3.455 \(\int (c \cos (e+f x))^m (a+b \cos (e+f x))^{3/2} (A+B \cos (e+f x)) \, dx\)

Optimal. Leaf size=180 \[ \frac{2 \text{Unintegrable}\left (\frac{(c \cos (e+f x))^m \left (\frac{1}{2} c \cos (e+f x) \left (a (2 m+5) (a B+2 A b)+b^2 B (2 m+3)\right )+\frac{1}{2} b c \cos ^2(e+f x) (2 a B (m+3)+A b (2 m+5))+\frac{1}{2} a c \left (2 a A \left (m+\frac{5}{2}\right )+2 b B (m+1)\right )\right )}{\sqrt{a+b \cos (e+f x)}},x\right )}{c (2 m+5)}+\frac{2 b B \sin (e+f x) \sqrt{a+b \cos (e+f x)} (c \cos (e+f x))^{m+1}}{c f (2 m+5)} \]

[Out]

(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)) + (2*Unintegrable[((c*C
os[e + f*x])^m*((a*c*(2*b*B*(1 + m) + 2*a*A*(5/2 + m)))/2 + (c*(b^2*B*(3 + 2*m) + a*(2*A*b + a*B)*(5 + 2*m))*C
os[e + f*x])/2 + (b*c*(2*a*B*(3 + m) + A*b*(5 + 2*m))*Cos[e + f*x]^2)/2))/Sqrt[a + b*Cos[e + f*x]], x])/(c*(5
+ 2*m))

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Rubi [A]  time = 0.526552, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., 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.

[In]

Int[(c*Cos[e + f*x])^m*(a + b*Cos[e + f*x])^(3/2)*(A + B*Cos[e + f*x]),x]

[Out]

(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)) + (2*Defer[Int][((c*Cos
[e + f*x])^m*((a*c*(2*b*B*(1 + m) + 2*a*A*(5/2 + m)))/2 + (c*(b^2*B*(3 + 2*m) + a*(2*A*b + a*B)*(5 + 2*m))*Cos
[e + f*x])/2 + (b*c*(2*a*B*(3 + m) + A*b*(5 + 2*m))*Cos[e + f*x]^2)/2))/Sqrt[a + b*Cos[e + f*x]], x])/(c*(5 +
2*m))

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*}

Mathematica [A]  time = 66.2563, size = 0, normalized size = 0. \[ \int (c \cos (e+f x))^m (a+b \cos (e+f x))^{3/2} (A+B \cos (e+f x)) \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(c*Cos[e + f*x])^m*(a + b*Cos[e + f*x])^(3/2)*(A + B*Cos[e + f*x]),x]

[Out]

Integrate[(c*Cos[e + f*x])^m*(a + b*Cos[e + f*x])^(3/2)*(A + B*Cos[e + f*x]), x]

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Maple [A]  time = 0.497, size = 0, normalized size = 0. \begin{align*} \int \left ( c\cos \left ( fx+e \right ) \right ) ^{m} \left ( a+b\cos \left ( fx+e \right ) \right ) ^{{\frac{3}{2}}} \left ( A+B\cos \left ( fx+e \right ) \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*cos(f*x+e))^m*(a+b*cos(f*x+e))^(3/2)*(A+B*cos(f*x+e)),x)

[Out]

int((c*cos(f*x+e))^m*(a+b*cos(f*x+e))^(3/2)*(A+B*cos(f*x+e)),x)

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*cos(f*x+e))^m*(a+b*cos(f*x+e))^(3/2)*(A+B*cos(f*x+e)),x, algorithm="maxima")

[Out]

integrate((B*cos(f*x + e) + A)*(b*cos(f*x + e) + a)^(3/2)*(c*cos(f*x + e))^m, x)

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*cos(f*x+e))^m*(a+b*cos(f*x+e))^(3/2)*(A+B*cos(f*x+e)),x, algorithm="fricas")

[Out]

integral((B*b*cos(f*x + e)^2 + A*a + (B*a + A*b)*cos(f*x + e))*sqrt(b*cos(f*x + e) + a)*(c*cos(f*x + e))^m, x)

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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.

[In]

integrate((c*cos(f*x+e))**m*(a+b*cos(f*x+e))**(3/2)*(A+B*cos(f*x+e)),x)

[Out]

Timed out

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: AttributeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*cos(f*x+e))^m*(a+b*cos(f*x+e))^(3/2)*(A+B*cos(f*x+e)),x, algorithm="giac")

[Out]

Exception raised: AttributeError