Integrand size = 31, antiderivative size = 125 \[ \int F^{c (a+b x)} \sqrt {g \cosh (d+e x)+f \sinh (d+e x)} \, dx=-\frac {2 F^{c (a+b x)} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{4} \left (-1+\frac {2 b c \log (F)}{e}\right ),\frac {1}{4} \left (3+\frac {2 b c \log (F)}{e}\right ),\frac {e^{2 d+2 e x} (f+g)}{f-g}\right ) \sqrt {g \cosh (d+e x)+f \sinh (d+e x)}}{\sqrt {1-\frac {e^{2 d+2 e x} (f+g)}{f-g}} (e-2 b c \log (F))} \] Output:
-2*F^(c*(b*x+a))*hypergeom([-1/2, -1/4+1/2*b*c*ln(F)/e],[3/4+1/2*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))^(1/2)/(1-exp (2*e*x+2*d)*(f+g)/(f-g))^(1/2)/(e-2*b*c*ln(F))
Time = 5.04 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.86 \[ \int F^{c (a+b x)} \sqrt {g \cosh (d+e x)+f \sinh (d+e x)} \, dx=\frac {2 F^{c (a+b x)} \sqrt {g \cosh (d+e x)+f \sinh (d+e x)} \left (g-\frac {e (f+g) \left (\operatorname {Hypergeometric2F1}\left (1,\frac {e+2 b c \log (F)}{4 e},\frac {3}{4}+\frac {b c \log (F)}{2 e},\frac {(f+g) (\cosh (2 (d+e x))+\sinh (2 (d+e x)))}{f-g}\right ) (3 e+2 b c \log (F))+\operatorname {Hypergeometric2F1}\left (1,\frac {5}{4}+\frac {b c \log (F)}{2 e},\frac {7}{4}+\frac {b c \log (F)}{2 e},\frac {(f+g) (\cosh (2 (d+e x))+\sinh (2 (d+e x)))}{f-g}\right ) (e-2 b c \log (F)) (\cosh (2 (d+e x))+\sinh (2 (d+e x)))\right )}{(e-2 b c \log (F)) (3 e+2 b c \log (F))}\right )}{e f+2 b c g \log (F)} \] Input:
Integrate[F^(c*(a + b*x))*Sqrt[g*Cosh[d + e*x] + f*Sinh[d + e*x]],x]
Output:
(2*F^(c*(a + b*x))*Sqrt[g*Cosh[d + e*x] + f*Sinh[d + e*x]]*(g - (e*(f + g) *(Hypergeometric2F1[1, (e + 2*b*c*Log[F])/(4*e), 3/4 + (b*c*Log[F])/(2*e), ((f + g)*(Cosh[2*(d + e*x)] + Sinh[2*(d + e*x)]))/(f - g)]*(3*e + 2*b*c*L og[F]) + Hypergeometric2F1[1, 5/4 + (b*c*Log[F])/(2*e), 7/4 + (b*c*Log[F]) /(2*e), ((f + g)*(Cosh[2*(d + e*x)] + Sinh[2*(d + e*x)]))/(f - g)]*(e - 2* b*c*Log[F])*(Cosh[2*(d + e*x)] + Sinh[2*(d + e*x)])))/((e - 2*b*c*Log[F])* (3*e + 2*b*c*Log[F]))))/(e*f + 2*b*c*g*Log[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.
| \(\displaystyle \int F^{c (a+b x)} \sqrt {f \sinh (d+e x)+g \cosh (d+e x)} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int F^{a c+b c x} \sqrt {f \sinh (d+e x)+g \cosh (d+e x)}dx\) |
| \(\Big \downarrow \) 7299 |
| \(\displaystyle \int F^{a c+b c x} \sqrt {f \sinh (d+e x)+g \cosh (d+e x)}dx\) |
Input:
Int[F^(c*(a + b*x))*Sqrt[g*Cosh[d + e*x] + f*Sinh[d + e*x]],x]
Output:
$Aborted
\[\int F^{c \left (b x +a \right )} \sqrt {g \cosh \left (e x +d \right )+f \sinh \left (e x +d \right )}d x\]
Input:
int(F^(c*(b*x+a))*(g*cosh(e*x+d)+f*sinh(e*x+d))^(1/2),x)
Output:
int(F^(c*(b*x+a))*(g*cosh(e*x+d)+f*sinh(e*x+d))^(1/2),x)
\[ \int F^{c (a+b x)} \sqrt {g \cosh (d+e x)+f \sinh (d+e x)} \, dx=\int { \sqrt {g \cosh \left (e x + d\right ) + f \sinh \left (e x + d\right )} F^{{\left (b x + a\right )} c} \,d x } \] Input:
integrate(F^(c*(b*x+a))*(g*cosh(e*x+d)+f*sinh(e*x+d))^(1/2),x, algorithm=" fricas")
Output:
integral(sqrt(g*cosh(e*x + d) + f*sinh(e*x + d))*F^(b*c*x + a*c), x)
\[ \int F^{c (a+b x)} \sqrt {g \cosh (d+e x)+f \sinh (d+e x)} \, dx=\int F^{c \left (a + b x\right )} \sqrt {f \sinh {\left (d + e x \right )} + g \cosh {\left (d + e x \right )}}\, dx \] Input:
integrate(F**(c*(b*x+a))*(g*cosh(e*x+d)+f*sinh(e*x+d))**(1/2),x)
Output:
Integral(F**(c*(a + b*x))*sqrt(f*sinh(d + e*x) + g*cosh(d + e*x)), x)
\[ \int F^{c (a+b x)} \sqrt {g \cosh (d+e x)+f \sinh (d+e x)} \, dx=\int { \sqrt {g \cosh \left (e x + d\right ) + f \sinh \left (e x + d\right )} F^{{\left (b x + a\right )} c} \,d x } \] Input:
integrate(F^(c*(b*x+a))*(g*cosh(e*x+d)+f*sinh(e*x+d))^(1/2),x, algorithm=" maxima")
Output:
integrate(sqrt(g*cosh(e*x + d) + f*sinh(e*x + d))*F^((b*x + a)*c), x)
\[ \int F^{c (a+b x)} \sqrt {g \cosh (d+e x)+f \sinh (d+e x)} \, dx=\int { \sqrt {g \cosh \left (e x + d\right ) + f \sinh \left (e x + d\right )} F^{{\left (b x + a\right )} c} \,d x } \] Input:
integrate(F^(c*(b*x+a))*(g*cosh(e*x+d)+f*sinh(e*x+d))^(1/2),x, algorithm=" giac")
Output:
integrate(sqrt(g*cosh(e*x + d) + f*sinh(e*x + d))*F^((b*x + a)*c), x)
Timed out. \[ \int F^{c (a+b x)} \sqrt {g \cosh (d+e x)+f \sinh (d+e x)} \, dx=\int F^{c\,\left (a+b\,x\right )}\,\sqrt {g\,\mathrm {cosh}\left (d+e\,x\right )+f\,\mathrm {sinh}\left (d+e\,x\right )} \,d x \] Input:
int(F^(c*(a + b*x))*(g*cosh(d + e*x) + f*sinh(d + e*x))^(1/2),x)
Output:
int(F^(c*(a + b*x))*(g*cosh(d + e*x) + f*sinh(d + e*x))^(1/2), x)
\[ \int F^{c (a+b x)} \sqrt {g \cosh (d+e x)+f \sinh (d+e x)} \, dx=\frac {f^{a c} \left (2 f^{b c x} \sqrt {\cosh \left (e x +d \right ) g +\sinh \left (e x +d \right ) f}\, f -2 \left (\int \frac {f^{b c x} \sqrt {\cosh \left (e x +d \right ) g +\sinh \left (e x +d \right ) f}\, \cosh \left (e x +d \right )}{2 \cosh \left (e x +d \right ) \mathrm {log}\left (f \right ) b c f g +\cosh \left (e x +d \right ) e \,g^{2}+2 \,\mathrm {log}\left (f \right ) \sinh \left (e x +d \right ) b c \,f^{2}+\sinh \left (e x +d \right ) e f g}d x \right ) \mathrm {log}\left (f \right ) b c e \,f^{3}+2 \left (\int \frac {f^{b c x} \sqrt {\cosh \left (e x +d \right ) g +\sinh \left (e x +d \right ) f}\, \cosh \left (e x +d \right )}{2 \cosh \left (e x +d \right ) \mathrm {log}\left (f \right ) b c f g +\cosh \left (e x +d \right ) e \,g^{2}+2 \,\mathrm {log}\left (f \right ) \sinh \left (e x +d \right ) b c \,f^{2}+\sinh \left (e x +d \right ) e f g}d x \right ) \mathrm {log}\left (f \right ) b c e f \,g^{2}-\left (\int \frac {f^{b c x} \sqrt {\cosh \left (e x +d \right ) g +\sinh \left (e x +d \right ) f}\, \cosh \left (e x +d \right )}{2 \cosh \left (e x +d \right ) \mathrm {log}\left (f \right ) b c f g +\cosh \left (e x +d \right ) e \,g^{2}+2 \,\mathrm {log}\left (f \right ) \sinh \left (e x +d \right ) b c \,f^{2}+\sinh \left (e x +d \right ) e f g}d x \right ) e^{2} f^{2} g +\left (\int \frac {f^{b c x} \sqrt {\cosh \left (e x +d \right ) g +\sinh \left (e x +d \right ) f}\, \cosh \left (e x +d \right )}{2 \cosh \left (e x +d \right ) \mathrm {log}\left (f \right ) b c f g +\cosh \left (e x +d \right ) e \,g^{2}+2 \,\mathrm {log}\left (f \right ) \sinh \left (e x +d \right ) b c \,f^{2}+\sinh \left (e x +d \right ) e f g}d x \right ) e^{2} g^{3}\right )}{2 \,\mathrm {log}\left (f \right ) b c f +e g} \] Input:
int(F^(c*(b*x+a))*(g*cosh(e*x+d)+f*sinh(e*x+d))^(1/2),x)
Output:
(f**(a*c)*(2*f**(b*c*x)*sqrt(cosh(d + e*x)*g + sinh(d + e*x)*f)*f - 2*int( (f**(b*c*x)*sqrt(cosh(d + e*x)*g + sinh(d + e*x)*f)*cosh(d + e*x))/(2*cosh (d + e*x)*log(f)*b*c*f*g + cosh(d + e*x)*e*g**2 + 2*log(f)*sinh(d + e*x)*b *c*f**2 + sinh(d + e*x)*e*f*g),x)*log(f)*b*c*e*f**3 + 2*int((f**(b*c*x)*sq rt(cosh(d + e*x)*g + sinh(d + e*x)*f)*cosh(d + e*x))/(2*cosh(d + e*x)*log( f)*b*c*f*g + cosh(d + e*x)*e*g**2 + 2*log(f)*sinh(d + e*x)*b*c*f**2 + sinh (d + e*x)*e*f*g),x)*log(f)*b*c*e*f*g**2 - int((f**(b*c*x)*sqrt(cosh(d + e* x)*g + sinh(d + e*x)*f)*cosh(d + e*x))/(2*cosh(d + e*x)*log(f)*b*c*f*g + c osh(d + e*x)*e*g**2 + 2*log(f)*sinh(d + e*x)*b*c*f**2 + sinh(d + e*x)*e*f* g),x)*e**2*f**2*g + int((f**(b*c*x)*sqrt(cosh(d + e*x)*g + sinh(d + e*x)*f )*cosh(d + e*x))/(2*cosh(d + e*x)*log(f)*b*c*f*g + cosh(d + e*x)*e*g**2 + 2*log(f)*sinh(d + e*x)*b*c*f**2 + sinh(d + e*x)*e*f*g),x)*e**2*g**3))/(2*l og(f)*b*c*f + e*g)