\(\int \frac {98 x+308 x^2+242 x^3+e^{\frac {1}{7 x+11 x^2}} (49 x+154 x^2+121 x^3)+(98 x+308 x^2+242 x^3+e^{\frac {1}{7 x+11 x^2}} (7+71 x+154 x^2+121 x^3)) \log (x)}{196 x+616 x^2+484 x^3+e^{\frac {2}{7 x+11 x^2}} (49 x+154 x^2+121 x^3)+e^{\frac {1}{7 x+11 x^2}} (196 x+616 x^2+484 x^3)} \, dx\) [392]

Optimal result

Integrand size = 163, antiderivative size = 21 \[ \int \frac {98 x+308 x^2+242 x^3+e^{\frac {1}{7 x+11 x^2}} \left (49 x+154 x^2+121 x^3\right )+\left (98 x+308 x^2+242 x^3+e^{\frac {1}{7 x+11 x^2}} \left (7+71 x+154 x^2+121 x^3\right )\right ) \log (x)}{196 x+616 x^2+484 x^3+e^{\frac {2}{7 x+11 x^2}} \left (49 x+154 x^2+121 x^3\right )+e^{\frac {1}{7 x+11 x^2}} \left (196 x+616 x^2+484 x^3\right )} \, dx=\frac {x \log (x)}{2+e^{\frac {1}{x (7+11 x)}}} \] Output:

ln(x)/(2+exp(1/x/(11*x+7)))*x
 

Mathematica [A] (verified)

Time = 0.50 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int \frac {98 x+308 x^2+242 x^3+e^{\frac {1}{7 x+11 x^2}} \left (49 x+154 x^2+121 x^3\right )+\left (98 x+308 x^2+242 x^3+e^{\frac {1}{7 x+11 x^2}} \left (7+71 x+154 x^2+121 x^3\right )\right ) \log (x)}{196 x+616 x^2+484 x^3+e^{\frac {2}{7 x+11 x^2}} \left (49 x+154 x^2+121 x^3\right )+e^{\frac {1}{7 x+11 x^2}} \left (196 x+616 x^2+484 x^3\right )} \, dx=\frac {x \log (x)}{2+e^{\frac {1}{7 x+11 x^2}}} \] Input:

Integrate[(98*x + 308*x^2 + 242*x^3 + E^(7*x + 11*x^2)^(-1)*(49*x + 154*x^ 
2 + 121*x^3) + (98*x + 308*x^2 + 242*x^3 + E^(7*x + 11*x^2)^(-1)*(7 + 71*x 
 + 154*x^2 + 121*x^3))*Log[x])/(196*x + 616*x^2 + 484*x^3 + E^(2/(7*x + 11 
*x^2))*(49*x + 154*x^2 + 121*x^3) + E^(7*x + 11*x^2)^(-1)*(196*x + 616*x^2 
 + 484*x^3)),x]
 

Output:

(x*Log[x])/(2 + E^(7*x + 11*x^2)^(-1))
 

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.

\(\displaystyle \int \frac {242 x^3+308 x^2+e^{\frac {1}{11 x^2+7 x}} \left (121 x^3+154 x^2+49 x\right )+\left (242 x^3+308 x^2+e^{\frac {1}{11 x^2+7 x}} \left (121 x^3+154 x^2+71 x+7\right )+98 x\right ) \log (x)+98 x}{484 x^3+616 x^2+e^{\frac {2}{11 x^2+7 x}} \left (121 x^3+154 x^2+49 x\right )+e^{\frac {1}{11 x^2+7 x}} \left (484 x^3+616 x^2+196 x\right )+196 x} \, dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {e^{\frac {1}{11 x^2+7 x}}+\frac {e^{\frac {1}{11 x^2+7 x}} \left (121 x^3+154 x^2+71 x+7\right ) \log (x)}{x (11 x+7)^2}+2 \log (x)+2}{\left (e^{\frac {1}{11 x^2+7 x}}+2\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {121 x^3+121 x^3 \log (x)+154 x^2+154 x^2 \log (x)+49 x+71 x \log (x)+7 \log (x)}{\left (e^{\frac {1}{11 x^2+7 x}}+2\right ) x (11 x+7)^2}-\frac {2 (22 x+7) \log (x)}{\left (e^{\frac {1}{11 x^2+7 x}}+2\right )^2 x (11 x+7)^2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \int \frac {1}{2+e^{\frac {1}{11 x^2+7 x}}}dx-\int \frac {\int \frac {1}{2+e^{\frac {1}{11 x^2+7 x}}}dx}{x}dx+\frac {2}{7} \int \frac {\int \frac {1}{\left (2+e^{\frac {1}{11 x^2+7 x}}\right )^2 x}dx}{x}dx+22 \int \frac {\int \frac {1}{\left (2+e^{\frac {1}{11 x^2+7 x}}\right )^2 (11 x+7)^2}dx}{x}dx-11 \int \frac {\int \frac {1}{\left (2+e^{\frac {1}{11 x^2+7 x}}\right ) (11 x+7)^2}dx}{x}dx-\frac {22}{7} \int \frac {\int \frac {1}{\left (2+e^{\frac {1}{11 x^2+7 x}}\right )^2 (11 x+7)}dx}{x}dx+\frac {11}{7} \int \frac {\int \frac {1}{\left (2+e^{\frac {1}{11 x^2+7 x}}\right ) (11 x+7)}dx}{x}dx-\frac {1}{7} \int \frac {\int \frac {1}{e^{\frac {1}{11 x^2+7 x}} x+2 x}dx}{x}dx+\log (x) \int \frac {1}{2+e^{\frac {1}{11 x^2+7 x}}}dx-\frac {2}{7} \log (x) \int \frac {1}{\left (2+e^{\frac {1}{11 x^2+7 x}}\right )^2 x}dx+\frac {1}{7} \log (x) \int \frac {1}{\left (2+e^{\frac {1}{11 x^2+7 x}}\right ) x}dx-22 \log (x) \int \frac {1}{\left (2+e^{\frac {1}{11 x^2+7 x}}\right )^2 (11 x+7)^2}dx+11 \log (x) \int \frac {1}{\left (2+e^{\frac {1}{11 x^2+7 x}}\right ) (11 x+7)^2}dx+\frac {22}{7} \log (x) \int \frac {1}{\left (2+e^{\frac {1}{11 x^2+7 x}}\right )^2 (11 x+7)}dx-\frac {11}{7} \log (x) \int \frac {1}{\left (2+e^{\frac {1}{11 x^2+7 x}}\right ) (11 x+7)}dx\)

Input:

Int[(98*x + 308*x^2 + 242*x^3 + E^(7*x + 11*x^2)^(-1)*(49*x + 154*x^2 + 12 
1*x^3) + (98*x + 308*x^2 + 242*x^3 + E^(7*x + 11*x^2)^(-1)*(7 + 71*x + 154 
*x^2 + 121*x^3))*Log[x])/(196*x + 616*x^2 + 484*x^3 + E^(2/(7*x + 11*x^2)) 
*(49*x + 154*x^2 + 121*x^3) + E^(7*x + 11*x^2)^(-1)*(196*x + 616*x^2 + 484 
*x^3)),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 148.38 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00

method	result	size
risch	\(\frac {\ln \left (x \right ) x}{2+{\mathrm e}^{\frac {1}{x \left (11 x +7\right )}}}\)	\(21\)
parallelrisch	\(\frac {\ln \left (x \right ) x}{2+{\mathrm e}^{\frac {1}{x \left (11 x +7\right )}}}\)	\(21\)

Input:

int((((121*x^3+154*x^2+71*x+7)*exp(1/(11*x^2+7*x))+242*x^3+308*x^2+98*x)*l 
n(x)+(121*x^3+154*x^2+49*x)*exp(1/(11*x^2+7*x))+242*x^3+308*x^2+98*x)/((12 
1*x^3+154*x^2+49*x)*exp(1/(11*x^2+7*x))^2+(484*x^3+616*x^2+196*x)*exp(1/(1 
1*x^2+7*x))+484*x^3+616*x^2+196*x),x,method=_RETURNVERBOSE)
 

Output:

ln(x)/(2+exp(1/x/(11*x+7)))*x
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.95 \[ \int \frac {98 x+308 x^2+242 x^3+e^{\frac {1}{7 x+11 x^2}} \left (49 x+154 x^2+121 x^3\right )+\left (98 x+308 x^2+242 x^3+e^{\frac {1}{7 x+11 x^2}} \left (7+71 x+154 x^2+121 x^3\right )\right ) \log (x)}{196 x+616 x^2+484 x^3+e^{\frac {2}{7 x+11 x^2}} \left (49 x+154 x^2+121 x^3\right )+e^{\frac {1}{7 x+11 x^2}} \left (196 x+616 x^2+484 x^3\right )} \, dx=\frac {x \log \left (x\right )}{e^{\left (\frac {1}{11 \, x^{2} + 7 \, x}\right )} + 2} \] Input:

integrate((((121*x^3+154*x^2+71*x+7)*exp(1/(11*x^2+7*x))+242*x^3+308*x^2+9 
8*x)*log(x)+(121*x^3+154*x^2+49*x)*exp(1/(11*x^2+7*x))+242*x^3+308*x^2+98* 
x)/((121*x^3+154*x^2+49*x)*exp(1/(11*x^2+7*x))^2+(484*x^3+616*x^2+196*x)*e 
xp(1/(11*x^2+7*x))+484*x^3+616*x^2+196*x),x, algorithm="fricas")
 

Output:

x*log(x)/(e^(1/(11*x^2 + 7*x)) + 2)
 

Sympy [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.81 \[ \int \frac {98 x+308 x^2+242 x^3+e^{\frac {1}{7 x+11 x^2}} \left (49 x+154 x^2+121 x^3\right )+\left (98 x+308 x^2+242 x^3+e^{\frac {1}{7 x+11 x^2}} \left (7+71 x+154 x^2+121 x^3\right )\right ) \log (x)}{196 x+616 x^2+484 x^3+e^{\frac {2}{7 x+11 x^2}} \left (49 x+154 x^2+121 x^3\right )+e^{\frac {1}{7 x+11 x^2}} \left (196 x+616 x^2+484 x^3\right )} \, dx=\frac {x \log {\left (x \right )}}{e^{\frac {1}{11 x^{2} + 7 x}} + 2} \] Input:

integrate((((121*x**3+154*x**2+71*x+7)*exp(1/(11*x**2+7*x))+242*x**3+308*x 
**2+98*x)*ln(x)+(121*x**3+154*x**2+49*x)*exp(1/(11*x**2+7*x))+242*x**3+308 
*x**2+98*x)/((121*x**3+154*x**2+49*x)*exp(1/(11*x**2+7*x))**2+(484*x**3+61 
6*x**2+196*x)*exp(1/(11*x**2+7*x))+484*x**3+616*x**2+196*x),x)
 

Output:

x*log(x)/(exp(1/(11*x**2 + 7*x)) + 2)
 

Maxima [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.67 \[ \int \frac {98 x+308 x^2+242 x^3+e^{\frac {1}{7 x+11 x^2}} \left (49 x+154 x^2+121 x^3\right )+\left (98 x+308 x^2+242 x^3+e^{\frac {1}{7 x+11 x^2}} \left (7+71 x+154 x^2+121 x^3\right )\right ) \log (x)}{196 x+616 x^2+484 x^3+e^{\frac {2}{7 x+11 x^2}} \left (49 x+154 x^2+121 x^3\right )+e^{\frac {1}{7 x+11 x^2}} \left (196 x+616 x^2+484 x^3\right )} \, dx=\frac {x e^{\left (\frac {11}{7 \, {\left (11 \, x + 7\right )}}\right )} \log \left (x\right )}{2 \, e^{\left (\frac {11}{7 \, {\left (11 \, x + 7\right )}}\right )} + e^{\left (\frac {1}{7 \, x}\right )}} \] Input:

integrate((((121*x^3+154*x^2+71*x+7)*exp(1/(11*x^2+7*x))+242*x^3+308*x^2+9 
8*x)*log(x)+(121*x^3+154*x^2+49*x)*exp(1/(11*x^2+7*x))+242*x^3+308*x^2+98* 
x)/((121*x^3+154*x^2+49*x)*exp(1/(11*x^2+7*x))^2+(484*x^3+616*x^2+196*x)*e 
xp(1/(11*x^2+7*x))+484*x^3+616*x^2+196*x),x, algorithm="maxima")
 

Output:

x*e^(11/7/(11*x + 7))*log(x)/(2*e^(11/7/(11*x + 7)) + e^(1/7/x))
 

Giac [A] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.95 \[ \int \frac {98 x+308 x^2+242 x^3+e^{\frac {1}{7 x+11 x^2}} \left (49 x+154 x^2+121 x^3\right )+\left (98 x+308 x^2+242 x^3+e^{\frac {1}{7 x+11 x^2}} \left (7+71 x+154 x^2+121 x^3\right )\right ) \log (x)}{196 x+616 x^2+484 x^3+e^{\frac {2}{7 x+11 x^2}} \left (49 x+154 x^2+121 x^3\right )+e^{\frac {1}{7 x+11 x^2}} \left (196 x+616 x^2+484 x^3\right )} \, dx=\frac {x \log \left (x\right )}{e^{\left (\frac {1}{11 \, x^{2} + 7 \, x}\right )} + 2} \] Input:

integrate((((121*x^3+154*x^2+71*x+7)*exp(1/(11*x^2+7*x))+242*x^3+308*x^2+9 
8*x)*log(x)+(121*x^3+154*x^2+49*x)*exp(1/(11*x^2+7*x))+242*x^3+308*x^2+98* 
x)/((121*x^3+154*x^2+49*x)*exp(1/(11*x^2+7*x))^2+(484*x^3+616*x^2+196*x)*e 
xp(1/(11*x^2+7*x))+484*x^3+616*x^2+196*x),x, algorithm="giac")
 

Output:

x*log(x)/(e^(1/(11*x^2 + 7*x)) + 2)
 

Mupad [B] (verification not implemented)

Time = 4.33 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.95 \[ \int \frac {98 x+308 x^2+242 x^3+e^{\frac {1}{7 x+11 x^2}} \left (49 x+154 x^2+121 x^3\right )+\left (98 x+308 x^2+242 x^3+e^{\frac {1}{7 x+11 x^2}} \left (7+71 x+154 x^2+121 x^3\right )\right ) \log (x)}{196 x+616 x^2+484 x^3+e^{\frac {2}{7 x+11 x^2}} \left (49 x+154 x^2+121 x^3\right )+e^{\frac {1}{7 x+11 x^2}} \left (196 x+616 x^2+484 x^3\right )} \, dx=\frac {x\,\ln \left (x\right )}{{\mathrm {e}}^{\frac {1}{11\,x^2+7\,x}}+2} \] Input:

int((98*x + log(x)*(98*x + exp(1/(7*x + 11*x^2))*(71*x + 154*x^2 + 121*x^3 
 + 7) + 308*x^2 + 242*x^3) + exp(1/(7*x + 11*x^2))*(49*x + 154*x^2 + 121*x 
^3) + 308*x^2 + 242*x^3)/(196*x + exp(1/(7*x + 11*x^2))*(196*x + 616*x^2 + 
 484*x^3) + exp(2/(7*x + 11*x^2))*(49*x + 154*x^2 + 121*x^3) + 616*x^2 + 4 
84*x^3),x)
 

Output:

(x*log(x))/(exp(1/(7*x + 11*x^2)) + 2)
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int \frac {98 x+308 x^2+242 x^3+e^{\frac {1}{7 x+11 x^2}} \left (49 x+154 x^2+121 x^3\right )+\left (98 x+308 x^2+242 x^3+e^{\frac {1}{7 x+11 x^2}} \left (7+71 x+154 x^2+121 x^3\right )\right ) \log (x)}{196 x+616 x^2+484 x^3+e^{\frac {2}{7 x+11 x^2}} \left (49 x+154 x^2+121 x^3\right )+e^{\frac {1}{7 x+11 x^2}} \left (196 x+616 x^2+484 x^3\right )} \, dx=\frac {\mathrm {log}\left (x \right ) x}{e^{\frac {1}{11 x^{2}+7 x}}+2} \] Input:

int((((121*x^3+154*x^2+71*x+7)*exp(1/(11*x^2+7*x))+242*x^3+308*x^2+98*x)*l 
og(x)+(121*x^3+154*x^2+49*x)*exp(1/(11*x^2+7*x))+242*x^3+308*x^2+98*x)/((1 
21*x^3+154*x^2+49*x)*exp(1/(11*x^2+7*x))^2+(484*x^3+616*x^2+196*x)*exp(1/( 
11*x^2+7*x))+484*x^3+616*x^2+196*x),x)
 

Output:

(log(x)*x)/(e**(1/(11*x**2 + 7*x)) + 2)