Integrand size = 123, antiderivative size = 28 \[ \int \frac {e^{\frac {-2 x+2 x^4+e^x \left (-x+x^4\right )+\left (-x+x^4\right ) \log (\log (5))}{35+7 x}} \left (175+60 x+7 x^2+40 x^4+6 x^5+e^x \left (-5 x-5 x^2-x^3+20 x^4+8 x^5+x^6\right )+\left (-5 x+20 x^4+3 x^5\right ) \log (\log (5))\right )}{175+70 x+7 x^2} \, dx=e^{\frac {\left (-x+x^4\right ) \left (2+e^x+\log (\log (5))\right )}{7 (5+x)}} x \] Output:
x*exp(1/7*(x^4-x)*(2+exp(x)+ln(ln(5)))/(5+x))
Time = 0.15 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.50 \[ \int \frac {e^{\frac {-2 x+2 x^4+e^x \left (-x+x^4\right )+\left (-x+x^4\right ) \log (\log (5))}{35+7 x}} \left (175+60 x+7 x^2+40 x^4+6 x^5+e^x \left (-5 x-5 x^2-x^3+20 x^4+8 x^5+x^6\right )+\left (-5 x+20 x^4+3 x^5\right ) \log (\log (5))\right )}{175+70 x+7 x^2} \, dx=e^{\frac {\left (2+e^x\right ) x \left (-1+x^3\right )}{7 (5+x)}} x \log ^{\frac {x \left (-1+x^3\right )}{7 (5+x)}}(5) \] Input:
Integrate[(E^((-2*x + 2*x^4 + E^x*(-x + x^4) + (-x + x^4)*Log[Log[5]])/(35 + 7*x))*(175 + 60*x + 7*x^2 + 40*x^4 + 6*x^5 + E^x*(-5*x - 5*x^2 - x^3 + 20*x^4 + 8*x^5 + x^6) + (-5*x + 20*x^4 + 3*x^5)*Log[Log[5]]))/(175 + 70*x + 7*x^2),x]
Output:
E^(((2 + E^x)*x*(-1 + x^3))/(7*(5 + x)))*x*Log[5]^((x*(-1 + x^3))/(7*(5 + 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.
| \(\displaystyle \int \frac {\left (6 x^5+40 x^4+7 x^2+\left (3 x^5+20 x^4-5 x\right ) \log (\log (5))+e^x \left (x^6+8 x^5+20 x^4-x^3-5 x^2-5 x\right )+60 x+175\right ) \exp \left (\frac {2 x^4+e^x \left (x^4-x\right )+\left (x^4-x\right ) \log (\log (5))-2 x}{7 x+35}\right )}{7 x^2+70 x+175} \, dx\) |
| \(\Big \downarrow \) 2007 |
| \(\displaystyle \int \frac {\left (6 x^5+40 x^4+7 x^2+\left (3 x^5+20 x^4-5 x\right ) \log (\log (5))+e^x \left (x^6+8 x^5+20 x^4-x^3-5 x^2-5 x\right )+60 x+175\right ) \exp \left (\frac {2 x^4+e^x \left (x^4-x\right )+\left (x^4-x\right ) \log (\log (5))-2 x}{7 x+35}\right )}{\left (\sqrt {7} x+5 \sqrt {7}\right )^2}dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {\left (6 x^5+40 x^4+7 x^2+\left (3 x^5+20 x^4-5 x\right ) \log (\log (5))+e^x \left (x^6+8 x^5+20 x^4-x^3-5 x^2-5 x\right )+60 x+175\right ) \exp \left (\frac {x \left (x^3-1\right ) \left (e^x+2 \left (1+\frac {1}{2} \log (\log (5))\right )\right )}{7 x+35}\right )}{\left (\sqrt {7} x+5 \sqrt {7}\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {60 x \exp \left (\frac {x \left (x^3-1\right ) \left (e^x+2 \left (1+\frac {1}{2} \log (\log (5))\right )\right )}{7 x+35}\right )}{7 (x+5)^2}+\frac {25 \exp \left (\frac {x \left (x^3-1\right ) \left (e^x+2 \left (1+\frac {1}{2} \log (\log (5))\right )\right )}{7 x+35}\right )}{(x+5)^2}+\frac {6 x^5 \exp \left (\frac {x \left (x^3-1\right ) \left (e^x+2 \left (1+\frac {1}{2} \log (\log (5))\right )\right )}{7 x+35}\right )}{7 (x+5)^2}+\frac {40 x^4 \exp \left (\frac {x \left (x^3-1\right ) \left (e^x+2 \left (1+\frac {1}{2} \log (\log (5))\right )\right )}{7 x+35}\right )}{7 (x+5)^2}+\frac {\left (3 x^4+20 x^3-5\right ) x \log (\log (5)) \exp \left (\frac {x \left (x^3-1\right ) \left (e^x+2 \left (1+\frac {1}{2} \log (\log (5))\right )\right )}{7 x+35}\right )}{7 (x+5)^2}+\frac {x^2 \exp \left (\frac {x \left (x^3-1\right ) \left (e^x+2 \left (1+\frac {1}{2} \log (\log (5))\right )\right )}{7 x+35}\right )}{(x+5)^2}+\frac {\left (x^5+8 x^4+20 x^3-x^2-5 x-5\right ) x \exp \left (\frac {\left (x^3-1\right ) x \left (e^x+2 \left (1+\frac {1}{2} \log (\log (5))\right )\right )}{7 x+35}+x\right )}{7 (x+5)^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 90 \int \exp \left (\frac {\left (x^3-1\right ) \left (e^x+2 \left (1+\frac {1}{2} \log (\log (5))\right )\right ) x}{7 x+35}+x\right )dx+\int \exp \left (\frac {x \left (x^3-1\right ) \left (e^x+2 \left (1+\frac {1}{2} \log (\log (5))\right )\right )}{7 x+35}\right )dx-\frac {101}{7} \int \exp \left (\frac {\left (x^3-1\right ) \left (e^x+2 \left (1+\frac {1}{2} \log (\log (5))\right )\right ) x}{7 x+35}+x\right ) xdx+\frac {25}{7} \log (\log (5)) \int \exp \left (\frac {x \left (x^3-1\right ) \left (e^x+2 \left (1+\frac {1}{2} \log (\log (5))\right )\right )}{7 x+35}\right ) xdx+\frac {50}{7} \int \exp \left (\frac {x \left (x^3-1\right ) \left (e^x+2 \left (1+\frac {1}{2} \log (\log (5))\right )\right )}{7 x+35}\right ) xdx-\frac {2}{7} \int \exp \left (\frac {\left (x^3-1\right ) \left (e^x+2 \left (1+\frac {1}{2} \log (\log (5))\right )\right ) x}{7 x+35}+x\right ) x^3dx+\frac {3}{7} \log (\log (5)) \int \exp \left (\frac {x \left (x^3-1\right ) \left (e^x+2 \left (1+\frac {1}{2} \log (\log (5))\right )\right )}{7 x+35}\right ) x^3dx+\frac {6}{7} \int \exp \left (\frac {x \left (x^3-1\right ) \left (e^x+2 \left (1+\frac {1}{2} \log (\log (5))\right )\right )}{7 x+35}\right ) x^3dx+450 \int \frac {\exp \left (\frac {\left (x^3-1\right ) \left (e^x+2 \left (1+\frac {1}{2} \log (\log (5))\right )\right ) x}{7 x+35}+x\right )}{(x+5)^2}dx+450 \log (\log (5)) \int \frac {\exp \left (\frac {x \left (x^3-1\right ) \left (e^x+2 \left (1+\frac {1}{2} \log (\log (5))\right )\right )}{7 x+35}\right )}{(x+5)^2}dx+900 \int \frac {\exp \left (\frac {x \left (x^3-1\right ) \left (e^x+2 \left (1+\frac {1}{2} \log (\log (5))\right )\right )}{7 x+35}\right )}{(x+5)^2}dx-540 \int \frac {\exp \left (\frac {\left (x^3-1\right ) \left (e^x+2 \left (1+\frac {1}{2} \log (\log (5))\right )\right ) x}{7 x+35}+x\right )}{x+5}dx-90 \log (\log (5)) \int \frac {\exp \left (\frac {x \left (x^3-1\right ) \left (e^x+2 \left (1+\frac {1}{2} \log (\log (5))\right )\right )}{7 x+35}\right )}{x+5}dx-180 \int \frac {\exp \left (\frac {x \left (x^3-1\right ) \left (e^x+2 \left (1+\frac {1}{2} \log (\log (5))\right )\right )}{7 x+35}\right )}{x+5}dx+\frac {1}{7} \int \exp \left (\frac {\left (x^3-1\right ) \left (e^x+2 \left (1+\frac {1}{2} \log (\log (5))\right )\right ) x}{7 x+35}+x\right ) x^4dx+\frac {15}{7} \int \exp \left (\frac {\left (x^3-1\right ) \left (e^x+2 \left (1+\frac {1}{2} \log (\log (5))\right )\right ) x}{7 x+35}+x\right ) x^2dx-\frac {10}{7} \log (\log (5)) \int \exp \left (\frac {x \left (x^3-1\right ) \left (e^x+2 \left (1+\frac {1}{2} \log (\log (5))\right )\right )}{7 x+35}\right ) x^2dx-\frac {20}{7} \int \exp \left (\frac {x \left (x^3-1\right ) \left (e^x+2 \left (1+\frac {1}{2} \log (\log (5))\right )\right )}{7 x+35}\right ) x^2dx\) |
Input:
Int[(E^((-2*x + 2*x^4 + E^x*(-x + x^4) + (-x + x^4)*Log[Log[5]])/(35 + 7*x ))*(175 + 60*x + 7*x^2 + 40*x^4 + 6*x^5 + E^x*(-5*x - 5*x^2 - x^3 + 20*x^4 + 8*x^5 + x^6) + (-5*x + 20*x^4 + 3*x^5)*Log[Log[5]]))/(175 + 70*x + 7*x^ 2),x]
Output:
$Aborted
Time = 4.06 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00
| method | result | size |
| risch | \(x \,{\mathrm e}^{\frac {\left (2+{\mathrm e}^{x}+\ln \left (\ln \left (5\right )\right )\right ) \left (x^{2}+x +1\right ) x \left (-1+x \right )}{7 x +35}}\) | \(28\) |
| parallelrisch | \(x \,{\mathrm e}^{\frac {\left (x^{4}-x \right ) \ln \left (\ln \left (5\right )\right )+\left (x^{4}-x \right ) {\mathrm e}^{x}+2 x^{4}-2 x}{7 x +35}}\) | \(41\) |
| norman | \(\frac {x^{2} {\mathrm e}^{\frac {\left (x^{4}-x \right ) \ln \left (\ln \left (5\right )\right )+\left (x^{4}-x \right ) {\mathrm e}^{x}+2 x^{4}-2 x}{7 x +35}}+5 x \,{\mathrm e}^{\frac {\left (x^{4}-x \right ) \ln \left (\ln \left (5\right )\right )+\left (x^{4}-x \right ) {\mathrm e}^{x}+2 x^{4}-2 x}{7 x +35}}}{5+x}\) | \(93\) |
Input:
int(((3*x^5+20*x^4-5*x)*ln(ln(5))+(x^6+8*x^5+20*x^4-x^3-5*x^2-5*x)*exp(x)+ 6*x^5+40*x^4+7*x^2+60*x+175)*exp(((x^4-x)*ln(ln(5))+(x^4-x)*exp(x)+2*x^4-2 *x)/(7*x+35))/(7*x^2+70*x+175),x,method=_RETURNVERBOSE)
Output:
x*exp(1/7*x*(-1+x)*(x^2+x+1)*(2+exp(x)+ln(ln(5)))/(5+x))
Time = 0.09 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.43 \[ \int \frac {e^{\frac {-2 x+2 x^4+e^x \left (-x+x^4\right )+\left (-x+x^4\right ) \log (\log (5))}{35+7 x}} \left (175+60 x+7 x^2+40 x^4+6 x^5+e^x \left (-5 x-5 x^2-x^3+20 x^4+8 x^5+x^6\right )+\left (-5 x+20 x^4+3 x^5\right ) \log (\log (5))\right )}{175+70 x+7 x^2} \, dx=x e^{\left (\frac {2 \, x^{4} + {\left (x^{4} - x\right )} e^{x} + {\left (x^{4} - x\right )} \log \left (\log \left (5\right )\right ) - 2 \, x}{7 \, {\left (x + 5\right )}}\right )} \] Input:
integrate(((3*x^5+20*x^4-5*x)*log(log(5))+(x^6+8*x^5+20*x^4-x^3-5*x^2-5*x) *exp(x)+6*x^5+40*x^4+7*x^2+60*x+175)*exp(((x^4-x)*log(log(5))+(x^4-x)*exp( x)+2*x^4-2*x)/(7*x+35))/(7*x^2+70*x+175),x, algorithm="fricas")
Output:
x*e^(1/7*(2*x^4 + (x^4 - x)*e^x + (x^4 - x)*log(log(5)) - 2*x)/(x + 5))
Time = 6.13 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.21 \[ \int \frac {e^{\frac {-2 x+2 x^4+e^x \left (-x+x^4\right )+\left (-x+x^4\right ) \log (\log (5))}{35+7 x}} \left (175+60 x+7 x^2+40 x^4+6 x^5+e^x \left (-5 x-5 x^2-x^3+20 x^4+8 x^5+x^6\right )+\left (-5 x+20 x^4+3 x^5\right ) \log (\log (5))\right )}{175+70 x+7 x^2} \, dx=x e^{\frac {2 x^{4} - 2 x + \left (x^{4} - x\right ) e^{x} + \left (x^{4} - x\right ) \log {\left (\log {\left (5 \right )} \right )}}{7 x + 35}} \] Input:
integrate(((3*x**5+20*x**4-5*x)*ln(ln(5))+(x**6+8*x**5+20*x**4-x**3-5*x**2 -5*x)*exp(x)+6*x**5+40*x**4+7*x**2+60*x+175)*exp(((x**4-x)*ln(ln(5))+(x**4 -x)*exp(x)+2*x**4-2*x)/(7*x+35))/(7*x**2+70*x+175),x)
Output:
x*exp((2*x**4 - 2*x + (x**4 - x)*exp(x) + (x**4 - x)*log(log(5)))/(7*x + 3 5))
\[ \int \frac {e^{\frac {-2 x+2 x^4+e^x \left (-x+x^4\right )+\left (-x+x^4\right ) \log (\log (5))}{35+7 x}} \left (175+60 x+7 x^2+40 x^4+6 x^5+e^x \left (-5 x-5 x^2-x^3+20 x^4+8 x^5+x^6\right )+\left (-5 x+20 x^4+3 x^5\right ) \log (\log (5))\right )}{175+70 x+7 x^2} \, dx=\int { \frac {{\left (6 \, x^{5} + 40 \, x^{4} + 7 \, x^{2} + {\left (x^{6} + 8 \, x^{5} + 20 \, x^{4} - x^{3} - 5 \, x^{2} - 5 \, x\right )} e^{x} + {\left (3 \, x^{5} + 20 \, x^{4} - 5 \, x\right )} \log \left (\log \left (5\right )\right ) + 60 \, x + 175\right )} e^{\left (\frac {2 \, x^{4} + {\left (x^{4} - x\right )} e^{x} + {\left (x^{4} - x\right )} \log \left (\log \left (5\right )\right ) - 2 \, x}{7 \, {\left (x + 5\right )}}\right )}}{7 \, {\left (x^{2} + 10 \, x + 25\right )}} \,d x } \] Input:
integrate(((3*x^5+20*x^4-5*x)*log(log(5))+(x^6+8*x^5+20*x^4-x^3-5*x^2-5*x) *exp(x)+6*x^5+40*x^4+7*x^2+60*x+175)*exp(((x^4-x)*log(log(5))+(x^4-x)*exp( x)+2*x^4-2*x)/(7*x+35))/(7*x^2+70*x+175),x, algorithm="maxima")
Output:
1/7*integrate((6*x^5 + 40*x^4 + 7*x^2 + (x^6 + 8*x^5 + 20*x^4 - x^3 - 5*x^ 2 - 5*x)*e^x + (3*x^5 + 20*x^4 - 5*x)*log(log(5)) + 60*x + 175)*e^(1/7*(2* x^4 + (x^4 - x)*e^x + (x^4 - x)*log(log(5)) - 2*x)/(x + 5))/(x^2 + 10*x + 25), x)
\[ \int \frac {e^{\frac {-2 x+2 x^4+e^x \left (-x+x^4\right )+\left (-x+x^4\right ) \log (\log (5))}{35+7 x}} \left (175+60 x+7 x^2+40 x^4+6 x^5+e^x \left (-5 x-5 x^2-x^3+20 x^4+8 x^5+x^6\right )+\left (-5 x+20 x^4+3 x^5\right ) \log (\log (5))\right )}{175+70 x+7 x^2} \, dx=\int { \frac {{\left (6 \, x^{5} + 40 \, x^{4} + 7 \, x^{2} + {\left (x^{6} + 8 \, x^{5} + 20 \, x^{4} - x^{3} - 5 \, x^{2} - 5 \, x\right )} e^{x} + {\left (3 \, x^{5} + 20 \, x^{4} - 5 \, x\right )} \log \left (\log \left (5\right )\right ) + 60 \, x + 175\right )} e^{\left (\frac {2 \, x^{4} + {\left (x^{4} - x\right )} e^{x} + {\left (x^{4} - x\right )} \log \left (\log \left (5\right )\right ) - 2 \, x}{7 \, {\left (x + 5\right )}}\right )}}{7 \, {\left (x^{2} + 10 \, x + 25\right )}} \,d x } \] Input:
integrate(((3*x^5+20*x^4-5*x)*log(log(5))+(x^6+8*x^5+20*x^4-x^3-5*x^2-5*x) *exp(x)+6*x^5+40*x^4+7*x^2+60*x+175)*exp(((x^4-x)*log(log(5))+(x^4-x)*exp( x)+2*x^4-2*x)/(7*x+35))/(7*x^2+70*x+175),x, algorithm="giac")
Output:
undef
Time = 3.49 (sec) , antiderivative size = 73, normalized size of antiderivative = 2.61 \[ \int \frac {e^{\frac {-2 x+2 x^4+e^x \left (-x+x^4\right )+\left (-x+x^4\right ) \log (\log (5))}{35+7 x}} \left (175+60 x+7 x^2+40 x^4+6 x^5+e^x \left (-5 x-5 x^2-x^3+20 x^4+8 x^5+x^6\right )+\left (-5 x+20 x^4+3 x^5\right ) \log (\log (5))\right )}{175+70 x+7 x^2} \, dx=\frac {x\,{\mathrm {e}}^{\frac {x^4\,{\mathrm {e}}^x}{7\,x+35}}\,{\mathrm {e}}^{\frac {2\,x^4}{7\,x+35}}\,{\mathrm {e}}^{-\frac {x\,{\mathrm {e}}^x}{7\,x+35}}\,{\mathrm {e}}^{-\frac {2\,x}{7\,x+35}}}{{\ln \left (5\right )}^{\frac {x-x^4}{7\,\left (x+5\right )}}} \] Input:
int((exp(-(2*x + log(log(5))*(x - x^4) + exp(x)*(x - x^4) - 2*x^4)/(7*x + 35))*(60*x - exp(x)*(5*x + 5*x^2 + x^3 - 20*x^4 - 8*x^5 - x^6) + 7*x^2 + 4 0*x^4 + 6*x^5 + log(log(5))*(20*x^4 - 5*x + 3*x^5) + 175))/(70*x + 7*x^2 + 175),x)
Output:
(x*exp((x^4*exp(x))/(7*x + 35))*exp((2*x^4)/(7*x + 35))*exp(-(x*exp(x))/(7 *x + 35))*exp(-(2*x)/(7*x + 35)))/log(5)^((x - x^4)/(7*(x + 5)))
Time = 0.19 (sec) , antiderivative size = 58, normalized size of antiderivative = 2.07 \[ \int \frac {e^{\frac {-2 x+2 x^4+e^x \left (-x+x^4\right )+\left (-x+x^4\right ) \log (\log (5))}{35+7 x}} \left (175+60 x+7 x^2+40 x^4+6 x^5+e^x \left (-5 x-5 x^2-x^3+20 x^4+8 x^5+x^6\right )+\left (-5 x+20 x^4+3 x^5\right ) \log (\log (5))\right )}{175+70 x+7 x^2} \, dx=\frac {e^{\frac {e^{x} x^{4}+\mathrm {log}\left (\mathrm {log}\left (5\right )\right ) x^{4}+2 x^{4}}{7 x +35}} x}{e^{\frac {e^{x} x +\mathrm {log}\left (\mathrm {log}\left (5\right )\right ) x +2 x}{7 x +35}}} \] Input:
int(((3*x^5+20*x^4-5*x)*log(log(5))+(x^6+8*x^5+20*x^4-x^3-5*x^2-5*x)*exp(x )+6*x^5+40*x^4+7*x^2+60*x+175)*exp(((x^4-x)*log(log(5))+(x^4-x)*exp(x)+2*x ^4-2*x)/(7*x+35))/(7*x^2+70*x+175),x)
Output:
(e**((e**x*x**4 + log(log(5))*x**4 + 2*x**4)/(7*x + 35))*x)/e**((e**x*x + log(log(5))*x + 2*x)/(7*x + 35))