x))^{5/2} \sin (c+d x)}{77 d}+\frac{2 (b \cos (c+d x))^{9/2} \sin (c+d x)}{11 b^2 d}+\frac{1}{77} (45 b) \int (b \cos (c+d x))^{3/2} \, dx\\ &=\frac{30 b^2 \sqrt{b \cos (c+d x)} \sin (c+d x)}{77 d}+\frac{18 (b \cos (c+d x))^{5/2} \sin (c+d x)}{77 d}+\frac{2 (b \cos (c+d x))^{9/2} \sin (c+d x)}{11 b^2 d}+\frac{1}{77} \left (15 b^3\right ) \int \frac{1}{\sqrt{b \cos (c+d x)}} \, dx\\ &=\frac{30 b^2 \sqrt{b \cos (c+d x)} \sin (c+d x)}{77 d}+\frac{18 (b \cos (c+d x))^{5/2} \sin (c+d x)}{77 d}+\frac{2 (b \cos (c+d x))^{9/2} \sin (c+d x)}{11 b^2 d}+\frac{\left (15 b^3 \sqrt{\cos (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx}{77 \sqrt{b \cos (c+d x)}}\\ &=\frac{30 b^3 \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{77 d \sqrt{b \cos (c+d x)}}+\frac{30 b^2 \sqrt{b \cos (c+d x)} \sin (c+d x)}{77 d}+\frac{18 (b \cos (c+d x))^{5/2} \sin (c+d x)}{77 d}+\frac{2 (b \cos (c+d x))^{9/2} \sin (c+d x)}{11 b^2 d}\\ \end{align*}
Mathematica [A] time = 0.195242, size = 83, normalized size = 0.66 \[ \frac{(b \cos (c+d x))^{5/2} \left (240 F\left (\left .\frac{1}{2} (c+d x)\right |2\right )+(290 \sin (c+d x)+57 \sin (3 (c+d x))+7 \sin (5 (c+d x))) \sqrt{\cos (c+d x)}\right )}{616 d \cos ^{\frac{5}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 2.044, size = 236, normalized size = 1.9 \begin{align*} -{\frac{2\,{b}^{3}}{77\,d}\sqrt{b \left ( 2\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1 \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}} \left ( 448\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{13}-1568\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{11}+2384\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{9}-2040\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{7}+1084\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{5}-370\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{3}+15\,\sqrt{ \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}\sqrt{-2\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}+1}{\it EllipticF} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) +62\,\cos \left ( 1/2\,dx+c/2 \right ) \right ){\frac{1}{\sqrt{-b \left ( 2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}- \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2} \right ) }}} \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-1}{\frac{1}{\sqrt{b \left ( 2\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1 \right ) }}}} \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 )\right )^{\frac{5}{2}} \cos \left (d x + c\right )^{3}\,{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 (\sqrt{b \cos \left (d x + c\right )} b^{2} \cos \left (d x + c\right )^{5}, 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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (b \cos \left (d x + c\right )\right )^{\frac{5}{2}} \cos \left (d x + c\right )^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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