Integrand size = 196, antiderivative size = 32 \[ \int \frac {-10 x^2+e^{\frac {3-x}{x}} \left (-15+8 x+x^2\right )+e^{-2+x} \left (-e^{\frac {2 (3-x)}{x}} x^3+2 e^{\frac {3-x}{x}} x^4-x^5\right )+\left (3 e^{\frac {3-x}{x}} x+x^3\right ) \log (x)}{5 x^3-x^4+e^{\frac {3-x}{x}} \left (-5 x^2+x^3\right )+e^{-2+x} \left (-e^{\frac {2 (3-x)}{x}} x^3+2 e^{\frac {3-x}{x}} x^4-x^5\right )+\left (e^{\frac {3-x}{x}} x^3-x^4\right ) \log (x)} \, dx=\log \left (e^{-2+x}+\frac {1-\frac {5}{x}+\log (x)}{-e^{-1+\frac {3}{x}}+x}\right ) \]
\[ \int \frac {-10 x^2+e^{\frac {3-x}{x}} \left (-15+8 x+x^2\right )+e^{-2+x} \left (-e^{\frac {2 (3-x)}{x}} x^3+2 e^{\frac {3-x}{x}} x^4-x^5\right )+\left (3 e^{\frac {3-x}{x}} x+x^3\right ) \log (x)}{5 x^3-x^4+e^{\frac {3-x}{x}} \left (-5 x^2+x^3\right )+e^{-2+x} \left (-e^{\frac {2 (3-x)}{x}} x^3+2 e^{\frac {3-x}{x}} x^4-x^5\right )+\left (e^{\frac {3-x}{x}} x^3-x^4\right ) \log (x)} \, dx=\int \frac {-10 x^2+e^{\frac {3-x}{x}} \left (-15+8 x+x^2\right )+e^{-2+x} \left (-e^{\frac {2 (3-x)}{x}} x^3+2 e^{\frac {3-x}{x}} x^4-x^5\right )+\left (3 e^{\frac {3-x}{x}} x+x^3\right ) \log (x)}{5 x^3-x^4+e^{\frac {3-x}{x}} \left (-5 x^2+x^3\right )+e^{-2+x} \left (-e^{\frac {2 (3-x)}{x}} x^3+2 e^{\frac {3-x}{x}} x^4-x^5\right )+\left (e^{\frac {3-x}{x}} x^3-x^4\right ) \log (x)} \, dx \]
Integrate[(-10*x^2 + E^((3 - x)/x)*(-15 + 8*x + x^2) + E^(-2 + x)*(-(E^((2 *(3 - x))/x)*x^3) + 2*E^((3 - x)/x)*x^4 - x^5) + (3*E^((3 - x)/x)*x + x^3) *Log[x])/(5*x^3 - x^4 + E^((3 - x)/x)*(-5*x^2 + x^3) + E^(-2 + x)*(-(E^((2 *(3 - x))/x)*x^3) + 2*E^((3 - x)/x)*x^4 - x^5) + (E^((3 - x)/x)*x^3 - x^4) *Log[x]),x]
Integrate[(-10*x^2 + E^((3 - x)/x)*(-15 + 8*x + x^2) + E^(-2 + x)*(-(E^((2 *(3 - x))/x)*x^3) + 2*E^((3 - x)/x)*x^4 - x^5) + (3*E^((3 - x)/x)*x + x^3) *Log[x])/(5*x^3 - x^4 + E^((3 - x)/x)*(-5*x^2 + x^3) + E^(-2 + x)*(-(E^((2 *(3 - x))/x)*x^3) + 2*E^((3 - x)/x)*x^4 - x^5) + (E^((3 - x)/x)*x^3 - x^4) *Log[x]), 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.
| \(\displaystyle \int \frac {\left (x^3+3 e^{\frac {3-x}{x}} x\right ) \log (x)-10 x^2+e^{\frac {3-x}{x}} \left (x^2+8 x-15\right )+e^{x-2} \left (-x^5+2 e^{\frac {3-x}{x}} x^4-e^{\frac {2 (3-x)}{x}} x^3\right )}{-x^4+5 x^3+\left (e^{\frac {3-x}{x}} x^3-x^4\right ) \log (x)+e^{\frac {3-x}{x}} \left (x^3-5 x^2\right )+e^{x-2} \left (-x^5+2 e^{\frac {3-x}{x}} x^4-e^{\frac {2 (3-x)}{x}} x^3\right )} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {e^4 \left (-\left (x^3+3 e^{\frac {3-x}{x}} x\right ) \log (x)+10 x^2-e^{\frac {3-x}{x}} \left (x^2+8 x-15\right )-e^{x-2} \left (-x^5+2 e^{\frac {3-x}{x}} x^4-e^{\frac {2 (3-x)}{x}} x^3\right )\right )}{x^2 \left (e^{3/x}-e x\right ) \left (-e^{x+1} x^2+e^{x+\frac {3}{x}} x-e^3 x-e^3 x \log (x)+5 e^3\right )}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle e^4 \int \frac {10 x^2+e^{\frac {3-x}{x}} \left (-x^2-8 x+15\right )+e^{x-2} \left (x^5-2 e^{\frac {3-x}{x}} x^4+e^{\frac {2 (3-x)}{x}} x^3\right )-\left (x^3+3 e^{\frac {3-x}{x}} x\right ) \log (x)}{x^2 \left (e^{3/x}-e x\right ) \left (-e^{x+1} x^2+e^{x+\frac {3}{x}} x-e^3 \log (x) x-e^3 x+5 e^3\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle e^4 \int \left (\frac {1}{e^4}-\frac {e \log (x) x^4+e x^4-e^{3/x} x^3-e^{3/x} \log (x) x^3+e \log (x) x^3-5 e x^3+6 e^{3/x} x^2-10 e x^2+8 e^{3/x} x+3 e^{3/x} \log (x) x-15 e^{3/x}}{e x^2 \left (e x-e^{3/x}\right ) \left (e^{x+1} x^2-e^{x+\frac {3}{x}} x+e^3 \log (x) x+e^3 x-5 e^3\right )}\right )dx\) |
| \(\Big \downarrow \) 7299 |
| \(\displaystyle e^4 \int \left (\frac {1}{e^4}-\frac {e \log (x) x^4+e x^4-e^{3/x} x^3-e^{3/x} \log (x) x^3+e \log (x) x^3-5 e x^3+6 e^{3/x} x^2-10 e x^2+8 e^{3/x} x+3 e^{3/x} \log (x) x-15 e^{3/x}}{e x^2 \left (e x-e^{3/x}\right ) \left (e^{x+1} x^2-e^{x+\frac {3}{x}} x+e^3 \log (x) x+e^3 x-5 e^3\right )}\right )dx\) |
Int[(-10*x^2 + E^((3 - x)/x)*(-15 + 8*x + x^2) + E^(-2 + x)*(-(E^((2*(3 - x))/x)*x^3) + 2*E^((3 - x)/x)*x^4 - x^5) + (3*E^((3 - x)/x)*x + x^3)*Log[x ])/(5*x^3 - x^4 + E^((3 - x)/x)*(-5*x^2 + x^3) + E^(-2 + x)*(-(E^((2*(3 - x))/x)*x^3) + 2*E^((3 - x)/x)*x^4 - x^5) + (E^((3 - x)/x)*x^3 - x^4)*Log[x ]),x]
3.23.82.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Time = 6.29 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.69
| method | result | size |
| risch | \(-\ln \left ({\mathrm e}^{-\frac {-3+x}{x}}-x \right )-1+\ln \left (\ln \left (x \right )+\frac {x^{2} {\mathrm e}^{-2+x}-x \,{\mathrm e}^{\frac {x^{2}-3 x +3}{x}}+x -5}{x}\right )\) | \(54\) |
| parallelrisch | \(-\ln \left (x \right )-\ln \left (x -{\mathrm e}^{-\frac {-3+x}{x}}\right )+\ln \left (x^{2} {\mathrm e}^{-2+x}-x \,{\mathrm e}^{-2+x} {\mathrm e}^{-\frac {-3+x}{x}}+x \ln \left (x \right )+x -5\right )\) | \(54\) |
int(((3*x*exp((-x+3)/x)+x^3)*ln(x)+(-x^3*exp((-x+3)/x)^2+2*x^4*exp((-x+3)/ x)-x^5)*exp(-2+x)+(x^2+8*x-15)*exp((-x+3)/x)-10*x^2)/((x^3*exp((-x+3)/x)-x ^4)*ln(x)+(-x^3*exp((-x+3)/x)^2+2*x^4*exp((-x+3)/x)-x^5)*exp(-2+x)+(x^3-5* x^2)*exp((-x+3)/x)-x^4+5*x^3),x,method=_RETURNVERBOSE)
Time = 0.25 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.56 \[ \int \frac {-10 x^2+e^{\frac {3-x}{x}} \left (-15+8 x+x^2\right )+e^{-2+x} \left (-e^{\frac {2 (3-x)}{x}} x^3+2 e^{\frac {3-x}{x}} x^4-x^5\right )+\left (3 e^{\frac {3-x}{x}} x+x^3\right ) \log (x)}{5 x^3-x^4+e^{\frac {3-x}{x}} \left (-5 x^2+x^3\right )+e^{-2+x} \left (-e^{\frac {2 (3-x)}{x}} x^3+2 e^{\frac {3-x}{x}} x^4-x^5\right )+\left (e^{\frac {3-x}{x}} x^3-x^4\right ) \log (x)} \, dx=-\log \left (-x + e^{\left (-\frac {x - 3}{x}\right )}\right ) + \log \left (\frac {{\left (x^{2} - x e^{\left (-\frac {x - 3}{x}\right )}\right )} e^{\left (x - 2\right )} + x \log \left (x\right ) + x - 5}{x}\right ) \]
integrate(((3*x*exp((-x+3)/x)+x^3)*log(x)+(-x^3*exp((-x+3)/x)^2+2*x^4*exp( (-x+3)/x)-x^5)*exp(-2+x)+(x^2+8*x-15)*exp((-x+3)/x)-10*x^2)/((x^3*exp((-x+ 3)/x)-x^4)*log(x)+(-x^3*exp((-x+3)/x)^2+2*x^4*exp((-x+3)/x)-x^5)*exp(-2+x) +(x^3-5*x^2)*exp((-x+3)/x)-x^4+5*x^3),x, algorithm=\
Time = 0.54 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.81 \[ \int \frac {-10 x^2+e^{\frac {3-x}{x}} \left (-15+8 x+x^2\right )+e^{-2+x} \left (-e^{\frac {2 (3-x)}{x}} x^3+2 e^{\frac {3-x}{x}} x^4-x^5\right )+\left (3 e^{\frac {3-x}{x}} x+x^3\right ) \log (x)}{5 x^3-x^4+e^{\frac {3-x}{x}} \left (-5 x^2+x^3\right )+e^{-2+x} \left (-e^{\frac {2 (3-x)}{x}} x^3+2 e^{\frac {3-x}{x}} x^4-x^5\right )+\left (e^{\frac {3-x}{x}} x^3-x^4\right ) \log (x)} \, dx=\log {\left (e^{x - 2} + \frac {x \log {\left (x \right )} + x - 5}{x^{2} - x e^{\frac {3 - x}{x}}} \right )} \]
integrate(((3*x*exp((-x+3)/x)+x**3)*ln(x)+(-x**3*exp((-x+3)/x)**2+2*x**4*e xp((-x+3)/x)-x**5)*exp(-2+x)+(x**2+8*x-15)*exp((-x+3)/x)-10*x**2)/((x**3*e xp((-x+3)/x)-x**4)*ln(x)+(-x**3*exp((-x+3)/x)**2+2*x**4*exp((-x+3)/x)-x**5 )*exp(-2+x)+(x**3-5*x**2)*exp((-x+3)/x)-x**4+5*x**3),x)
Time = 0.24 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.91 \[ \int \frac {-10 x^2+e^{\frac {3-x}{x}} \left (-15+8 x+x^2\right )+e^{-2+x} \left (-e^{\frac {2 (3-x)}{x}} x^3+2 e^{\frac {3-x}{x}} x^4-x^5\right )+\left (3 e^{\frac {3-x}{x}} x+x^3\right ) \log (x)}{5 x^3-x^4+e^{\frac {3-x}{x}} \left (-5 x^2+x^3\right )+e^{-2+x} \left (-e^{\frac {2 (3-x)}{x}} x^3+2 e^{\frac {3-x}{x}} x^4-x^5\right )+\left (e^{\frac {3-x}{x}} x^3-x^4\right ) \log (x)} \, dx=x - \log \left (-x e + e^{\frac {3}{x}}\right ) + \log \left (-\frac {{\left (x^{2} e^{\left (x + 1\right )} + x e^{3} \log \left (x\right ) + x e^{3} - x e^{\left (x + \frac {3}{x}\right )} - 5 \, e^{3}\right )} e^{\left (-x\right )}}{x}\right ) \]
integrate(((3*x*exp((-x+3)/x)+x^3)*log(x)+(-x^3*exp((-x+3)/x)^2+2*x^4*exp( (-x+3)/x)-x^5)*exp(-2+x)+(x^2+8*x-15)*exp((-x+3)/x)-10*x^2)/((x^3*exp((-x+ 3)/x)-x^4)*log(x)+(-x^3*exp((-x+3)/x)^2+2*x^4*exp((-x+3)/x)-x^5)*exp(-2+x) +(x^3-5*x^2)*exp((-x+3)/x)-x^4+5*x^3),x, algorithm=\
x - log(-x*e + e^(3/x)) + log(-(x^2*e^(x + 1) + x*e^3*log(x) + x*e^3 - x*e ^(x + 3/x) - 5*e^3)*e^(-x)/x)
Timed out. \[ \int \frac {-10 x^2+e^{\frac {3-x}{x}} \left (-15+8 x+x^2\right )+e^{-2+x} \left (-e^{\frac {2 (3-x)}{x}} x^3+2 e^{\frac {3-x}{x}} x^4-x^5\right )+\left (3 e^{\frac {3-x}{x}} x+x^3\right ) \log (x)}{5 x^3-x^4+e^{\frac {3-x}{x}} \left (-5 x^2+x^3\right )+e^{-2+x} \left (-e^{\frac {2 (3-x)}{x}} x^3+2 e^{\frac {3-x}{x}} x^4-x^5\right )+\left (e^{\frac {3-x}{x}} x^3-x^4\right ) \log (x)} \, dx=\text {Timed out} \]
integrate(((3*x*exp((-x+3)/x)+x^3)*log(x)+(-x^3*exp((-x+3)/x)^2+2*x^4*exp( (-x+3)/x)-x^5)*exp(-2+x)+(x^2+8*x-15)*exp((-x+3)/x)-10*x^2)/((x^3*exp((-x+ 3)/x)-x^4)*log(x)+(-x^3*exp((-x+3)/x)^2+2*x^4*exp((-x+3)/x)-x^5)*exp(-2+x) +(x^3-5*x^2)*exp((-x+3)/x)-x^4+5*x^3),x, algorithm=\
Timed out. \[ \int \frac {-10 x^2+e^{\frac {3-x}{x}} \left (-15+8 x+x^2\right )+e^{-2+x} \left (-e^{\frac {2 (3-x)}{x}} x^3+2 e^{\frac {3-x}{x}} x^4-x^5\right )+\left (3 e^{\frac {3-x}{x}} x+x^3\right ) \log (x)}{5 x^3-x^4+e^{\frac {3-x}{x}} \left (-5 x^2+x^3\right )+e^{-2+x} \left (-e^{\frac {2 (3-x)}{x}} x^3+2 e^{\frac {3-x}{x}} x^4-x^5\right )+\left (e^{\frac {3-x}{x}} x^3-x^4\right ) \log (x)} \, dx=\int \frac {{\mathrm {e}}^{x-2}\,\left (x^3\,{\mathrm {e}}^{-\frac {2\,\left (x-3\right )}{x}}-2\,x^4\,{\mathrm {e}}^{-\frac {x-3}{x}}+x^5\right )-\ln \left (x\right )\,\left (3\,x\,{\mathrm {e}}^{-\frac {x-3}{x}}+x^3\right )+10\,x^2-{\mathrm {e}}^{-\frac {x-3}{x}}\,\left (x^2+8\,x-15\right )}{{\mathrm {e}}^{x-2}\,\left (x^3\,{\mathrm {e}}^{-\frac {2\,\left (x-3\right )}{x}}-2\,x^4\,{\mathrm {e}}^{-\frac {x-3}{x}}+x^5\right )-\ln \left (x\right )\,\left (x^3\,{\mathrm {e}}^{-\frac {x-3}{x}}-x^4\right )+{\mathrm {e}}^{-\frac {x-3}{x}}\,\left (5\,x^2-x^3\right )-5\,x^3+x^4} \,d x \]
int((exp(x - 2)*(x^3*exp(-(2*(x - 3))/x) - 2*x^4*exp(-(x - 3)/x) + x^5) - log(x)*(3*x*exp(-(x - 3)/x) + x^3) + 10*x^2 - exp(-(x - 3)/x)*(8*x + x^2 - 15))/(exp(x - 2)*(x^3*exp(-(2*(x - 3))/x) - 2*x^4*exp(-(x - 3)/x) + x^5) - log(x)*(x^3*exp(-(x - 3)/x) - x^4) + exp(-(x - 3)/x)*(5*x^2 - x^3) - 5*x ^3 + x^4),x)
int((exp(x - 2)*(x^3*exp(-(2*(x - 3))/x) - 2*x^4*exp(-(x - 3)/x) + x^5) - log(x)*(3*x*exp(-(x - 3)/x) + x^3) + 10*x^2 - exp(-(x - 3)/x)*(8*x + x^2 - 15))/(exp(x - 2)*(x^3*exp(-(2*(x - 3))/x) - 2*x^4*exp(-(x - 3)/x) + x^5) - log(x)*(x^3*exp(-(x - 3)/x) - x^4) + exp(-(x - 3)/x)*(5*x^2 - x^3) - 5*x ^3 + x^4), x)