\(\int F^{c (a+b x)} (g \cosh (d+e x)+f \sinh (d+e x))^{3/2} \, dx\) [28]

Optimal result

Integrand size = 31, antiderivative size = 154 \[ \int F^{c (a+b x)} (g \cosh (d+e x)+f \sinh (d+e x))^{3/2} \, dx=-\frac {2 F^{c (a+b x)} (f-g) \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {1}{4} \left (-3+\frac {2 b c \log (F)}{e}\right ),\frac {e+2 b c \log (F)}{4 e},\frac {e^{2 d+2 e x} (f+g)}{f-g}\right ) (g \cosh (d+e x)+f \sinh (d+e x))^{3/2}}{\left (f-g-e^{2 d+2 e x} (f+g)\right ) \sqrt {1-\frac {e^{2 d+2 e x} (f+g)}{f-g}} (3 e-2 b c \log (F))} \] Output:

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

Mathematica [A] (warning: unable to verify)

Time = 2.59 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.18 \[ \int F^{c (a+b x)} (g \cosh (d+e x)+f \sinh (d+e x))^{3/2} \, dx=\frac {2 F^{c (a+b x)} \sqrt {g \cosh (d+e x)+f \sinh (d+e x)} \left (\cosh (d+e x) (3 e f-2 b c g \log (F))+(3 e g-2 b c f \log (F)) \sinh (d+e x)+\frac {6 e^2 (f+g) \operatorname {Hypergeometric2F1}\left (1,\frac {3}{4}+\frac {b c \log (F)}{2 e},\frac {5}{4}+\frac {b c \log (F)}{2 e},\frac {(f+g) (\cosh (2 (d+e x))+\sinh (2 (d+e x)))}{f-g}\right ) (\cosh (d+e x)+\sinh (d+e x))}{e+2 b c \log (F)}\right )}{9 e^2-4 b^2 c^2 \log ^2(F)} \] Input:

Integrate[F^(c*(a + b*x))*(g*Cosh[d + e*x] + f*Sinh[d + e*x])^(3/2),x]
 

Output:

(2*F^(c*(a + b*x))*Sqrt[g*Cosh[d + e*x] + f*Sinh[d + e*x]]*(Cosh[d + e*x]* 
(3*e*f - 2*b*c*g*Log[F]) + (3*e*g - 2*b*c*f*Log[F])*Sinh[d + e*x] + (6*e^2 
*(f + g)*Hypergeometric2F1[1, 3/4 + (b*c*Log[F])/(2*e), 5/4 + (b*c*Log[F]) 
/(2*e), ((f + g)*(Cosh[2*(d + e*x)] + Sinh[2*(d + e*x)]))/(f - g)]*(Cosh[d 
 + e*x] + Sinh[d + e*x]))/(e + 2*b*c*Log[F])))/(9*e^2 - 4*b^2*c^2*Log[F]^2 
)
 

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.

\(\displaystyle \int F^{c (a+b x)} (f \sinh (d+e x)+g \cosh (d+e x))^{3/2} \, dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int F^{a c+b c x} (f \sinh (d+e x)+g \cosh (d+e x))^{3/2}dx\)

\(\Big \downarrow \) 7299

\(\displaystyle \int F^{a c+b c x} (f \sinh (d+e x)+g \cosh (d+e x))^{3/2}dx\)

Input:

Int[F^(c*(a + b*x))*(g*Cosh[d + e*x] + f*Sinh[d + e*x])^(3/2),x]
 

Output:

$Aborted
 

Maple [F]

\[\int F^{c \left (b x +a \right )} \left (g \cosh \left (e x +d \right )+f \sinh \left (e x +d \right )\right )^{\frac {3}{2}}d x\]

Input:

int(F^(c*(b*x+a))*(g*cosh(e*x+d)+f*sinh(e*x+d))^(3/2),x)
 

Output:

int(F^(c*(b*x+a))*(g*cosh(e*x+d)+f*sinh(e*x+d))^(3/2),x)
 

Fricas [F]

\[ \int F^{c (a+b x)} (g \cosh (d+e x)+f \sinh (d+e x))^{3/2} \, dx=\int { {\left (g \cosh \left (e x + d\right ) + f \sinh \left (e x + d\right )\right )}^{\frac {3}{2}} F^{{\left (b x + a\right )} c} \,d x } \] Input:

integrate(F^(c*(b*x+a))*(g*cosh(e*x+d)+f*sinh(e*x+d))^(3/2),x, algorithm=" 
fricas")
                                                                                    
                                                                                    
 

Output:

integral((g*cosh(e*x + d) + f*sinh(e*x + d))^(3/2)*F^(b*c*x + a*c), x)
 

Sympy [F]

\[ \int F^{c (a+b x)} (g \cosh (d+e x)+f \sinh (d+e x))^{3/2} \, dx=\int F^{c \left (a + b x\right )} \left (f \sinh {\left (d + e x \right )} + g \cosh {\left (d + e x \right )}\right )^{\frac {3}{2}}\, dx \] Input:

integrate(F**(c*(b*x+a))*(g*cosh(e*x+d)+f*sinh(e*x+d))**(3/2),x)
 

Output:

Integral(F**(c*(a + b*x))*(f*sinh(d + e*x) + g*cosh(d + e*x))**(3/2), x)
 

Maxima [F]

\[ \int F^{c (a+b x)} (g \cosh (d+e x)+f \sinh (d+e x))^{3/2} \, dx=\int { {\left (g \cosh \left (e x + d\right ) + f \sinh \left (e x + d\right )\right )}^{\frac {3}{2}} F^{{\left (b x + a\right )} c} \,d x } \] Input:

integrate(F^(c*(b*x+a))*(g*cosh(e*x+d)+f*sinh(e*x+d))^(3/2),x, algorithm=" 
maxima")
 

Output:

integrate((g*cosh(e*x + d) + f*sinh(e*x + d))^(3/2)*F^((b*x + a)*c), x)
 

Giac [F]

\[ \int F^{c (a+b x)} (g \cosh (d+e x)+f \sinh (d+e x))^{3/2} \, dx=\int { {\left (g \cosh \left (e x + d\right ) + f \sinh \left (e x + d\right )\right )}^{\frac {3}{2}} F^{{\left (b x + a\right )} c} \,d x } \] Input:

integrate(F^(c*(b*x+a))*(g*cosh(e*x+d)+f*sinh(e*x+d))^(3/2),x, algorithm=" 
giac")
 

Output:

integrate((g*cosh(e*x + d) + f*sinh(e*x + d))^(3/2)*F^((b*x + a)*c), x)
 

Mupad [F(-1)]

Timed out. \[ \int F^{c (a+b x)} (g \cosh (d+e x)+f \sinh (d+e x))^{3/2} \, dx=\int F^{c\,\left (a+b\,x\right )}\,{\left (g\,\mathrm {cosh}\left (d+e\,x\right )+f\,\mathrm {sinh}\left (d+e\,x\right )\right )}^{3/2} \,d x \] Input:

int(F^(c*(a + b*x))*(g*cosh(d + e*x) + f*sinh(d + e*x))^(3/2),x)
 

Output:

int(F^(c*(a + b*x))*(g*cosh(d + e*x) + f*sinh(d + e*x))^(3/2), x)
 

Reduce [F]

\[ \int F^{c (a+b x)} (g \cosh (d+e x)+f \sinh (d+e x))^{3/2} \, dx=\text {too large to display} \] Input:

int(F^(c*(b*x+a))*(g*cosh(e*x+d)+f*sinh(e*x+d))^(3/2),x)
 

Output:

(f**(a*c)*(4*f**(b*c*x)*sqrt(cosh(d + e*x)*g + sinh(d + e*x)*f)*cosh(d + e 
*x)*log(f)*b*c*g**3 - 2*f**(b*c*x)*sqrt(cosh(d + e*x)*g + sinh(d + e*x)*f) 
*cosh(d + e*x)*e*f*g**2 + 4*f**(b*c*x)*sqrt(cosh(d + e*x)*g + sinh(d + e*x 
)*f)*log(f)*sinh(d + e*x)*b*c*f*g**2 + 4*f**(b*c*x)*sqrt(cosh(d + e*x)*g + 
 sinh(d + e*x)*f)*sinh(d + e*x)*e*f**2*g - 6*f**(b*c*x)*sqrt(cosh(d + e*x) 
*g + sinh(d + e*x)*f)*sinh(d + e*x)*e*g**3 + 12*int((f**(b*c*x)*sqrt(cosh( 
d + e*x)*g + sinh(d + e*x)*f)*sinh(d + e*x)**2)/(4*cosh(d + e*x)*log(f)**2 
*b**2*c**2*g**3 + 4*cosh(d + e*x)*log(f)*b*c*e*f*g**2 + 3*cosh(d + e*x)*e* 
*2*f**2*g - 6*cosh(d + e*x)*e**2*g**3 + 4*log(f)**2*sinh(d + e*x)*b**2*c** 
2*f*g**2 + 4*log(f)*sinh(d + e*x)*b*c*e*f**2*g + 3*sinh(d + e*x)*e**2*f**3 
 - 6*sinh(d + e*x)*e**2*f*g**2),x)*log(f)**2*b**2*c**2*e**2*f**4*g**2 - 24 
*int((f**(b*c*x)*sqrt(cosh(d + e*x)*g + sinh(d + e*x)*f)*sinh(d + e*x)**2) 
/(4*cosh(d + e*x)*log(f)**2*b**2*c**2*g**3 + 4*cosh(d + e*x)*log(f)*b*c*e* 
f*g**2 + 3*cosh(d + e*x)*e**2*f**2*g - 6*cosh(d + e*x)*e**2*g**3 + 4*log(f 
)**2*sinh(d + e*x)*b**2*c**2*f*g**2 + 4*log(f)*sinh(d + e*x)*b*c*e*f**2*g 
+ 3*sinh(d + e*x)*e**2*f**3 - 6*sinh(d + e*x)*e**2*f*g**2),x)*log(f)**2*b* 
*2*c**2*e**2*f**2*g**4 + 12*int((f**(b*c*x)*sqrt(cosh(d + e*x)*g + sinh(d 
+ e*x)*f)*sinh(d + e*x)**2)/(4*cosh(d + e*x)*log(f)**2*b**2*c**2*g**3 + 4* 
cosh(d + e*x)*log(f)*b*c*e*f*g**2 + 3*cosh(d + e*x)*e**2*f**2*g - 6*cosh(d 
 + e*x)*e**2*g**3 + 4*log(f)**2*sinh(d + e*x)*b**2*c**2*f*g**2 + 4*log(...