\(\int \frac {3^{-\frac {2}{-5-x-x^2+\log (x)}} (-4050-1620 x-1782 x^2-324 x^3-162 x^4+(54-54 x-108 x^2) \log (3)+(1620+324 x+324 x^2) \log (x)-162 \log ^2(x))}{25 x^7+10 x^8+11 x^9+2 x^{10}+x^{11}+(-10 x^7-2 x^8-2 x^9) \log (x)+x^7 \log ^2(x)} \, dx\) [1221]

Optimal result

Integrand size = 124, antiderivative size = 24 \[ \int \frac {3^{-\frac {2}{-5-x-x^2+\log (x)}} \left (-4050-1620 x-1782 x^2-324 x^3-162 x^4+\left (54-54 x-108 x^2\right ) \log (3)+\left (1620+324 x+324 x^2\right ) \log (x)-162 \log ^2(x)\right )}{25 x^7+10 x^8+11 x^9+2 x^{10}+x^{11}+\left (-10 x^7-2 x^8-2 x^9\right ) \log (x)+x^7 \log ^2(x)} \, dx=-3+\frac {3^{3+\frac {2}{5+x+x^2-\log (x)}}}{x^6} \] Output:

27/x^6*exp(ln(3)/(5+x^2-ln(x)+x))^2-3
 

Mathematica [F]

\[ \int \frac {3^{-\frac {2}{-5-x-x^2+\log (x)}} \left (-4050-1620 x-1782 x^2-324 x^3-162 x^4+\left (54-54 x-108 x^2\right ) \log (3)+\left (1620+324 x+324 x^2\right ) \log (x)-162 \log ^2(x)\right )}{25 x^7+10 x^8+11 x^9+2 x^{10}+x^{11}+\left (-10 x^7-2 x^8-2 x^9\right ) \log (x)+x^7 \log ^2(x)} \, dx=\int \frac {3^{-\frac {2}{-5-x-x^2+\log (x)}} \left (-4050-1620 x-1782 x^2-324 x^3-162 x^4+\left (54-54 x-108 x^2\right ) \log (3)+\left (1620+324 x+324 x^2\right ) \log (x)-162 \log ^2(x)\right )}{25 x^7+10 x^8+11 x^9+2 x^{10}+x^{11}+\left (-10 x^7-2 x^8-2 x^9\right ) \log (x)+x^7 \log ^2(x)} \, dx \] Input:

Integrate[(-4050 - 1620*x - 1782*x^2 - 324*x^3 - 162*x^4 + (54 - 54*x - 10 
8*x^2)*Log[3] + (1620 + 324*x + 324*x^2)*Log[x] - 162*Log[x]^2)/(3^(2/(-5 
- x - x^2 + Log[x]))*(25*x^7 + 10*x^8 + 11*x^9 + 2*x^10 + x^11 + (-10*x^7 
- 2*x^8 - 2*x^9)*Log[x] + x^7*Log[x]^2)),x]
 

Output:

Integrate[(-4050 - 1620*x - 1782*x^2 - 324*x^3 - 162*x^4 + (54 - 54*x - 10 
8*x^2)*Log[3] + (1620 + 324*x + 324*x^2)*Log[x] - 162*Log[x]^2)/(3^(2/(-5 
- x - x^2 + Log[x]))*(25*x^7 + 10*x^8 + 11*x^9 + 2*x^10 + x^11 + (-10*x^7 
- 2*x^8 - 2*x^9)*Log[x] + x^7*Log[x]^2)), x]
 

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(104\) vs. \(2(24)=48\).

Time = 0.41 (sec) , antiderivative size = 104, normalized size of antiderivative = 4.33, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.008, Rules used = {2726}

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.

\(\displaystyle \int \frac {3^{-\frac {2}{-x^2-x+\log (x)-5}} \left (-162 x^4-324 x^3-1782 x^2+\left (324 x^2+324 x+1620\right ) \log (x)+\left (-108 x^2-54 x+54\right ) \log (3)-1620 x-162 \log ^2(x)-4050\right )}{x^{11}+2 x^{10}+11 x^9+10 x^8+25 x^7+x^7 \log ^2(x)+\left (-2 x^9-2 x^8-10 x^7\right ) \log (x)} \, dx\)

\(\Big \downarrow \) 2726

\(\displaystyle -\frac {\left (-2 x^2-x+1\right ) 3^{\frac {2}{x^2+x-\log (x)+5}+3} \left (x^2+x-\log (x)+5\right )^2}{\left (2 x-\frac {1}{x}+1\right ) \left (x^{11}+2 x^{10}+11 x^9+10 x^8+25 x^7+x^7 \log ^2(x)-2 \left (x^9+x^8+5 x^7\right ) \log (x)\right )}\)

Input:

Int[(-4050 - 1620*x - 1782*x^2 - 324*x^3 - 162*x^4 + (54 - 54*x - 108*x^2) 
*Log[3] + (1620 + 324*x + 324*x^2)*Log[x] - 162*Log[x]^2)/(3^(2/(-5 - x - 
x^2 + Log[x]))*(25*x^7 + 10*x^8 + 11*x^9 + 2*x^10 + x^11 + (-10*x^7 - 2*x^ 
8 - 2*x^9)*Log[x] + x^7*Log[x]^2)),x]
 

Output:

-((3^(3 + 2/(5 + x + x^2 - Log[x]))*(1 - x - 2*x^2)*(5 + x + x^2 - Log[x]) 
^2)/((1 - x^(-1) + 2*x)*(25*x^7 + 10*x^8 + 11*x^9 + 2*x^10 + x^11 - 2*(5*x 
^7 + x^8 + x^9)*Log[x] + x^7*Log[x]^2)))
 

Defintions of rubi rules used

rule 2726

Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = v*(y/(Log[F]*D[u, 
 x]))}, Simp[F^u*z, x] /; EqQ[D[z, x], w*y]] /; FreeQ[F, x]
 

Maple [A] (verified)

Time = 31.74 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00

method	result	size
risch	\(\frac {27 \,3^{-\frac {2}{\ln \left (x \right )-x^{2}-x -5}}}{x^{6}}\)	\(24\)
parallelrisch	\(\frac {27 \,{\mathrm e}^{-\frac {2 \ln \left (3\right )}{\ln \left (x \right )-x^{2}-x -5}}}{x^{6}}\)	\(27\)

Input:

int((-162*ln(x)^2+(324*x^2+324*x+1620)*ln(x)+(-108*x^2-54*x+54)*ln(3)-162* 
x^4-324*x^3-1782*x^2-1620*x-4050)*exp(-ln(3)/(ln(x)-x^2-x-5))^2/(x^7*ln(x) 
^2+(-2*x^9-2*x^8-10*x^7)*ln(x)+x^11+2*x^10+11*x^9+10*x^8+25*x^7),x,method= 
_RETURNVERBOSE)
 

Output:

27/x^6*((1/3)^(1/(ln(x)-x^2-x-5)))^2
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.88 \[ \int \frac {3^{-\frac {2}{-5-x-x^2+\log (x)}} \left (-4050-1620 x-1782 x^2-324 x^3-162 x^4+\left (54-54 x-108 x^2\right ) \log (3)+\left (1620+324 x+324 x^2\right ) \log (x)-162 \log ^2(x)\right )}{25 x^7+10 x^8+11 x^9+2 x^{10}+x^{11}+\left (-10 x^7-2 x^8-2 x^9\right ) \log (x)+x^7 \log ^2(x)} \, dx=\frac {27 \cdot 3^{\frac {2}{x^{2} + x - \log \left (x\right ) + 5}}}{x^{6}} \] Input:

integrate((-162*log(x)^2+(324*x^2+324*x+1620)*log(x)+(-108*x^2-54*x+54)*lo 
g(3)-162*x^4-324*x^3-1782*x^2-1620*x-4050)*exp(-log(3)/(log(x)-x^2-x-5))^2 
/(x^7*log(x)^2+(-2*x^9-2*x^8-10*x^7)*log(x)+x^11+2*x^10+11*x^9+10*x^8+25*x 
^7),x, algorithm="fricas")
 

Output:

27*3^(2/(x^2 + x - log(x) + 5))/x^6
 

Sympy [A] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.83 \[ \int \frac {3^{-\frac {2}{-5-x-x^2+\log (x)}} \left (-4050-1620 x-1782 x^2-324 x^3-162 x^4+\left (54-54 x-108 x^2\right ) \log (3)+\left (1620+324 x+324 x^2\right ) \log (x)-162 \log ^2(x)\right )}{25 x^7+10 x^8+11 x^9+2 x^{10}+x^{11}+\left (-10 x^7-2 x^8-2 x^9\right ) \log (x)+x^7 \log ^2(x)} \, dx=\frac {27 e^{- \frac {2 \log {\left (3 \right )}}{- x^{2} - x + \log {\left (x \right )} - 5}}}{x^{6}} \] Input:

integrate((-162*ln(x)**2+(324*x**2+324*x+1620)*ln(x)+(-108*x**2-54*x+54)*l 
n(3)-162*x**4-324*x**3-1782*x**2-1620*x-4050)*exp(-ln(3)/(ln(x)-x**2-x-5)) 
**2/(x**7*ln(x)**2+(-2*x**9-2*x**8-10*x**7)*ln(x)+x**11+2*x**10+11*x**9+10 
*x**8+25*x**7),x)
 

Output:

27*exp(-2*log(3)/(-x**2 - x + log(x) - 5))/x**6
                                                                                    
                                                                                    
 

Maxima [F]

\[ \int \frac {3^{-\frac {2}{-5-x-x^2+\log (x)}} \left (-4050-1620 x-1782 x^2-324 x^3-162 x^4+\left (54-54 x-108 x^2\right ) \log (3)+\left (1620+324 x+324 x^2\right ) \log (x)-162 \log ^2(x)\right )}{25 x^7+10 x^8+11 x^9+2 x^{10}+x^{11}+\left (-10 x^7-2 x^8-2 x^9\right ) \log (x)+x^7 \log ^2(x)} \, dx=\int { -\frac {54 \, {\left (3 \, x^{4} + 6 \, x^{3} + 33 \, x^{2} + {\left (2 \, x^{2} + x - 1\right )} \log \left (3\right ) - 6 \, {\left (x^{2} + x + 5\right )} \log \left (x\right ) + 3 \, \log \left (x\right )^{2} + 30 \, x + 75\right )} 3^{\frac {2}{x^{2} + x - \log \left (x\right ) + 5}}}{x^{11} + 2 \, x^{10} + 11 \, x^{9} + x^{7} \log \left (x\right )^{2} + 10 \, x^{8} + 25 \, x^{7} - 2 \, {\left (x^{9} + x^{8} + 5 \, x^{7}\right )} \log \left (x\right )} \,d x } \] Input:

integrate((-162*log(x)^2+(324*x^2+324*x+1620)*log(x)+(-108*x^2-54*x+54)*lo 
g(3)-162*x^4-324*x^3-1782*x^2-1620*x-4050)*exp(-log(3)/(log(x)-x^2-x-5))^2 
/(x^7*log(x)^2+(-2*x^9-2*x^8-10*x^7)*log(x)+x^11+2*x^10+11*x^9+10*x^8+25*x 
^7),x, algorithm="maxima")
 

Output:

-27*3^(2/(x^2 + x - log(x) + 5))*log(3)/(2*x^8*log(3) + x^7*log(3) - x^6*l 
og(3)) + 27*3^(2/(x^2 + x - log(x) + 5))*log(3)/(2*x^7*log(3) + x^6*log(3) 
 - x^5*log(3)) + 54*3^(2/(x^2 + x - log(x) + 5))*log(3)/(2*x^6*log(3) + x^ 
5*log(3) - x^4*log(3)) + 2025*3^(2/(x^2 + x - log(x) + 5))/(2*x^8*log(3) + 
 x^7*log(3) - x^6*log(3)) + 810*3^(2/(x^2 + x - log(x) + 5))/(2*x^7*log(3) 
 + x^6*log(3) - x^5*log(3)) + 891*3^(2/(x^2 + x - log(x) + 5))/(2*x^6*log( 
3) + x^5*log(3) - x^4*log(3)) + 162*3^(2/(x^2 + x - log(x) + 5))/(2*x^5*lo 
g(3) + x^4*log(3) - x^3*log(3)) + 81*3^(2/(x^2 + x - log(x) + 5))/(2*x^4*l 
og(3) + x^3*log(3) - x^2*log(3)) - 2*integrate(-81*(2*(x^2 + x + 5)*log(x) 
 - log(x)^2)*3^(2/(x^2 + x - log(x) + 5))/(x^11 + 2*x^10 + 11*x^9 + x^7*lo 
g(x)^2 + 10*x^8 + 25*x^7 - 2*(x^9 + x^8 + 5*x^7)*log(x)), x)
 

Giac [F]

\[ \int \frac {3^{-\frac {2}{-5-x-x^2+\log (x)}} \left (-4050-1620 x-1782 x^2-324 x^3-162 x^4+\left (54-54 x-108 x^2\right ) \log (3)+\left (1620+324 x+324 x^2\right ) \log (x)-162 \log ^2(x)\right )}{25 x^7+10 x^8+11 x^9+2 x^{10}+x^{11}+\left (-10 x^7-2 x^8-2 x^9\right ) \log (x)+x^7 \log ^2(x)} \, dx=\int { -\frac {54 \, {\left (3 \, x^{4} + 6 \, x^{3} + 33 \, x^{2} + {\left (2 \, x^{2} + x - 1\right )} \log \left (3\right ) - 6 \, {\left (x^{2} + x + 5\right )} \log \left (x\right ) + 3 \, \log \left (x\right )^{2} + 30 \, x + 75\right )} 3^{\frac {2}{x^{2} + x - \log \left (x\right ) + 5}}}{x^{11} + 2 \, x^{10} + 11 \, x^{9} + x^{7} \log \left (x\right )^{2} + 10 \, x^{8} + 25 \, x^{7} - 2 \, {\left (x^{9} + x^{8} + 5 \, x^{7}\right )} \log \left (x\right )} \,d x } \] Input:

integrate((-162*log(x)^2+(324*x^2+324*x+1620)*log(x)+(-108*x^2-54*x+54)*lo 
g(3)-162*x^4-324*x^3-1782*x^2-1620*x-4050)*exp(-log(3)/(log(x)-x^2-x-5))^2 
/(x^7*log(x)^2+(-2*x^9-2*x^8-10*x^7)*log(x)+x^11+2*x^10+11*x^9+10*x^8+25*x 
^7),x, algorithm="giac")
 

Output:

integrate(-54*(3*x^4 + 6*x^3 + 33*x^2 + (2*x^2 + x - 1)*log(3) - 6*(x^2 + 
x + 5)*log(x) + 3*log(x)^2 + 30*x + 75)*3^(2/(x^2 + x - log(x) + 5))/(x^11 
 + 2*x^10 + 11*x^9 + x^7*log(x)^2 + 10*x^8 + 25*x^7 - 2*(x^9 + x^8 + 5*x^7 
)*log(x)), x)
 

Mupad [B] (verification not implemented)

Time = 7.67 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.88 \[ \int \frac {3^{-\frac {2}{-5-x-x^2+\log (x)}} \left (-4050-1620 x-1782 x^2-324 x^3-162 x^4+\left (54-54 x-108 x^2\right ) \log (3)+\left (1620+324 x+324 x^2\right ) \log (x)-162 \log ^2(x)\right )}{25 x^7+10 x^8+11 x^9+2 x^{10}+x^{11}+\left (-10 x^7-2 x^8-2 x^9\right ) \log (x)+x^7 \log ^2(x)} \, dx=\frac {27\,3^{\frac {2}{x-\ln \left (x\right )+x^2+5}}}{x^6} \] Input:

int(-(exp((2*log(3))/(x - log(x) + x^2 + 5))*(1620*x + log(3)*(54*x + 108* 
x^2 - 54) + 162*log(x)^2 - log(x)*(324*x + 324*x^2 + 1620) + 1782*x^2 + 32 
4*x^3 + 162*x^4 + 4050))/(x^7*log(x)^2 - log(x)*(10*x^7 + 2*x^8 + 2*x^9) + 
 25*x^7 + 10*x^8 + 11*x^9 + 2*x^10 + x^11),x)
 

Output:

(27*3^(2/(x - log(x) + x^2 + 5)))/x^6
 

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.12 \[ \int \frac {3^{-\frac {2}{-5-x-x^2+\log (x)}} \left (-4050-1620 x-1782 x^2-324 x^3-162 x^4+\left (54-54 x-108 x^2\right ) \log (3)+\left (1620+324 x+324 x^2\right ) \log (x)-162 \log ^2(x)\right )}{25 x^7+10 x^8+11 x^9+2 x^{10}+x^{11}+\left (-10 x^7-2 x^8-2 x^9\right ) \log (x)+x^7 \log ^2(x)} \, dx=\frac {27}{e^{\frac {2 \,\mathrm {log}\left (3\right )}{\mathrm {log}\left (x \right )-x^{2}-x -5}} x^{6}} \] Input:

int((-162*log(x)^2+(324*x^2+324*x+1620)*log(x)+(-108*x^2-54*x+54)*log(3)-1 
62*x^4-324*x^3-1782*x^2-1620*x-4050)*exp(-log(3)/(log(x)-x^2-x-5))^2/(x^7* 
log(x)^2+(-2*x^9-2*x^8-10*x^7)*log(x)+x^11+2*x^10+11*x^9+10*x^8+25*x^7),x)
 

Output:

27/(e**((2*log(3))/(log(x) - x**2 - x - 5))*x**6)