\(\int (c \cos (e+f x))^m (a+b \cos (e+f x))^{3/2} (A+B \cos (e+f x)) \, dx\) [455]

Optimal result

Integrand size = 35, antiderivative size = 35 \[ \int (c \cos (e+f x))^m (a+b \cos (e+f x))^{3/2} (A+B \cos (e+f x)) \, dx=\frac {2 b B (c \cos (e+f x))^{1+m} \sqrt {a+b \cos (e+f x)} \sin (e+f x)}{c f (5+2 m)}+\frac {2 \text {Int}\left (\frac {(c \cos (e+f x))^m \left (\frac {1}{2} a c \left (2 b B (1+m)+2 a A \left (\frac {5}{2}+m\right )\right )+\frac {1}{2} c \left (b^2 B (3+2 m)+a (2 A b+a B) (5+2 m)\right ) \cos (e+f x)+\frac {1}{2} b c (2 a B (3+m)+A b (5+2 m)) \cos ^2(e+f x)\right )}{\sqrt {a+b \cos (e+f x)}},x\right )}{c (5+2 m)} \]

[Out]

2*b*B*(c*cos(f*x+e))^(1+m)*sin(f*x+e)*(a+b*cos(f*x+e))^(1/2)/c/f/(5+2*m)+2*Unintegrable((c*cos(f*x+e))^m*(1/2*
a*c*(2*b*B*(1+m)+2*a*A*(5/2+m))+1/2*c*(b^2*B*(3+2*m)+a*(2*A*b+B*a)*(5+2*m))*cos(f*x+e)+1/2*b*c*(2*a*B*(3+m)+A*
b*(5+2*m))*cos(f*x+e)^2)/(a+b*cos(f*x+e))^(1/2),x)/c/(5+2*m)

Rubi [N/A]

Not integrable

Time = 0.54 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int (c \cos (e+f x))^m (a+b \cos (e+f x))^{3/2} (A+B \cos (e+f x)) \, dx=\int (c \cos (e+f x))^m (a+b \cos (e+f x))^{3/2} (A+B \cos (e+f x)) \, dx \]

[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*} \text {integral}& = \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 [N/A]

Not integrable

Time = 79.33 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.06 \[ \int (c \cos (e+f x))^m (a+b \cos (e+f x))^{3/2} (A+B \cos (e+f x)) \, dx=\int (c \cos (e+f x))^m (a+b \cos (e+f x))^{3/2} (A+B \cos (e+f x)) \, dx \]

[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]

Maple [N/A] (verified)

Not integrable

Time = 0.79 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.94

\[\int \left (c \cos \left (f x +e \right )\right )^{m} \left (a +b \cos \left (f x +e \right )\right )^{\frac {3}{2}} \left (A +\cos \left (f x +e \right ) B \right )d x\]

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.30 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.54 \[ \int (c \cos (e+f x))^m (a+b \cos (e+f x))^{3/2} (A+B \cos (e+f x)) \, dx=\int { {\left (B \cos \left (f x + e\right ) + A\right )} {\left (b \cos \left (f x + e\right ) + a\right )}^{\frac {3}{2}} \left (c \cos \left (f x + e\right )\right )^{m} \,d x } \]

[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)

Sympy [N/A]

Not integrable

Time = 147.50 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.97 \[ \int (c \cos (e+f x))^m (a+b \cos (e+f x))^{3/2} (A+B \cos (e+f x)) \, dx=\int \left (c \cos {\left (e + f x \right )}\right )^{m} \left (A + B \cos {\left (e + f x \right )}\right ) \left (a + b \cos {\left (e + f x \right )}\right )^{\frac {3}{2}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 2.59 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00 \[ \int (c \cos (e+f x))^m (a+b \cos (e+f x))^{3/2} (A+B \cos (e+f x)) \, dx=\int { {\left (B \cos \left (f x + e\right ) + A\right )} {\left (b \cos \left (f x + e\right ) + a\right )}^{\frac {3}{2}} \left (c \cos \left (f x + e\right )\right )^{m} \,d x } \]

[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)

Giac [N/A]

Not integrable

Time = 16.26 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00 \[ \int (c \cos (e+f x))^m (a+b \cos (e+f x))^{3/2} (A+B \cos (e+f x)) \, dx=\int { {\left (B \cos \left (f x + e\right ) + A\right )} {\left (b \cos \left (f x + e\right ) + a\right )}^{\frac {3}{2}} \left (c \cos \left (f x + e\right )\right )^{m} \,d x } \]

[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]

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

Mupad [N/A]

Not integrable

Time = 4.40 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00 \[ \int (c \cos (e+f x))^m (a+b \cos (e+f x))^{3/2} (A+B \cos (e+f x)) \, dx=\int {\left (c\,\cos \left (e+f\,x\right )\right )}^m\,\left (A+B\,\cos \left (e+f\,x\right )\right )\,{\left (a+b\,\cos \left (e+f\,x\right )\right )}^{3/2} \,d x \]

[In]

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

[Out]

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