Optimal. Leaf size=181 \[ \frac {2 \text {Int}\left (\frac {(c \cos (e+f x))^m \left (\frac {1}{2} c \cos (e+f x) \left (a (2 m+5) (a B+2 A b)+b^2 B (2 m+3)\right )+\frac {1}{2} b c \cos ^2(e+f x) (2 a B (m+3)+A b (2 m+5))+\frac {1}{2} a c \left (2 a A \left (m+\frac {5}{2}\right )+2 b B (m+1)\right )\right )}{\sqrt {a+b \cos (e+f x)}},x\right )}{c (2 m+5)}+\frac {2 b B \sin (e+f x) \sqrt {a+b \cos (e+f x)} (c \cos (e+f x))^{m+1}}{c f (2 m+5)} \]
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Rubi [A] time = 0.53, 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 = {} \[ \int (c \cos (e+f x))^m (a+b \cos (e+f x))^{3/2} (A+B \cos (e+f x)) \, dx \]
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 = 66.75, size = 0, normalized size = 0.00 \[ \int (c \cos (e+f x))^m (a+b \cos (e+f x))^{3/2} (A+B \cos (e+f x)) \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 1.98, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (B b \cos \left (f x + e\right )^{2} + A a + {\left (B a + A b\right )} \cos \left (f x + e\right )\right )} \sqrt {b \cos \left (f x + e\right ) + a} \left (c \cos \left (f x + e\right )\right )^{m}, x\right ) \]
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 [A] time = 0.46, 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 \[ \int {\left (B \cos \left (f x + e\right ) + A\right )} {\left (b \cos \left (f x + e\right ) + a\right )}^{\frac {3}{2}} \left (c \cos \left (f x + e\right )\right )^{m}\,{d x} \]
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 \[ \int {\left (c\,\cos \left (e+f\,x\right )\right )}^m\,\left (A+B\,\cos \left (e+f\,x\right )\right )\,{\left (a+b\,\cos \left (e+f\,x\right )\right )}^{3/2} \,d x \]
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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