Optimal. Leaf size=138 \[ \frac{64 a^3 (7 A+5 B) \sin (c+d x)}{105 d \sqrt{a \cos (c+d x)+a}}+\frac{16 a^2 (7 A+5 B) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{105 d}+\frac{2 a (7 A+5 B) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{35 d}+\frac{2 B \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{7 d} \]
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Rubi [A] time = 0.10898, antiderivative size = 138, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {2751, 2647, 2646} \[ \frac{64 a^3 (7 A+5 B) \sin (c+d x)}{105 d \sqrt{a \cos (c+d x)+a}}+\frac{16 a^2 (7 A+5 B) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{105 d}+\frac{2 a (7 A+5 B) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{35 d}+\frac{2 B \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{7 d} \]
Antiderivative was successfully verified.
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Rule 2751
Rule 2647
Rule 2646
Rubi steps
\begin{align*} \int (a+a \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \, dx &=\frac{2 B (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{7 d}+\frac{1}{7} (7 A+5 B) \int (a+a \cos (c+d x))^{5/2} \, dx\\ &=\frac{2 a (7 A+5 B) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{35 d}+\frac{2 B (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{7 d}+\frac{1}{35} (8 a (7 A+5 B)) \int (a+a \cos (c+d x))^{3/2} \, dx\\ &=\frac{16 a^2 (7 A+5 B) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{105 d}+\frac{2 a (7 A+5 B) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{35 d}+\frac{2 B (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{7 d}+\frac{1}{105} \left (32 a^2 (7 A+5 B)\right ) \int \sqrt{a+a \cos (c+d x)} \, dx\\ &=\frac{64 a^3 (7 A+5 B) \sin (c+d x)}{105 d \sqrt{a+a \cos (c+d x)}}+\frac{16 a^2 (7 A+5 B) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{105 d}+\frac{2 a (7 A+5 B) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{35 d}+\frac{2 B (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{7 d}\\ \end{align*}
Mathematica [A] time = 0.319035, size = 83, normalized size = 0.6 \[ \frac{a^2 \tan \left (\frac{1}{2} (c+d x)\right ) \sqrt{a (\cos (c+d x)+1)} ((392 A+505 B) \cos (c+d x)+6 (7 A+20 B) \cos (2 (c+d x))+1246 A+15 B \cos (3 (c+d x))+1040 B)}{210 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 1., size = 104, normalized size = 0.8 \begin{align*}{\frac{8\,{a}^{3}\sqrt{2}}{105\,d}\cos \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \left ( -30\,B \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{6}+ \left ( 21\,A+105\,B \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{4}+ \left ( -70\,A-140\,B \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}+105\,A+105\,B \right ){\frac{1}{\sqrt{ \left ( \cos \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}a}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 2.64998, size = 188, normalized size = 1.36 \begin{align*} \frac{14 \,{\left (3 \, \sqrt{2} a^{2} \sin \left (\frac{5}{2} \, d x + \frac{5}{2} \, c\right ) + 25 \, \sqrt{2} a^{2} \sin \left (\frac{3}{2} \, d x + \frac{3}{2} \, c\right ) + 150 \, \sqrt{2} a^{2} \sin \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )} A \sqrt{a} + 5 \,{\left (3 \, \sqrt{2} a^{2} \sin \left (\frac{7}{2} \, d x + \frac{7}{2} \, c\right ) + 21 \, \sqrt{2} a^{2} \sin \left (\frac{5}{2} \, d x + \frac{5}{2} \, c\right ) + 77 \, \sqrt{2} a^{2} \sin \left (\frac{3}{2} \, d x + \frac{3}{2} \, c\right ) + 315 \, \sqrt{2} a^{2} \sin \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )} B \sqrt{a}}{420 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.61758, size = 248, normalized size = 1.8 \begin{align*} \frac{2 \,{\left (15 \, B a^{2} \cos \left (d x + c\right )^{3} + 3 \,{\left (7 \, A + 20 \, B\right )} a^{2} \cos \left (d x + c\right )^{2} +{\left (98 \, A + 115 \, B\right )} a^{2} \cos \left (d x + c\right ) +{\left (301 \, A + 230 \, B\right )} a^{2}\right )} \sqrt{a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{105 \,{\left (d \cos \left (d x + c\right ) + d\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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