Optimal. Leaf size=285 \[ \frac{2 b \left (-41 a^2 C+56 a b B+25 b^2 C\right ) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{105 d}-\frac{2 \left (a^2-b^2\right ) \left (-41 a^2 C+56 a b B+25 b^2 C\right ) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{105 d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left (161 a^2 b B-146 a^3 C+82 a b^2 C+63 b^3 B\right ) \sqrt{a+b \cos (c+d x)} E\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{105 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 b (7 b B-2 a C) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{35 d}+\frac{2 b C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d} \]
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Rubi [A] time = 0.6555, antiderivative size = 285, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 50, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.14, Rules used = {3015, 2753, 2752, 2663, 2661, 2655, 2653} \[ \frac{2 b \left (-41 a^2 C+56 a b B+25 b^2 C\right ) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{105 d}-\frac{2 \left (a^2-b^2\right ) \left (-41 a^2 C+56 a b B+25 b^2 C\right ) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{105 d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left (161 a^2 b B-146 a^3 C+82 a b^2 C+63 b^3 B\right ) \sqrt{a+b \cos (c+d x)} E\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{105 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 b (7 b B-2 a C) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{35 d}+\frac{2 b C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d} \]
Antiderivative was successfully verified.
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Rule 3015
Rule 2753
Rule 2752
Rule 2663
Rule 2661
Rule 2655
Rule 2653
Rubi steps
\begin{align*} \int (a+b \cos (c+d x))^{3/2} \left (a b B-a^2 C+b^2 B \cos (c+d x)+b^2 C \cos ^2(c+d x)\right ) \, dx &=\frac{\int (a+b \cos (c+d x))^{5/2} \left (b^2 (b B-a C)+b^3 C \cos (c+d x)\right ) \, dx}{b^2}\\ &=\frac{2 b C (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{7 d}+\frac{2 \int (a+b \cos (c+d x))^{3/2} \left (\frac{1}{2} b^2 \left (7 a b B-7 a^2 C+5 b^2 C\right )+\frac{1}{2} b^3 (7 b B-2 a C) \cos (c+d x)\right ) \, dx}{7 b^2}\\ &=\frac{2 b (7 b B-2 a C) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{35 d}+\frac{2 b C (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{7 d}+\frac{4 \int \sqrt{a+b \cos (c+d x)} \left (\frac{1}{4} b^2 \left (35 a^2 b B+21 b^3 B-35 a^3 C+19 a b^2 C\right )+\frac{1}{4} b^3 \left (56 a b B-41 a^2 C+25 b^2 C\right ) \cos (c+d x)\right ) \, dx}{35 b^2}\\ &=\frac{2 b \left (56 a b B-41 a^2 C+25 b^2 C\right ) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{105 d}+\frac{2 b (7 b B-2 a C) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{35 d}+\frac{2 b C (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{7 d}+\frac{8 \int \frac{\frac{1}{8} b^2 \left (105 a^3 b B+119 a b^3 B-105 a^4 C+16 a^2 b^2 C+25 b^4 C\right )+\frac{1}{8} b^3 \left (161 a^2 b B+63 b^3 B-146 a^3 C+82 a b^2 C\right ) \cos (c+d x)}{\sqrt{a+b \cos (c+d x)}} \, dx}{105 b^2}\\ &=\frac{2 b \left (56 a b B-41 a^2 C+25 b^2 C\right ) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{105 d}+\frac{2 b (7 b B-2 a C) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{35 d}+\frac{2 b C (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{7 d}-\frac{1}{105} \left (\left (a^2-b^2\right ) \left (56 a b B-41 a^2 C+25 b^2 C\right )\right ) \int \frac{1}{\sqrt{a+b \cos (c+d x)}} \, dx+\frac{1}{105} \left (161 a^2 b B+63 b^3 B-146 a^3 C+82 a b^2 C\right ) \int \sqrt{a+b \cos (c+d x)} \, dx\\ &=\frac{2 b \left (56 a b B-41 a^2 C+25 b^2 C\right ) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{105 d}+\frac{2 b (7 b B-2 a C) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{35 d}+\frac{2 b C (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{7 d}+\frac{\left (\left (161 a^2 b B+63 b^3 B-146 a^3 C+82 a b^2 C\right ) \sqrt{a+b \cos (c+d x)}\right ) \int \sqrt{\frac{a}{a+b}+\frac{b \cos (c+d x)}{a+b}} \, dx}{105 \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{\left (\left (a^2-b^2\right ) \left (56 a b B-41 a^2 C+25 b^2 C\right ) \sqrt{\frac{a+b \cos (c+d x)}{a+b}}\right ) \int \frac{1}{\sqrt{\frac{a}{a+b}+\frac{b \cos (c+d x)}{a+b}}} \, dx}{105 \sqrt{a+b \cos (c+d x)}}\\ &=\frac{2 \left (161 a^2 b B+63 b^3 B-146 a^3 C+82 a b^2 C\right ) \sqrt{a+b \cos (c+d x)} E\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{105 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{2 \left (a^2-b^2\right ) \left (56 a b B-41 a^2 C+25 b^2 C\right ) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{105 d \sqrt{a+b \cos (c+d x)}}+\frac{2 b \left (56 a b B-41 a^2 C+25 b^2 C\right ) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{105 d}+\frac{2 b (7 b B-2 a C) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{35 d}+\frac{2 b C (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{7 d}\\ \end{align*}
Mathematica [A] time = 1.22072, size = 259, normalized size = 0.91 \[ \frac{b \sin (c+d x) (a+b \cos (c+d x)) \left (-64 a^2 C+6 b (8 a C+7 b B) \cos (c+d x)+154 a b B+15 b^2 C \cos (2 (c+d x))+65 b^2 C\right )+2 \left (16 a^2 b^2 C+105 a^3 b B-105 a^4 C+119 a b^3 B+25 b^4 C\right ) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )+2 \left (161 a^2 b B-146 a^3 C+82 a b^2 C+63 b^3 B\right ) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \left ((a+b) E\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )-a F\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )\right )}{105 d \sqrt{a+b \cos (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 1.026, size = 1302, normalized size = 4.6 \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 (C b^{2} \cos \left (d x + c\right )^{2} + B b^{2} \cos \left (d x + c\right ) - C a^{2} + B a b\right )}{\left (b \cos \left (d x + c\right ) + a\right )}^{\frac{3}{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 (C b^{3} \cos \left (d x + c\right )^{3} - C a^{3} + B a^{2} b +{\left (C a b^{2} + B b^{3}\right )} \cos \left (d x + c\right )^{2} -{\left (C a^{2} b - 2 \, B a b^{2}\right )} \cos \left (d x + c\right )\right )} \sqrt{b \cos \left (d x + c\right ) + a}, 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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