Integrand size = 127, antiderivative size = 24 \[ \int \frac {e^{\frac {3-x^3+x^2 \log (11+5 x)}{-x+\log (11+5 x)}} \left (18+15 x+22 x^3+10 x^4+\left (-44 x^2-20 x^3\right ) \log (11+5 x)+\left (22 x+10 x^2\right ) \log ^2(11+5 x)\right )}{11 x^2+5 x^3+\left (-22 x-10 x^2\right ) \log (11+5 x)+(11+5 x) \log ^2(11+5 x)} \, dx=1+e^{x^2+\frac {3}{-x+\log (1+5 (2+x))}} \]
Time = 0.13 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.83 \[ \int \frac {e^{\frac {3-x^3+x^2 \log (11+5 x)}{-x+\log (11+5 x)}} \left (18+15 x+22 x^3+10 x^4+\left (-44 x^2-20 x^3\right ) \log (11+5 x)+\left (22 x+10 x^2\right ) \log ^2(11+5 x)\right )}{11 x^2+5 x^3+\left (-22 x-10 x^2\right ) \log (11+5 x)+(11+5 x) \log ^2(11+5 x)} \, dx=e^{\frac {-3+x^3}{x-\log (11+5 x)}} (11+5 x)^{-\frac {x^2}{x-\log (11+5 x)}} \]
Integrate[(E^((3 - x^3 + x^2*Log[11 + 5*x])/(-x + Log[11 + 5*x]))*(18 + 15 *x + 22*x^3 + 10*x^4 + (-44*x^2 - 20*x^3)*Log[11 + 5*x] + (22*x + 10*x^2)* Log[11 + 5*x]^2))/(11*x^2 + 5*x^3 + (-22*x - 10*x^2)*Log[11 + 5*x] + (11 + 5*x)*Log[11 + 5*x]^2),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.
\(\displaystyle \int \frac {\left (10 x^4+22 x^3+\left (10 x^2+22 x\right ) \log ^2(5 x+11)+\left (-20 x^3-44 x^2\right ) \log (5 x+11)+15 x+18\right ) \exp \left (\frac {-x^3+x^2 \log (5 x+11)+3}{\log (5 x+11)-x}\right )}{5 x^3+11 x^2+\left (-10 x^2-22 x\right ) \log (5 x+11)+(5 x+11) \log ^2(5 x+11)} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {\left (10 x^4+22 x^3+\left (10 x^2+22 x\right ) \log ^2(5 x+11)+\left (-20 x^3-44 x^2\right ) \log (5 x+11)+15 x+18\right ) \exp \left (\frac {-x^3+x^2 \log (5 x+11)+3}{\log (5 x+11)-x}\right )}{(5 x+11) (x-\log (5 x+11))^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (2 x \exp \left (\frac {-x^3+x^2 \log (5 x+11)+3}{\log (5 x+11)-x}\right )+\frac {3 (5 x+6) \exp \left (\frac {-x^3+x^2 \log (5 x+11)+3}{\log (5 x+11)-x}\right )}{(5 x+11) (x-\log (5 x+11))^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 2 \int e^{\frac {-x^3+\log (5 x+11) x^2+3}{\log (5 x+11)-x}} xdx+3 \int \frac {e^{\frac {-x^3+\log (5 x+11) x^2+3}{\log (5 x+11)-x}}}{(x-\log (5 x+11))^2}dx-15 \int \frac {e^{\frac {-x^3+\log (5 x+11) x^2+3}{\log (5 x+11)-x}}}{(5 x+11) (x-\log (5 x+11))^2}dx\) |
Int[(E^((3 - x^3 + x^2*Log[11 + 5*x])/(-x + Log[11 + 5*x]))*(18 + 15*x + 2 2*x^3 + 10*x^4 + (-44*x^2 - 20*x^3)*Log[11 + 5*x] + (22*x + 10*x^2)*Log[11 + 5*x]^2))/(11*x^2 + 5*x^3 + (-22*x - 10*x^2)*Log[11 + 5*x] + (11 + 5*x)* Log[11 + 5*x]^2),x]
3.16.68.3.1 Defintions of rubi rules used
Time = 0.76 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.29
method | result | size |
risch | \({\mathrm e}^{\frac {-x^{2} \ln \left (5 x +11\right )+x^{3}-3}{-\ln \left (5 x +11\right )+x}}\) | \(31\) |
parallelrisch | \({\mathrm e}^{\frac {x^{2} \ln \left (5 x +11\right )-x^{3}+3}{\ln \left (5 x +11\right )-x}}\) | \(32\) |
int(((10*x^2+22*x)*ln(5*x+11)^2+(-20*x^3-44*x^2)*ln(5*x+11)+10*x^4+22*x^3+ 15*x+18)*exp((x^2*ln(5*x+11)-x^3+3)/(ln(5*x+11)-x))/((5*x+11)*ln(5*x+11)^2 +(-10*x^2-22*x)*ln(5*x+11)+5*x^3+11*x^2),x,method=_RETURNVERBOSE)
Time = 0.23 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.25 \[ \int \frac {e^{\frac {3-x^3+x^2 \log (11+5 x)}{-x+\log (11+5 x)}} \left (18+15 x+22 x^3+10 x^4+\left (-44 x^2-20 x^3\right ) \log (11+5 x)+\left (22 x+10 x^2\right ) \log ^2(11+5 x)\right )}{11 x^2+5 x^3+\left (-22 x-10 x^2\right ) \log (11+5 x)+(11+5 x) \log ^2(11+5 x)} \, dx=e^{\left (\frac {x^{3} - x^{2} \log \left (5 \, x + 11\right ) - 3}{x - \log \left (5 \, x + 11\right )}\right )} \]
integrate(((10*x^2+22*x)*log(5*x+11)^2+(-20*x^3-44*x^2)*log(5*x+11)+10*x^4 +22*x^3+15*x+18)*exp((x^2*log(5*x+11)-x^3+3)/(log(5*x+11)-x))/((5*x+11)*lo g(5*x+11)^2+(-10*x^2-22*x)*log(5*x+11)+5*x^3+11*x^2),x, algorithm=\
Time = 0.26 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {e^{\frac {3-x^3+x^2 \log (11+5 x)}{-x+\log (11+5 x)}} \left (18+15 x+22 x^3+10 x^4+\left (-44 x^2-20 x^3\right ) \log (11+5 x)+\left (22 x+10 x^2\right ) \log ^2(11+5 x)\right )}{11 x^2+5 x^3+\left (-22 x-10 x^2\right ) \log (11+5 x)+(11+5 x) \log ^2(11+5 x)} \, dx=e^{\frac {- x^{3} + x^{2} \log {\left (5 x + 11 \right )} + 3}{- x + \log {\left (5 x + 11 \right )}}} \]
integrate(((10*x**2+22*x)*ln(5*x+11)**2+(-20*x**3-44*x**2)*ln(5*x+11)+10*x **4+22*x**3+15*x+18)*exp((x**2*ln(5*x+11)-x**3+3)/(ln(5*x+11)-x))/((5*x+11 )*ln(5*x+11)**2+(-10*x**2-22*x)*ln(5*x+11)+5*x**3+11*x**2),x)
\[ \int \frac {e^{\frac {3-x^3+x^2 \log (11+5 x)}{-x+\log (11+5 x)}} \left (18+15 x+22 x^3+10 x^4+\left (-44 x^2-20 x^3\right ) \log (11+5 x)+\left (22 x+10 x^2\right ) \log ^2(11+5 x)\right )}{11 x^2+5 x^3+\left (-22 x-10 x^2\right ) \log (11+5 x)+(11+5 x) \log ^2(11+5 x)} \, dx=\int { \frac {{\left (10 \, x^{4} + 22 \, x^{3} + 2 \, {\left (5 \, x^{2} + 11 \, x\right )} \log \left (5 \, x + 11\right )^{2} - 4 \, {\left (5 \, x^{3} + 11 \, x^{2}\right )} \log \left (5 \, x + 11\right ) + 15 \, x + 18\right )} e^{\left (\frac {x^{3} - x^{2} \log \left (5 \, x + 11\right ) - 3}{x - \log \left (5 \, x + 11\right )}\right )}}{5 \, x^{3} + {\left (5 \, x + 11\right )} \log \left (5 \, x + 11\right )^{2} + 11 \, x^{2} - 2 \, {\left (5 \, x^{2} + 11 \, x\right )} \log \left (5 \, x + 11\right )} \,d x } \]
integrate(((10*x^2+22*x)*log(5*x+11)^2+(-20*x^3-44*x^2)*log(5*x+11)+10*x^4 +22*x^3+15*x+18)*exp((x^2*log(5*x+11)-x^3+3)/(log(5*x+11)-x))/((5*x+11)*lo g(5*x+11)^2+(-10*x^2-22*x)*log(5*x+11)+5*x^3+11*x^2),x, algorithm=\
integrate((10*x^4 + 22*x^3 + 2*(5*x^2 + 11*x)*log(5*x + 11)^2 - 4*(5*x^3 + 11*x^2)*log(5*x + 11) + 15*x + 18)*e^((x^3 - x^2*log(5*x + 11) - 3)/(x - log(5*x + 11)))/(5*x^3 + (5*x + 11)*log(5*x + 11)^2 + 11*x^2 - 2*(5*x^2 + 11*x)*log(5*x + 11)), x)
Leaf count of result is larger than twice the leaf count of optimal. 55 vs. \(2 (21) = 42\).
Time = 0.71 (sec) , antiderivative size = 55, normalized size of antiderivative = 2.29 \[ \int \frac {e^{\frac {3-x^3+x^2 \log (11+5 x)}{-x+\log (11+5 x)}} \left (18+15 x+22 x^3+10 x^4+\left (-44 x^2-20 x^3\right ) \log (11+5 x)+\left (22 x+10 x^2\right ) \log ^2(11+5 x)\right )}{11 x^2+5 x^3+\left (-22 x-10 x^2\right ) \log (11+5 x)+(11+5 x) \log ^2(11+5 x)} \, dx=e^{\left (\frac {x^{3}}{x - \log \left (5 \, x + 11\right )} - \frac {x^{2} \log \left (5 \, x + 11\right )}{x - \log \left (5 \, x + 11\right )} - \frac {3}{x - \log \left (5 \, x + 11\right )}\right )} \]
integrate(((10*x^2+22*x)*log(5*x+11)^2+(-20*x^3-44*x^2)*log(5*x+11)+10*x^4 +22*x^3+15*x+18)*exp((x^2*log(5*x+11)-x^3+3)/(log(5*x+11)-x))/((5*x+11)*lo g(5*x+11)^2+(-10*x^2-22*x)*log(5*x+11)+5*x^3+11*x^2),x, algorithm=\
Time = 13.34 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.83 \[ \int \frac {e^{\frac {3-x^3+x^2 \log (11+5 x)}{-x+\log (11+5 x)}} \left (18+15 x+22 x^3+10 x^4+\left (-44 x^2-20 x^3\right ) \log (11+5 x)+\left (22 x+10 x^2\right ) \log ^2(11+5 x)\right )}{11 x^2+5 x^3+\left (-22 x-10 x^2\right ) \log (11+5 x)+(11+5 x) \log ^2(11+5 x)} \, dx=\frac {{\mathrm {e}}^{\frac {x^3-3}{x-\ln \left (5\,x+11\right )}}}{{\left (5\,x+11\right )}^{\frac {x^2}{x-\ln \left (5\,x+11\right )}}} \]