Optimal. Leaf size=171 \[ \frac{6 A b^2 E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{b \cos (c+d x)}}{5 d \sqrt{\cos (c+d x)}}+\frac{2 A b \sin (c+d x) (b \cos (c+d x))^{3/2}}{5 d}+\frac{10 b^2 B \sin (c+d x) \sqrt{b \cos (c+d x)}}{21 d}+\frac{10 b^3 B \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d \sqrt{b \cos (c+d x)}}+\frac{2 B \sin (c+d x) (b \cos (c+d x))^{5/2}}{7 d} \]
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Rubi [A] time = 0.119754, antiderivative size = 171, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261, Rules used = {2748, 2635, 2640, 2639, 2642, 2641} \[ \frac{6 A b^2 E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{b \cos (c+d x)}}{5 d \sqrt{\cos (c+d x)}}+\frac{2 A b \sin (c+d x) (b \cos (c+d x))^{3/2}}{5 d}+\frac{10 b^2 B \sin (c+d x) \sqrt{b \cos (c+d x)}}{21 d}+\frac{10 b^3 B \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d \sqrt{b \cos (c+d x)}}+\frac{2 B \sin (c+d x) (b \cos (c+d x))^{5/2}}{7 d} \]
Antiderivative was successfully verified.
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Rule 2748
Rule 2635
Rule 2640
Rule 2639
Rule 2642
Rule 2641
Rubi steps
\begin{align*} \int (b \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \, dx &=A \int (b \cos (c+d x))^{5/2} \, dx+\frac{B \int (b \cos (c+d x))^{7/2} \, dx}{b}\\ &=\frac{2 A b (b \cos (c+d x))^{3/2} \sin (c+d x)}{5 d}+\frac{2 B (b \cos (c+d x))^{5/2} \sin (c+d x)}{7 d}+\frac{1}{5} \left (3 A b^2\right ) \int \sqrt{b \cos (c+d x)} \, dx+\frac{1}{7} (5 b B) \int (b \cos (c+d x))^{3/2} \, dx\\ &=\frac{10 b^2 B \sqrt{b \cos (c+d x)} \sin (c+d x)}{21 d}+\frac{2 A b (b \cos (c+d x))^{3/2} \sin (c+d x)}{5 d}+\frac{2 B (b \cos (c+d x))^{5/2} \sin (c+d x)}{7 d}+\frac{1}{21} \left (5 b^3 B\right ) \int \frac{1}{\sqrt{b \cos (c+d x)}} \, dx+\frac{\left (3 A b^2 \sqrt{b \cos (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx}{5 \sqrt{\cos (c+d x)}}\\ &=\frac{6 A b^2 \sqrt{b \cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d \sqrt{\cos (c+d x)}}+\frac{10 b^2 B \sqrt{b \cos (c+d x)} \sin (c+d x)}{21 d}+\frac{2 A b (b \cos (c+d x))^{3/2} \sin (c+d x)}{5 d}+\frac{2 B (b \cos (c+d x))^{5/2} \sin (c+d x)}{7 d}+\frac{\left (5 b^3 B \sqrt{\cos (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx}{21 \sqrt{b \cos (c+d x)}}\\ &=\frac{6 A b^2 \sqrt{b \cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d \sqrt{\cos (c+d x)}}+\frac{10 b^3 B \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d \sqrt{b \cos (c+d x)}}+\frac{10 b^2 B \sqrt{b \cos (c+d x)} \sin (c+d x)}{21 d}+\frac{2 A b (b \cos (c+d x))^{3/2} \sin (c+d x)}{5 d}+\frac{2 B (b \cos (c+d x))^{5/2} \sin (c+d x)}{7 d}\\ \end{align*}
Mathematica [A] time = 0.0587152, size = 100, normalized size = 0.58 \[ \frac{(b \cos (c+d x))^{5/2} \left (2 \sin (c+d x) \sqrt{\cos (c+d x)} (42 A \cos (c+d x)+15 B \cos (2 (c+d x))+65 B)+252 A E\left (\left .\frac{1}{2} (c+d x)\right |2\right )+100 B F\left (\left .\frac{1}{2} (c+d x)\right |2\right )\right )}{210 d \cos ^{\frac{5}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 3.313, size = 301, normalized size = 1.8 \begin{align*} -{\frac{2\,{b}^{3}}{105\,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 ( 240\,B\cos \left ( 1/2\,dx+c/2 \right ) \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{8}+ \left ( -168\,A-360\,B \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{6}\cos \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) + \left ( 168\,A+280\,B \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{4}\cos \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) + \left ( -42\,A-80\,B \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}\cos \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -63\,A\sqrt{ \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}\sqrt{2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1}{\it EllipticE} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) +25\,B\sqrt{ \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}\sqrt{2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1}{\it EllipticF} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{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 ) + A\right )} \left (b \cos \left (d x + c\right )\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 )^{3} + A b^{2} \cos \left (d x + c\right )^{2}\right )} \sqrt{b \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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