\(\int F^{c (a+b x)} \sqrt {g \cos (d+e x)+f \sin (d+e x)} \, dx\) [29]

Optimal result

Integrand size = 31, antiderivative size = 147 \[ \int F^{c (a+b x)} \sqrt {g \cos (d+e x)+f \sin (d+e x)} \, dx=-\frac {2 F^{c (a+b x)} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},-\frac {e+2 i b c \log (F)}{4 e},\frac {1}{4} \left (3-\frac {2 i b c \log (F)}{e}\right ),\frac {e^{2 i (d+e x)} (f+i g)}{f-i g}\right ) \sqrt {g \cos (d+e x)+f \sin (d+e x)}}{\sqrt {1-\frac {e^{2 i (d+e x)} (f+i g)}{f-i g}} (i e-2 b c \log (F))} \] Output:

-2*F^(c*(b*x+a))*hypergeom([-1/2, -1/4*(2*I*b*c*ln(F)+e)/e],[3/4-1/2*I*b*c 
*ln(F)/e],exp(2*I*(e*x+d))*(f+I*g)/(f-I*g))*(g*cos(e*x+d)+f*sin(e*x+d))^(1 
/2)/(1-exp(2*I*(e*x+d))*(f+I*g)/(f-I*g))^(1/2)/(I*e-2*b*c*ln(F))
 

Mathematica [B] (warning: unable to verify)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(399\) vs. \(2(147)=294\).

Time = 6.01 (sec) , antiderivative size = 399, normalized size of antiderivative = 2.71 \[ \int F^{c (a+b x)} \sqrt {g \cos (d+e x)+f \sin (d+e x)} \, dx=\frac {F^{c \left (a-\frac {b d}{e}\right )} \left (\frac {\sqrt {2} e e^{-\frac {1}{2} i (d+e x)} (f+i g) \left (\left (-1+e^{2 i (d+e x)}\right ) f+i \left (1+e^{2 i (d+e x)}\right ) g\right ) \left (e^{(d+e x) \left (-\frac {i}{2}+\frac {b c \log (F)}{e}\right )} \operatorname {Hypergeometric2F1}\left (1,\frac {e-2 i b c \log (F)}{4 e},\frac {3}{4}-\frac {i b c \log (F)}{2 e},\frac {e^{2 i (d+e x)} (f+i g)}{f-i g}\right ) (3 e-2 i b c \log (F))+e^{(d+e x) \left (\frac {3 i}{2}+\frac {b c \log (F)}{e}\right )} \operatorname {Hypergeometric2F1}\left (1,\frac {5}{4}-\frac {i b c \log (F)}{2 e},\frac {7}{4}-\frac {i b c \log (F)}{2 e},\frac {e^{2 i (d+e x)} (f+i g)}{f-i g}\right ) (e+2 i b c \log (F))\right )}{\sqrt {e^{-i (d+e x)} \left (-i \left (-1+e^{2 i (d+e x)}\right ) f+\left (1+e^{2 i (d+e x)}\right ) g\right )} (3 e-2 i b c \log (F)) (e+2 i b c \log (F))}+2 F^{\frac {b c (d+e x)}{e}} g \sqrt {g \cos (d+e x)+f \sin (d+e x)}\right )}{e f+2 b c g \log (F)} \] Input:

Integrate[F^(c*(a + b*x))*Sqrt[g*Cos[d + e*x] + f*Sin[d + e*x]],x]
 

Output:

(F^(c*(a - (b*d)/e))*((Sqrt[2]*e*(f + I*g)*((-1 + E^((2*I)*(d + e*x)))*f + 
 I*(1 + E^((2*I)*(d + e*x)))*g)*(E^((d + e*x)*(-1/2*I + (b*c*Log[F])/e))*H 
ypergeometric2F1[1, (e - (2*I)*b*c*Log[F])/(4*e), 3/4 - ((I/2)*b*c*Log[F]) 
/e, (E^((2*I)*(d + e*x))*(f + I*g))/(f - I*g)]*(3*e - (2*I)*b*c*Log[F]) + 
E^((d + e*x)*((3*I)/2 + (b*c*Log[F])/e))*Hypergeometric2F1[1, 5/4 - ((I/2) 
*b*c*Log[F])/e, 7/4 - ((I/2)*b*c*Log[F])/e, (E^((2*I)*(d + e*x))*(f + I*g) 
)/(f - I*g)]*(e + (2*I)*b*c*Log[F])))/(E^((I/2)*(d + e*x))*Sqrt[((-I)*(-1 
+ E^((2*I)*(d + e*x)))*f + (1 + E^((2*I)*(d + e*x)))*g)/E^(I*(d + e*x))]*( 
3*e - (2*I)*b*c*Log[F])*(e + (2*I)*b*c*Log[F])) + 2*F^((b*c*(d + e*x))/e)* 
g*Sqrt[g*Cos[d + e*x] + f*Sin[d + e*x]]))/(e*f + 2*b*c*g*Log[F])
 

Rubi [F]

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int F^{c (a+b x)} \sqrt {f \sin (d+e x)+g \cos (d+e x)} \, dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int F^{a c+b c x} \sqrt {f \sin (d+e x)+g \cos (d+e x)}dx\)

\(\Big \downarrow \) 7299

\(\displaystyle \int F^{a c+b c x} \sqrt {f \sin (d+e x)+g \cos (d+e x)}dx\)

Input:

Int[F^(c*(a + b*x))*Sqrt[g*Cos[d + e*x] + f*Sin[d + e*x]],x]
 

Output:

$Aborted
 

Defintions of rubi rules used

rule 7292

Int[u_, x_Symbol] :> With[{v = NormalizeIntegrand[u, x]}, Int[v, x] /; v =! 
= u]
 

rule 7299

Int[u_, x_] :> CannotIntegrate[u, x]
 

Maple [F]

\[\int F^{c \left (b x +a \right )} \sqrt {g \cos \left (e x +d \right )+f \sin \left (e x +d \right )}d x\]

Input:

int(F^(c*(b*x+a))*(g*cos(e*x+d)+f*sin(e*x+d))^(1/2),x)
 

Output:

int(F^(c*(b*x+a))*(g*cos(e*x+d)+f*sin(e*x+d))^(1/2),x)
 

Fricas [F]

\[ \int F^{c (a+b x)} \sqrt {g \cos (d+e x)+f \sin (d+e x)} \, dx=\int { \sqrt {g \cos \left (e x + d\right ) + f \sin \left (e x + d\right )} F^{{\left (b x + a\right )} c} \,d x } \] Input:

integrate(F^(c*(b*x+a))*(g*cos(e*x+d)+f*sin(e*x+d))^(1/2),x, algorithm="fr 
icas")
 

Output:

integral(sqrt(g*cos(e*x + d) + f*sin(e*x + d))*F^(b*c*x + a*c), x)
 

Sympy [F]

\[ \int F^{c (a+b x)} \sqrt {g \cos (d+e x)+f \sin (d+e x)} \, dx=\int F^{c \left (a + b x\right )} \sqrt {f \sin {\left (d + e x \right )} + g \cos {\left (d + e x \right )}}\, dx \] Input:

integrate(F**(c*(b*x+a))*(g*cos(e*x+d)+f*sin(e*x+d))**(1/2),x)
 

Output:

Integral(F**(c*(a + b*x))*sqrt(f*sin(d + e*x) + g*cos(d + e*x)), x)
 

Maxima [F]

\[ \int F^{c (a+b x)} \sqrt {g \cos (d+e x)+f \sin (d+e x)} \, dx=\int { \sqrt {g \cos \left (e x + d\right ) + f \sin \left (e x + d\right )} F^{{\left (b x + a\right )} c} \,d x } \] Input:

integrate(F^(c*(b*x+a))*(g*cos(e*x+d)+f*sin(e*x+d))^(1/2),x, algorithm="ma 
xima")
 

Output:

integrate(sqrt(g*cos(e*x + d) + f*sin(e*x + d))*F^((b*x + a)*c), x)
 

Giac [F]

\[ \int F^{c (a+b x)} \sqrt {g \cos (d+e x)+f \sin (d+e x)} \, dx=\int { \sqrt {g \cos \left (e x + d\right ) + f \sin \left (e x + d\right )} F^{{\left (b x + a\right )} c} \,d x } \] Input:

integrate(F^(c*(b*x+a))*(g*cos(e*x+d)+f*sin(e*x+d))^(1/2),x, algorithm="gi 
ac")
 

Output:

integrate(sqrt(g*cos(e*x + d) + f*sin(e*x + d))*F^((b*x + a)*c), x)
 

Mupad [F(-1)]

Timed out. \[ \int F^{c (a+b x)} \sqrt {g \cos (d+e x)+f \sin (d+e x)} \, dx=\int F^{c\,\left (a+b\,x\right )}\,\sqrt {g\,\cos \left (d+e\,x\right )+f\,\sin \left (d+e\,x\right )} \,d x \] Input:

int(F^(c*(a + b*x))*(g*cos(d + e*x) + f*sin(d + e*x))^(1/2),x)
 

Output:

int(F^(c*(a + b*x))*(g*cos(d + e*x) + f*sin(d + e*x))^(1/2), x)
 

Reduce [F]

\[ \int F^{c (a+b x)} \sqrt {g \cos (d+e x)+f \sin (d+e x)} \, dx=\frac {f^{a c} \left (2 f^{b c x} \sqrt {\cos \left (e x +d \right ) g +\sin \left (e x +d \right ) f}\, f -2 \left (\int \frac {f^{b c x} \sqrt {\cos \left (e x +d \right ) g +\sin \left (e x +d \right ) f}\, \cos \left (e x +d \right )}{2 \cos \left (e x +d \right ) \mathrm {log}\left (f \right ) b c f g -\cos \left (e x +d \right ) e \,g^{2}+2 \,\mathrm {log}\left (f \right ) \sin \left (e x +d \right ) b c \,f^{2}-\sin \left (e x +d \right ) e f g}d x \right ) \mathrm {log}\left (f \right ) b c e \,f^{3}-2 \left (\int \frac {f^{b c x} \sqrt {\cos \left (e x +d \right ) g +\sin \left (e x +d \right ) f}\, \cos \left (e x +d \right )}{2 \cos \left (e x +d \right ) \mathrm {log}\left (f \right ) b c f g -\cos \left (e x +d \right ) e \,g^{2}+2 \,\mathrm {log}\left (f \right ) \sin \left (e x +d \right ) b c \,f^{2}-\sin \left (e x +d \right ) e f g}d x \right ) \mathrm {log}\left (f \right ) b c e f \,g^{2}+\left (\int \frac {f^{b c x} \sqrt {\cos \left (e x +d \right ) g +\sin \left (e x +d \right ) f}\, \cos \left (e x +d \right )}{2 \cos \left (e x +d \right ) \mathrm {log}\left (f \right ) b c f g -\cos \left (e x +d \right ) e \,g^{2}+2 \,\mathrm {log}\left (f \right ) \sin \left (e x +d \right ) b c \,f^{2}-\sin \left (e x +d \right ) e f g}d x \right ) e^{2} f^{2} g +\left (\int \frac {f^{b c x} \sqrt {\cos \left (e x +d \right ) g +\sin \left (e x +d \right ) f}\, \cos \left (e x +d \right )}{2 \cos \left (e x +d \right ) \mathrm {log}\left (f \right ) b c f g -\cos \left (e x +d \right ) e \,g^{2}+2 \,\mathrm {log}\left (f \right ) \sin \left (e x +d \right ) b c \,f^{2}-\sin \left (e x +d \right ) e f g}d x \right ) e^{2} g^{3}\right )}{2 \,\mathrm {log}\left (f \right ) b c f -e g} \] Input:

int(F^(c*(b*x+a))*(g*cos(e*x+d)+f*sin(e*x+d))^(1/2),x)
 

Output:

(f**(a*c)*(2*f**(b*c*x)*sqrt(cos(d + e*x)*g + sin(d + e*x)*f)*f - 2*int((f 
**(b*c*x)*sqrt(cos(d + e*x)*g + sin(d + e*x)*f)*cos(d + e*x))/(2*cos(d + e 
*x)*log(f)*b*c*f*g - cos(d + e*x)*e*g**2 + 2*log(f)*sin(d + e*x)*b*c*f**2 
- sin(d + e*x)*e*f*g),x)*log(f)*b*c*e*f**3 - 2*int((f**(b*c*x)*sqrt(cos(d 
+ e*x)*g + sin(d + e*x)*f)*cos(d + e*x))/(2*cos(d + e*x)*log(f)*b*c*f*g - 
cos(d + e*x)*e*g**2 + 2*log(f)*sin(d + e*x)*b*c*f**2 - sin(d + e*x)*e*f*g) 
,x)*log(f)*b*c*e*f*g**2 + int((f**(b*c*x)*sqrt(cos(d + e*x)*g + sin(d + e* 
x)*f)*cos(d + e*x))/(2*cos(d + e*x)*log(f)*b*c*f*g - cos(d + e*x)*e*g**2 + 
 2*log(f)*sin(d + e*x)*b*c*f**2 - sin(d + e*x)*e*f*g),x)*e**2*f**2*g + int 
((f**(b*c*x)*sqrt(cos(d + e*x)*g + sin(d + e*x)*f)*cos(d + e*x))/(2*cos(d 
+ e*x)*log(f)*b*c*f*g - cos(d + e*x)*e*g**2 + 2*log(f)*sin(d + e*x)*b*c*f* 
*2 - sin(d + e*x)*e*f*g),x)*e**2*g**3))/(2*log(f)*b*c*f - e*g)