Integrand size = 108, antiderivative size = 25 \[ \int \frac {e^{\frac {45+\left (25 x^2+10 x^3+x^4\right ) \log \left (\frac {19+x}{3}\right )}{x^2 \log \left (\frac {19+x}{3}\right )}} \left (-45 x+(-1710-90 x) \log \left (\frac {19+x}{3}\right )+\left (190 x^3+48 x^4+2 x^5\right ) \log ^2\left (\frac {19+x}{3}\right )\right )}{\left (19 x^3+x^4\right ) \log ^2\left (\frac {19+x}{3}\right )} \, dx=e^{(5+x)^2+\frac {45}{x^2 \log \left (4+\frac {7+x}{3}\right )}} \]
Time = 0.07 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {e^{\frac {45+\left (25 x^2+10 x^3+x^4\right ) \log \left (\frac {19+x}{3}\right )}{x^2 \log \left (\frac {19+x}{3}\right )}} \left (-45 x+(-1710-90 x) \log \left (\frac {19+x}{3}\right )+\left (190 x^3+48 x^4+2 x^5\right ) \log ^2\left (\frac {19+x}{3}\right )\right )}{\left (19 x^3+x^4\right ) \log ^2\left (\frac {19+x}{3}\right )} \, dx=e^{25+10 x+x^2+\frac {45}{x^2 \log \left (\frac {19+x}{3}\right )}} \]
Integrate[(E^((45 + (25*x^2 + 10*x^3 + x^4)*Log[(19 + x)/3])/(x^2*Log[(19 + x)/3]))*(-45*x + (-1710 - 90*x)*Log[(19 + x)/3] + (190*x^3 + 48*x^4 + 2* x^5)*Log[(19 + x)/3]^2))/((19*x^3 + x^4)*Log[(19 + x)/3]^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.
\(\displaystyle \int \frac {\left (\left (2 x^5+48 x^4+190 x^3\right ) \log ^2\left (\frac {x+19}{3}\right )-45 x+(-90 x-1710) \log \left (\frac {x+19}{3}\right )\right ) \exp \left (\frac {\left (x^4+10 x^3+25 x^2\right ) \log \left (\frac {x+19}{3}\right )+45}{x^2 \log \left (\frac {x+19}{3}\right )}\right )}{\left (x^4+19 x^3\right ) \log ^2\left (\frac {x+19}{3}\right )} \, dx\) |
\(\Big \downarrow \) 2026 |
\(\displaystyle \int \frac {\left (\left (2 x^5+48 x^4+190 x^3\right ) \log ^2\left (\frac {x+19}{3}\right )-45 x+(-90 x-1710) \log \left (\frac {x+19}{3}\right )\right ) \exp \left (\frac {\left (x^4+10 x^3+25 x^2\right ) \log \left (\frac {x+19}{3}\right )+45}{x^2 \log \left (\frac {x+19}{3}\right )}\right )}{x^3 (x+19) \log ^2\left (\frac {x+19}{3}\right )}dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {\left (\left (2 x^5+48 x^4+190 x^3\right ) \log ^2\left (\frac {x+19}{3}\right )-45 x+(-90 x-1710) \log \left (\frac {x+19}{3}\right )\right ) \exp \left (\frac {\left (x^4+10 x^3+25 x^2\right ) \log \left (\frac {x+19}{3}\right )+45}{x^2 \log \left (\frac {x}{3}+\frac {19}{3}\right )}\right )}{x^3 (x+19) \log ^2\left (\frac {x}{3}+\frac {19}{3}\right )}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {45 \exp \left (\frac {\left (x^4+10 x^3+25 x^2\right ) \log \left (\frac {x+19}{3}\right )+45}{x^2 \log \left (\frac {x}{3}+\frac {19}{3}\right )}\right )}{(-x-19) x^2 \log ^2\left (\frac {x}{3}+\frac {19}{3}\right )}+2 (x+5) \exp \left (\frac {\left (x^4+10 x^3+25 x^2\right ) \log \left (\frac {x+19}{3}\right )+45}{x^2 \log \left (\frac {x}{3}+\frac {19}{3}\right )}\right )-\frac {90 \exp \left (\frac {\left (x^4+10 x^3+25 x^2\right ) \log \left (\frac {x+19}{3}\right )+45}{x^2 \log \left (\frac {x}{3}+\frac {19}{3}\right )}\right )}{x^3 \log \left (\frac {x}{3}+\frac {19}{3}\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {45}{361} \int \frac {e^{(x+5)^2+\frac {45}{x^2 \log \left (\frac {x+19}{3}\right )}}}{(-x-19) \log ^2\left (\frac {x}{3}+\frac {19}{3}\right )}dx-\frac {45}{19} \int \frac {e^{(x+5)^2+\frac {45}{x^2 \log \left (\frac {x+19}{3}\right )}}}{x^2 \log ^2\left (\frac {x}{3}+\frac {19}{3}\right )}dx+\frac {45}{361} \int \frac {e^{(x+5)^2+\frac {45}{x^2 \log \left (\frac {x+19}{3}\right )}}}{x \log ^2\left (\frac {x}{3}+\frac {19}{3}\right )}dx+10 \int e^{(x+5)^2+\frac {45}{x^2 \log \left (\frac {x+19}{3}\right )}}dx+2 \int e^{(x+5)^2+\frac {45}{x^2 \log \left (\frac {x+19}{3}\right )}} xdx-90 \int \frac {e^{(x+5)^2+\frac {45}{x^2 \log \left (\frac {x+19}{3}\right )}}}{x^3 \log \left (\frac {x}{3}+\frac {19}{3}\right )}dx\) |
Int[(E^((45 + (25*x^2 + 10*x^3 + x^4)*Log[(19 + x)/3])/(x^2*Log[(19 + x)/3 ]))*(-45*x + (-1710 - 90*x)*Log[(19 + x)/3] + (190*x^3 + 48*x^4 + 2*x^5)*L og[(19 + x)/3]^2))/((19*x^3 + x^4)*Log[(19 + x)/3]^2),x]
3.7.3.3.1 Defintions of rubi rules used
Int[(Fx_.)*(Px_)^(p_.), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Int[x^(p *r)*ExpandToSum[Px/x^r, x]^p*Fx, x] /; IGtQ[r, 0]] /; PolyQ[Px, x] && Integ erQ[p] && !MonomialQ[Px, x] && (ILtQ[p, 0] || !PolyQ[u, x])
Time = 1.33 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.48
method | result | size |
parallelrisch | \({\mathrm e}^{\frac {\left (x^{4}+10 x^{3}+25 x^{2}\right ) \ln \left (\frac {x}{3}+\frac {19}{3}\right )+45}{x^{2} \ln \left (\frac {x}{3}+\frac {19}{3}\right )}}\) | \(37\) |
risch | \({\mathrm e}^{\frac {\ln \left (\frac {x}{3}+\frac {19}{3}\right ) x^{4}+10 \ln \left (\frac {x}{3}+\frac {19}{3}\right ) x^{3}+25 x^{2} \ln \left (\frac {x}{3}+\frac {19}{3}\right )+45}{x^{2} \ln \left (\frac {x}{3}+\frac {19}{3}\right )}}\) | \(48\) |
int(((2*x^5+48*x^4+190*x^3)*ln(1/3*x+19/3)^2+(-90*x-1710)*ln(1/3*x+19/3)-4 5*x)*exp(((x^4+10*x^3+25*x^2)*ln(1/3*x+19/3)+45)/x^2/ln(1/3*x+19/3))/(x^4+ 19*x^3)/ln(1/3*x+19/3)^2,x,method=_RETURNVERBOSE)
Time = 0.43 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.44 \[ \int \frac {e^{\frac {45+\left (25 x^2+10 x^3+x^4\right ) \log \left (\frac {19+x}{3}\right )}{x^2 \log \left (\frac {19+x}{3}\right )}} \left (-45 x+(-1710-90 x) \log \left (\frac {19+x}{3}\right )+\left (190 x^3+48 x^4+2 x^5\right ) \log ^2\left (\frac {19+x}{3}\right )\right )}{\left (19 x^3+x^4\right ) \log ^2\left (\frac {19+x}{3}\right )} \, dx=e^{\left (\frac {{\left (x^{4} + 10 \, x^{3} + 25 \, x^{2}\right )} \log \left (\frac {1}{3} \, x + \frac {19}{3}\right ) + 45}{x^{2} \log \left (\frac {1}{3} \, x + \frac {19}{3}\right )}\right )} \]
integrate(((2*x^5+48*x^4+190*x^3)*log(1/3*x+19/3)^2+(-90*x-1710)*log(1/3*x +19/3)-45*x)*exp(((x^4+10*x^3+25*x^2)*log(1/3*x+19/3)+45)/x^2/log(1/3*x+19 /3))/(x^4+19*x^3)/log(1/3*x+19/3)^2,x, algorithm=\
Time = 0.19 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.44 \[ \int \frac {e^{\frac {45+\left (25 x^2+10 x^3+x^4\right ) \log \left (\frac {19+x}{3}\right )}{x^2 \log \left (\frac {19+x}{3}\right )}} \left (-45 x+(-1710-90 x) \log \left (\frac {19+x}{3}\right )+\left (190 x^3+48 x^4+2 x^5\right ) \log ^2\left (\frac {19+x}{3}\right )\right )}{\left (19 x^3+x^4\right ) \log ^2\left (\frac {19+x}{3}\right )} \, dx=e^{\frac {\left (x^{4} + 10 x^{3} + 25 x^{2}\right ) \log {\left (\frac {x}{3} + \frac {19}{3} \right )} + 45}{x^{2} \log {\left (\frac {x}{3} + \frac {19}{3} \right )}}} \]
integrate(((2*x**5+48*x**4+190*x**3)*ln(1/3*x+19/3)**2+(-90*x-1710)*ln(1/3 *x+19/3)-45*x)*exp(((x**4+10*x**3+25*x**2)*ln(1/3*x+19/3)+45)/x**2/ln(1/3* x+19/3))/(x**4+19*x**3)/ln(1/3*x+19/3)**2,x)
Time = 0.47 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {e^{\frac {45+\left (25 x^2+10 x^3+x^4\right ) \log \left (\frac {19+x}{3}\right )}{x^2 \log \left (\frac {19+x}{3}\right )}} \left (-45 x+(-1710-90 x) \log \left (\frac {19+x}{3}\right )+\left (190 x^3+48 x^4+2 x^5\right ) \log ^2\left (\frac {19+x}{3}\right )\right )}{\left (19 x^3+x^4\right ) \log ^2\left (\frac {19+x}{3}\right )} \, dx=e^{\left (x^{2} + 10 \, x - \frac {45}{x^{2} {\left (\log \left (3\right ) - \log \left (x + 19\right )\right )}} + 25\right )} \]
integrate(((2*x^5+48*x^4+190*x^3)*log(1/3*x+19/3)^2+(-90*x-1710)*log(1/3*x +19/3)-45*x)*exp(((x^4+10*x^3+25*x^2)*log(1/3*x+19/3)+45)/x^2/log(1/3*x+19 /3))/(x^4+19*x^3)/log(1/3*x+19/3)^2,x, algorithm=\
Time = 0.65 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.88 \[ \int \frac {e^{\frac {45+\left (25 x^2+10 x^3+x^4\right ) \log \left (\frac {19+x}{3}\right )}{x^2 \log \left (\frac {19+x}{3}\right )}} \left (-45 x+(-1710-90 x) \log \left (\frac {19+x}{3}\right )+\left (190 x^3+48 x^4+2 x^5\right ) \log ^2\left (\frac {19+x}{3}\right )\right )}{\left (19 x^3+x^4\right ) \log ^2\left (\frac {19+x}{3}\right )} \, dx=e^{\left (x^{2} + 10 \, x + \frac {45}{x^{2} \log \left (\frac {1}{3} \, x + \frac {19}{3}\right )} + 25\right )} \]
integrate(((2*x^5+48*x^4+190*x^3)*log(1/3*x+19/3)^2+(-90*x-1710)*log(1/3*x +19/3)-45*x)*exp(((x^4+10*x^3+25*x^2)*log(1/3*x+19/3)+45)/x^2/log(1/3*x+19 /3))/(x^4+19*x^3)/log(1/3*x+19/3)^2,x, algorithm=\
Time = 11.40 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {e^{\frac {45+\left (25 x^2+10 x^3+x^4\right ) \log \left (\frac {19+x}{3}\right )}{x^2 \log \left (\frac {19+x}{3}\right )}} \left (-45 x+(-1710-90 x) \log \left (\frac {19+x}{3}\right )+\left (190 x^3+48 x^4+2 x^5\right ) \log ^2\left (\frac {19+x}{3}\right )\right )}{\left (19 x^3+x^4\right ) \log ^2\left (\frac {19+x}{3}\right )} \, dx={\mathrm {e}}^{10\,x}\,{\mathrm {e}}^{x^2}\,{\mathrm {e}}^{25}\,{\mathrm {e}}^{\frac {45}{x^2\,\ln \left (\frac {x}{3}+\frac {19}{3}\right )}} \]