Optimal. Leaf size=297 \[ \frac {2 \left (-6 a^2 B+21 a A b+25 b^2 B\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{105 b d}-\frac {2 \left (a^2-b^2\right ) \left (-6 a^2 B+21 a A b+25 b^2 B\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} F\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{105 b^2 d \sqrt {a+b \cos (c+d x)}}+\frac {2 \left (-6 a^3 B+21 a^2 A b+82 a b^2 B+63 A b^3\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{105 b^2 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}+\frac {2 (7 A b-2 a B) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{35 b d}+\frac {2 B \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 b d} \]
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Rubi [A] time = 0.53, antiderivative size = 297, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.258, Rules used = {2968, 3023, 2753, 2752, 2663, 2661, 2655, 2653} \[ \frac {2 \left (-6 a^2 B+21 a A b+25 b^2 B\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{105 b d}-\frac {2 \left (a^2-b^2\right ) \left (-6 a^2 B+21 a A b+25 b^2 B\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} F\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{105 b^2 d \sqrt {a+b \cos (c+d x)}}+\frac {2 \left (21 a^2 A b-6 a^3 B+82 a b^2 B+63 A b^3\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{105 b^2 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}+\frac {2 (7 A b-2 a B) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{35 b d}+\frac {2 B \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 b d} \]
Antiderivative was successfully verified.
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Rule 2653
Rule 2655
Rule 2661
Rule 2663
Rule 2752
Rule 2753
Rule 2968
Rule 3023
Rubi steps
\begin {align*} \int \cos (c+d x) (a+b \cos (c+d x))^{3/2} (A+B \cos (c+d x)) \, dx &=\int (a+b \cos (c+d x))^{3/2} \left (A \cos (c+d x)+B \cos ^2(c+d x)\right ) \, dx\\ &=\frac {2 B (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{7 b d}+\frac {2 \int (a+b \cos (c+d x))^{3/2} \left (\frac {5 b B}{2}+\frac {1}{2} (7 A b-2 a B) \cos (c+d x)\right ) \, dx}{7 b}\\ &=\frac {2 (7 A b-2 a B) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{35 b d}+\frac {2 B (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{7 b d}+\frac {4 \int \sqrt {a+b \cos (c+d x)} \left (\frac {1}{4} b (21 A b+19 a B)+\frac {1}{4} \left (21 a A b-6 a^2 B+25 b^2 B\right ) \cos (c+d x)\right ) \, dx}{35 b}\\ &=\frac {2 \left (21 a A b-6 a^2 B+25 b^2 B\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{105 b d}+\frac {2 (7 A b-2 a B) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{35 b d}+\frac {2 B (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{7 b d}+\frac {8 \int \frac {\frac {1}{8} b \left (84 a A b+51 a^2 B+25 b^2 B\right )+\frac {1}{8} \left (21 a^2 A b+63 A b^3-6 a^3 B+82 a b^2 B\right ) \cos (c+d x)}{\sqrt {a+b \cos (c+d x)}} \, dx}{105 b}\\ &=\frac {2 \left (21 a A b-6 a^2 B+25 b^2 B\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{105 b d}+\frac {2 (7 A b-2 a B) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{35 b d}+\frac {2 B (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{7 b d}-\frac {\left (\left (a^2-b^2\right ) \left (21 a A b-6 a^2 B+25 b^2 B\right )\right ) \int \frac {1}{\sqrt {a+b \cos (c+d x)}} \, dx}{105 b^2}+\frac {\left (21 a^2 A b+63 A b^3-6 a^3 B+82 a b^2 B\right ) \int \sqrt {a+b \cos (c+d x)} \, dx}{105 b^2}\\ &=\frac {2 \left (21 a A b-6 a^2 B+25 b^2 B\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{105 b d}+\frac {2 (7 A b-2 a B) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{35 b d}+\frac {2 B (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{7 b d}+\frac {\left (\left (21 a^2 A b+63 A b^3-6 a^3 B+82 a b^2 B\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 b^2 \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {\left (\left (a^2-b^2\right ) \left (21 a A b-6 a^2 B+25 b^2 B\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 b^2 \sqrt {a+b \cos (c+d x)}}\\ &=\frac {2 \left (21 a^2 A b+63 A b^3-6 a^3 B+82 a b^2 B\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{105 b^2 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {2 \left (a^2-b^2\right ) \left (21 a A b-6 a^2 B+25 b^2 B\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 b^2 d \sqrt {a+b \cos (c+d x)}}+\frac {2 \left (21 a A b-6 a^2 B+25 b^2 B\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{105 b d}+\frac {2 (7 A b-2 a B) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{35 b d}+\frac {2 B (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{7 b d}\\ \end {align*}
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Mathematica [A] time = 1.07, size = 233, normalized size = 0.78 \[ \frac {b (a+b \cos (c+d x)) \left (\left (12 a^2 B+168 a A b+115 b^2 B\right ) \sin (c+d x)+3 b (2 (8 a B+7 A b) \sin (2 (c+d x))+5 b B \sin (3 (c+d x)))\right )+4 \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \left (b^2 \left (51 a^2 B+84 a A b+25 b^2 B\right ) F\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )+\left (-6 a^3 B+21 a^2 A b+82 a b^2 B+63 A b^3\right ) \left ((a+b) E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )-a F\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )\right )\right )}{210 b^2 d \sqrt {a+b \cos (c+d x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.88, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (B b \cos \left (d x + c\right )^{3} + A a \cos \left (d x + c\right ) + {\left (B a + A b\right )} \cos \left (d x + c\right )^{2}\right )} \sqrt {b \cos \left (d x + c\right ) + a}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 1.58, size = 1305, normalized size = 4.39 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \cos \left (d x + c\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \cos \left (c+d\,x\right )\,\left (A+B\,\cos \left (c+d\,x\right )\right )\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^{3/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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