Optimal. Leaf size=181 \[ \frac {2 b B (c \cos (e+f x))^{1+m} \sqrt {a+b \cos (e+f x)} \sin (e+f x)}{c f (5+2 m)}+\frac {2 \text {Int}\left (\frac {(c \cos (e+f x))^m \left (\frac {1}{2} a c \left (2 b B (1+m)+2 a A \left (\frac {5}{2}+m\right )\right )+\frac {1}{2} c \left (b^2 B (3+2 m)+a (2 A b+a B) (5+2 m)\right ) \cos (e+f x)+\frac {1}{2} b c (2 a B (3+m)+A b (5+2 m)) \cos ^2(e+f x)\right )}{\sqrt {a+b \cos (e+f x)}},x\right )}{c (5+2 m)} \]
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Rubi [A]
time = 0.34, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps
used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {}
\begin {gather*} \int (c \cos (e+f x))^m (a+b \cos (e+f x))^{3/2} (A+B \cos (e+f x)) \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int (c \cos (e+f x))^m (a+b \cos (e+f x))^{3/2} (A+B \cos (e+f x)) \, dx &=\frac {2 b B (c \cos (e+f x))^{1+m} \sqrt {a+b \cos (e+f x)} \sin (e+f x)}{c f (5+2 m)}+\frac {2 \int \frac {(c \cos (e+f x))^m \left (\frac {1}{2} a c \left (2 b B (1+m)+2 a A \left (\frac {5}{2}+m\right )\right )+\frac {1}{2} c \left (b^2 B (3+2 m)+a (2 A b+a B) (5+2 m)\right ) \cos (e+f x)+\frac {1}{2} b c (2 a B (3+m)+A b (5+2 m)) \cos ^2(e+f x)\right )}{\sqrt {a+b \cos (e+f x)}} \, dx}{c (5+2 m)}\\ \end {align*}
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Mathematica [A]
time = 75.98, size = 0, normalized size = 0.00 \begin {gather*} \int (c \cos (e+f x))^m (a+b \cos (e+f x))^{3/2} (A+B \cos (e+f x)) \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [A]
time = 0.15, size = 0, normalized size = 0.00 \[\int \left (c \cos \left (f x +e \right )\right )^{m} \left (a +b \cos \left (f x +e \right )\right )^{\frac {3}{2}} \left (A +B \cos \left (f x +e \right )\right )\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [A]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\left (c\,\cos \left (e+f\,x\right )\right )}^m\,\left (A+B\,\cos \left (e+f\,x\right )\right )\,{\left (a+b\,\cos \left (e+f\,x\right )\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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