Integrand size = 163, antiderivative size = 21 \begin {dmath*} \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)}}} \end {dmath*}
Time = 0.69 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \begin {dmath*} \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}}} \end {dmath*}
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]
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 |
\(\displaystyle \int \frac {1}{2+e^{\frac {1}{11 x^2+7 x}}}dx-\int \frac {\int \frac {1}{2+e^{\frac {1}{11 x^2+7 x}}}dx}{x}dx+\frac {2}{7} \int \frac {\int \frac {1}{\left (2+e^{\frac {1}{11 x^2+7 x}}\right )^2 x}dx}{x}dx+22 \int \frac {\int \frac {1}{\left (2+e^{\frac {1}{11 x^2+7 x}}\right )^2 (11 x+7)^2}dx}{x}dx-11 \int \frac {\int \frac {1}{\left (2+e^{\frac {1}{11 x^2+7 x}}\right ) (11 x+7)^2}dx}{x}dx-\frac {22}{7} \int \frac {\int \frac {1}{\left (2+e^{\frac {1}{11 x^2+7 x}}\right )^2 (11 x+7)}dx}{x}dx+\frac {11}{7} \int \frac {\int \frac {1}{\left (2+e^{\frac {1}{11 x^2+7 x}}\right ) (11 x+7)}dx}{x}dx-\frac {1}{7} \int \frac {\int \frac {1}{e^{\frac {1}{11 x^2+7 x}} x+2 x}dx}{x}dx+\log (x) \int \frac {1}{2+e^{\frac {1}{11 x^2+7 x}}}dx-\frac {2}{7} \log (x) \int \frac {1}{\left (2+e^{\frac {1}{11 x^2+7 x}}\right )^2 x}dx+\frac {1}{7} \log (x) \int \frac {1}{\left (2+e^{\frac {1}{11 x^2+7 x}}\right ) x}dx-22 \log (x) \int \frac {1}{\left (2+e^{\frac {1}{11 x^2+7 x}}\right )^2 (11 x+7)^2}dx+11 \log (x) \int \frac {1}{\left (2+e^{\frac {1}{11 x^2+7 x}}\right ) (11 x+7)^2}dx+\frac {22}{7} \log (x) \int \frac {1}{\left (2+e^{\frac {1}{11 x^2+7 x}}\right )^2 (11 x+7)}dx-\frac {11}{7} \log (x) \int \frac {1}{\left (2+e^{\frac {1}{11 x^2+7 x}}\right ) (11 x+7)}dx\) |
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]
3.4.92.3.1 Defintions of rubi rules used
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 0.70 (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 | \(\frac {\ln \left (x \right ) x}{2+{\mathrm e}^{\frac {1}{x \left (11 x +7\right )}}}\) | \(21\) |
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)
Time = 0.28 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.95 \begin {dmath*} \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} \end {dmath*}
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=\
Time = 0.13 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.81 \begin {dmath*} \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} \end {dmath*}
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)
Time = 0.25 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.67 \begin {dmath*} \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 )}} \end {dmath*}
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=\
Time = 0.34 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.95 \begin {dmath*} \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} \end {dmath*}
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=\
Time = 14.29 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.95 \begin {dmath*} \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} \end {dmath*}
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)