Integrand size = 105, antiderivative size = 26 \[ \int \frac {-75 e^{4 x} x-150 e^{4 x} x \log (x)+\left (6 e^{4 x} x^2+e^{2 x} \left (-18 x-18 x^2\right )\right ) \log ^2(x)}{1250 e^{4 x}+\left (300 e^{2 x}-200 e^{4 x} x\right ) \log (x)+\left (18-24 e^{2 x} x+8 e^{4 x} x^2\right ) \log ^2(x)} \, dx=\frac {3 x^2}{2 \left (-3 e^{-2 x}+2 x-\frac {25}{\log (x)}\right )} \] Output:
3/2/(2*x-25/ln(x)-3/exp(x)^2)*x^2
Time = 1.03 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.42 \[ \int \frac {-75 e^{4 x} x-150 e^{4 x} x \log (x)+\left (6 e^{4 x} x^2+e^{2 x} \left (-18 x-18 x^2\right )\right ) \log ^2(x)}{1250 e^{4 x}+\left (300 e^{2 x}-200 e^{4 x} x\right ) \log (x)+\left (18-24 e^{2 x} x+8 e^{4 x} x^2\right ) \log ^2(x)} \, dx=\frac {3 e^{2 x} x^2 \log (x)}{2 \left (-25 e^{2 x}+\left (-3+2 e^{2 x} x\right ) \log (x)\right )} \] Input:
Integrate[(-75*E^(4*x)*x - 150*E^(4*x)*x*Log[x] + (6*E^(4*x)*x^2 + E^(2*x) *(-18*x - 18*x^2))*Log[x]^2)/(1250*E^(4*x) + (300*E^(2*x) - 200*E^(4*x)*x) *Log[x] + (18 - 24*E^(2*x)*x + 8*E^(4*x)*x^2)*Log[x]^2),x]
Output:
(3*E^(2*x)*x^2*Log[x])/(2*(-25*E^(2*x) + (-3 + 2*E^(2*x)*x)*Log[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.
| \(\displaystyle \int \frac {\left (6 e^{4 x} x^2+e^{2 x} \left (-18 x^2-18 x\right )\right ) \log ^2(x)-75 e^{4 x} x-150 e^{4 x} x \log (x)}{\left (8 e^{4 x} x^2-24 e^{2 x} x+18\right ) \log ^2(x)+1250 e^{4 x}+\left (300 e^{2 x}-200 e^{4 x} x\right ) \log (x)} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {3 e^{2 x} x \left (-25 e^{2 x}+2 e^{2 x} x \log ^2(x)-6 x \log ^2(x)-6 \log ^2(x)-50 e^{2 x} \log (x)\right )}{2 \left (25 e^{2 x}-2 e^{2 x} x \log (x)+3 \log (x)\right )^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3}{2} \int -\frac {e^{2 x} x \left (-2 e^{2 x} x \log ^2(x)+6 x \log ^2(x)+6 \log ^2(x)+50 e^{2 x} \log (x)+25 e^{2 x}\right )}{\left (-2 e^{2 x} x \log (x)+3 \log (x)+25 e^{2 x}\right )^2}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {3}{2} \int \frac {e^{2 x} x \left (-2 e^{2 x} x \log ^2(x)+6 x \log ^2(x)+6 \log ^2(x)+50 e^{2 x} \log (x)+25 e^{2 x}\right )}{\left (-2 e^{2 x} x \log (x)+3 \log (x)+25 e^{2 x}\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {3}{2} \int \left (\frac {3 e^{2 x} x \log (x) \left (4 x^2 \log ^2(x)+2 x \log ^2(x)-50 x \log (x)+25\right )}{(2 x \log (x)-25) \left (2 e^{2 x} x \log (x)-3 \log (x)-25 e^{2 x}\right )^2}-\frac {e^{2 x} x \left (2 x \log ^2(x)-50 \log (x)-25\right )}{(2 x \log (x)-25) \left (2 e^{2 x} x \log (x)-3 \log (x)-25 e^{2 x}\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {3}{2} \int \frac {e^{2 x} x \left (\left (6-2 \left (-3+e^{2 x}\right ) x\right ) \log ^2(x)+50 e^{2 x} \log (x)+25 e^{2 x}\right )}{\left (25 e^{2 x}-\left (2 e^{2 x} x-3\right ) \log (x)\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {3}{2} \int \left (\frac {3 e^{2 x} x \log (x) \left (4 x^2 \log ^2(x)+2 x \log ^2(x)-50 x \log (x)+25\right )}{(2 x \log (x)-25) \left (2 e^{2 x} x \log (x)-3 \log (x)-25 e^{2 x}\right )^2}-\frac {e^{2 x} x \left (2 x \log ^2(x)-50 \log (x)-25\right )}{(2 x \log (x)-25) \left (2 e^{2 x} x \log (x)-3 \log (x)-25 e^{2 x}\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {3}{2} \int \frac {e^{2 x} x \left (\left (6-2 \left (-3+e^{2 x}\right ) x\right ) \log ^2(x)+50 e^{2 x} \log (x)+25 e^{2 x}\right )}{\left (25 e^{2 x}-\left (2 e^{2 x} x-3\right ) \log (x)\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {3}{2} \int \left (\frac {3 e^{2 x} x \log (x) \left (4 x^2 \log ^2(x)+2 x \log ^2(x)-50 x \log (x)+25\right )}{(2 x \log (x)-25) \left (2 e^{2 x} x \log (x)-3 \log (x)-25 e^{2 x}\right )^2}-\frac {e^{2 x} x \left (2 x \log ^2(x)-50 \log (x)-25\right )}{(2 x \log (x)-25) \left (2 e^{2 x} x \log (x)-3 \log (x)-25 e^{2 x}\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {3}{2} \int \frac {e^{2 x} x \left (\left (6-2 \left (-3+e^{2 x}\right ) x\right ) \log ^2(x)+50 e^{2 x} \log (x)+25 e^{2 x}\right )}{\left (25 e^{2 x}-\left (2 e^{2 x} x-3\right ) \log (x)\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {3}{2} \int \left (\frac {3 e^{2 x} x \log (x) \left (4 x^2 \log ^2(x)+2 x \log ^2(x)-50 x \log (x)+25\right )}{(2 x \log (x)-25) \left (2 e^{2 x} x \log (x)-3 \log (x)-25 e^{2 x}\right )^2}-\frac {e^{2 x} x \left (2 x \log ^2(x)-50 \log (x)-25\right )}{(2 x \log (x)-25) \left (2 e^{2 x} x \log (x)-3 \log (x)-25 e^{2 x}\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {3}{2} \int \frac {e^{2 x} x \left (\left (6-2 \left (-3+e^{2 x}\right ) x\right ) \log ^2(x)+50 e^{2 x} \log (x)+25 e^{2 x}\right )}{\left (25 e^{2 x}-\left (2 e^{2 x} x-3\right ) \log (x)\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {3}{2} \int \left (\frac {3 e^{2 x} x \log (x) \left (4 x^2 \log ^2(x)+2 x \log ^2(x)-50 x \log (x)+25\right )}{(2 x \log (x)-25) \left (2 e^{2 x} x \log (x)-3 \log (x)-25 e^{2 x}\right )^2}-\frac {e^{2 x} x \left (2 x \log ^2(x)-50 \log (x)-25\right )}{(2 x \log (x)-25) \left (2 e^{2 x} x \log (x)-3 \log (x)-25 e^{2 x}\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {3}{2} \int \frac {e^{2 x} x \left (\left (6-2 \left (-3+e^{2 x}\right ) x\right ) \log ^2(x)+50 e^{2 x} \log (x)+25 e^{2 x}\right )}{\left (25 e^{2 x}-\left (2 e^{2 x} x-3\right ) \log (x)\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {3}{2} \int \left (\frac {3 e^{2 x} x \log (x) \left (4 x^2 \log ^2(x)+2 x \log ^2(x)-50 x \log (x)+25\right )}{(2 x \log (x)-25) \left (2 e^{2 x} x \log (x)-3 \log (x)-25 e^{2 x}\right )^2}-\frac {e^{2 x} x \left (2 x \log ^2(x)-50 \log (x)-25\right )}{(2 x \log (x)-25) \left (2 e^{2 x} x \log (x)-3 \log (x)-25 e^{2 x}\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {3}{2} \int \frac {e^{2 x} x \left (\left (6-2 \left (-3+e^{2 x}\right ) x\right ) \log ^2(x)+50 e^{2 x} \log (x)+25 e^{2 x}\right )}{\left (25 e^{2 x}-\left (2 e^{2 x} x-3\right ) \log (x)\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {3}{2} \int \left (\frac {3 e^{2 x} x \log (x) \left (4 x^2 \log ^2(x)+2 x \log ^2(x)-50 x \log (x)+25\right )}{(2 x \log (x)-25) \left (2 e^{2 x} x \log (x)-3 \log (x)-25 e^{2 x}\right )^2}-\frac {e^{2 x} x \left (2 x \log ^2(x)-50 \log (x)-25\right )}{(2 x \log (x)-25) \left (2 e^{2 x} x \log (x)-3 \log (x)-25 e^{2 x}\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {3}{2} \int \frac {e^{2 x} x \left (\left (6-2 \left (-3+e^{2 x}\right ) x\right ) \log ^2(x)+50 e^{2 x} \log (x)+25 e^{2 x}\right )}{\left (25 e^{2 x}-\left (2 e^{2 x} x-3\right ) \log (x)\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {3}{2} \int \left (\frac {3 e^{2 x} x \log (x) \left (4 x^2 \log ^2(x)+2 x \log ^2(x)-50 x \log (x)+25\right )}{(2 x \log (x)-25) \left (2 e^{2 x} x \log (x)-3 \log (x)-25 e^{2 x}\right )^2}-\frac {e^{2 x} x \left (2 x \log ^2(x)-50 \log (x)-25\right )}{(2 x \log (x)-25) \left (2 e^{2 x} x \log (x)-3 \log (x)-25 e^{2 x}\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {3}{2} \int \frac {e^{2 x} x \left (\left (6-2 \left (-3+e^{2 x}\right ) x\right ) \log ^2(x)+50 e^{2 x} \log (x)+25 e^{2 x}\right )}{\left (25 e^{2 x}-\left (2 e^{2 x} x-3\right ) \log (x)\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {3}{2} \int \left (\frac {3 e^{2 x} x \log (x) \left (4 x^2 \log ^2(x)+2 x \log ^2(x)-50 x \log (x)+25\right )}{(2 x \log (x)-25) \left (2 e^{2 x} x \log (x)-3 \log (x)-25 e^{2 x}\right )^2}-\frac {e^{2 x} x \left (2 x \log ^2(x)-50 \log (x)-25\right )}{(2 x \log (x)-25) \left (2 e^{2 x} x \log (x)-3 \log (x)-25 e^{2 x}\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {3}{2} \int \frac {e^{2 x} x \left (\left (6-2 \left (-3+e^{2 x}\right ) x\right ) \log ^2(x)+50 e^{2 x} \log (x)+25 e^{2 x}\right )}{\left (25 e^{2 x}-\left (2 e^{2 x} x-3\right ) \log (x)\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {3}{2} \int \left (\frac {3 e^{2 x} x \log (x) \left (4 x^2 \log ^2(x)+2 x \log ^2(x)-50 x \log (x)+25\right )}{(2 x \log (x)-25) \left (2 e^{2 x} x \log (x)-3 \log (x)-25 e^{2 x}\right )^2}-\frac {e^{2 x} x \left (2 x \log ^2(x)-50 \log (x)-25\right )}{(2 x \log (x)-25) \left (2 e^{2 x} x \log (x)-3 \log (x)-25 e^{2 x}\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {3}{2} \int \frac {e^{2 x} x \left (\left (6-2 \left (-3+e^{2 x}\right ) x\right ) \log ^2(x)+50 e^{2 x} \log (x)+25 e^{2 x}\right )}{\left (25 e^{2 x}-\left (2 e^{2 x} x-3\right ) \log (x)\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {3}{2} \int \left (\frac {3 e^{2 x} x \log (x) \left (4 x^2 \log ^2(x)+2 x \log ^2(x)-50 x \log (x)+25\right )}{(2 x \log (x)-25) \left (2 e^{2 x} x \log (x)-3 \log (x)-25 e^{2 x}\right )^2}-\frac {e^{2 x} x \left (2 x \log ^2(x)-50 \log (x)-25\right )}{(2 x \log (x)-25) \left (2 e^{2 x} x \log (x)-3 \log (x)-25 e^{2 x}\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {3}{2} \int \frac {e^{2 x} x \left (\left (6-2 \left (-3+e^{2 x}\right ) x\right ) \log ^2(x)+50 e^{2 x} \log (x)+25 e^{2 x}\right )}{\left (25 e^{2 x}-\left (2 e^{2 x} x-3\right ) \log (x)\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {3}{2} \int \left (\frac {3 e^{2 x} x \log (x) \left (4 x^2 \log ^2(x)+2 x \log ^2(x)-50 x \log (x)+25\right )}{(2 x \log (x)-25) \left (2 e^{2 x} x \log (x)-3 \log (x)-25 e^{2 x}\right )^2}-\frac {e^{2 x} x \left (2 x \log ^2(x)-50 \log (x)-25\right )}{(2 x \log (x)-25) \left (2 e^{2 x} x \log (x)-3 \log (x)-25 e^{2 x}\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {3}{2} \int \frac {e^{2 x} x \left (\left (6-2 \left (-3+e^{2 x}\right ) x\right ) \log ^2(x)+50 e^{2 x} \log (x)+25 e^{2 x}\right )}{\left (25 e^{2 x}-\left (2 e^{2 x} x-3\right ) \log (x)\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {3}{2} \int \left (\frac {3 e^{2 x} x \log (x) \left (4 x^2 \log ^2(x)+2 x \log ^2(x)-50 x \log (x)+25\right )}{(2 x \log (x)-25) \left (2 e^{2 x} x \log (x)-3 \log (x)-25 e^{2 x}\right )^2}-\frac {e^{2 x} x \left (2 x \log ^2(x)-50 \log (x)-25\right )}{(2 x \log (x)-25) \left (2 e^{2 x} x \log (x)-3 \log (x)-25 e^{2 x}\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {3}{2} \int \frac {e^{2 x} x \left (\left (6-2 \left (-3+e^{2 x}\right ) x\right ) \log ^2(x)+50 e^{2 x} \log (x)+25 e^{2 x}\right )}{\left (25 e^{2 x}-\left (2 e^{2 x} x-3\right ) \log (x)\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {3}{2} \int \left (\frac {3 e^{2 x} x \log (x) \left (4 x^2 \log ^2(x)+2 x \log ^2(x)-50 x \log (x)+25\right )}{(2 x \log (x)-25) \left (2 e^{2 x} x \log (x)-3 \log (x)-25 e^{2 x}\right )^2}-\frac {e^{2 x} x \left (2 x \log ^2(x)-50 \log (x)-25\right )}{(2 x \log (x)-25) \left (2 e^{2 x} x \log (x)-3 \log (x)-25 e^{2 x}\right )}\right )dx\) |
Input:
Int[(-75*E^(4*x)*x - 150*E^(4*x)*x*Log[x] + (6*E^(4*x)*x^2 + E^(2*x)*(-18* x - 18*x^2))*Log[x]^2)/(1250*E^(4*x) + (300*E^(2*x) - 200*E^(4*x)*x)*Log[x ] + (18 - 24*E^(2*x)*x + 8*E^(4*x)*x^2)*Log[x]^2),x]
Output:
$Aborted
Time = 0.82 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.31
| method | result | size |
| parallelrisch | \(\frac {3 x^{2} {\mathrm e}^{2 x} \ln \left (x \right )}{2 \left (2 \ln \left (x \right ) {\mathrm e}^{2 x} x -25 \,{\mathrm e}^{2 x}-3 \ln \left (x \right )\right )}\) | \(34\) |
| risch | \(\frac {3 x^{2} \left (2 x \,{\mathrm e}^{4 x} \ln \left (x \right )-3 \ln \left (x \right ) {\mathrm e}^{2 x}\right )}{2 \left (2 x \,{\mathrm e}^{2 x}-3\right ) \left (2 \ln \left (x \right ) {\mathrm e}^{2 x} x -25 \,{\mathrm e}^{2 x}-3 \ln \left (x \right )\right )}\) | \(57\) |
Input:
int(((6*x^2*exp(x)^4+(-18*x^2-18*x)*exp(x)^2)*ln(x)^2-150*x*exp(x)^4*ln(x) -75*x*exp(x)^4)/((8*x^2*exp(x)^4-24*x*exp(x)^2+18)*ln(x)^2+(-200*x*exp(x)^ 4+300*exp(x)^2)*ln(x)+1250*exp(x)^4),x,method=_RETURNVERBOSE)
Output:
3/2*x^2*exp(x)^2*ln(x)/(2*x*exp(x)^2*ln(x)-25*exp(x)^2-3*ln(x))
Time = 0.09 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.23 \[ \int \frac {-75 e^{4 x} x-150 e^{4 x} x \log (x)+\left (6 e^{4 x} x^2+e^{2 x} \left (-18 x-18 x^2\right )\right ) \log ^2(x)}{1250 e^{4 x}+\left (300 e^{2 x}-200 e^{4 x} x\right ) \log (x)+\left (18-24 e^{2 x} x+8 e^{4 x} x^2\right ) \log ^2(x)} \, dx=\frac {3 \, x^{2} e^{\left (2 \, x\right )} \log \left (x\right )}{2 \, {\left ({\left (2 \, x e^{\left (2 \, x\right )} - 3\right )} \log \left (x\right ) - 25 \, e^{\left (2 \, x\right )}\right )}} \] Input:
integrate(((6*x^2*exp(x)^4+(-18*x^2-18*x)*exp(x)^2)*log(x)^2-150*x*exp(x)^ 4*log(x)-75*x*exp(x)^4)/((8*x^2*exp(x)^4-24*x*exp(x)^2+18)*log(x)^2+(-200* x*exp(x)^4+300*exp(x)^2)*log(x)+1250*exp(x)^4),x, algorithm="fricas")
Output:
3/2*x^2*e^(2*x)*log(x)/((2*x*e^(2*x) - 3)*log(x) - 25*e^(2*x))
Leaf count of result is larger than twice the leaf count of optimal. 63 vs. \(2 (19) = 38\).
Time = 0.20 (sec) , antiderivative size = 63, normalized size of antiderivative = 2.42 \[ \int \frac {-75 e^{4 x} x-150 e^{4 x} x \log (x)+\left (6 e^{4 x} x^2+e^{2 x} \left (-18 x-18 x^2\right )\right ) \log ^2(x)}{1250 e^{4 x}+\left (300 e^{2 x}-200 e^{4 x} x\right ) \log (x)+\left (18-24 e^{2 x} x+8 e^{4 x} x^2\right ) \log ^2(x)} \, dx=\frac {9 x^{2} \log {\left (x \right )}^{2}}{- 12 x \log {\left (x \right )}^{2} + \left (8 x^{2} \log {\left (x \right )}^{2} - 200 x \log {\left (x \right )} + 1250\right ) e^{2 x} + 150 \log {\left (x \right )}} + \frac {3 x}{4} + \frac {75 x}{8 x \log {\left (x \right )} - 100} \] Input:
integrate(((6*x**2*exp(x)**4+(-18*x**2-18*x)*exp(x)**2)*ln(x)**2-150*x*exp (x)**4*ln(x)-75*x*exp(x)**4)/((8*x**2*exp(x)**4-24*x*exp(x)**2+18)*ln(x)** 2+(-200*x*exp(x)**4+300*exp(x)**2)*ln(x)+1250*exp(x)**4),x)
Output:
9*x**2*log(x)**2/(-12*x*log(x)**2 + (8*x**2*log(x)**2 - 200*x*log(x) + 125 0)*exp(2*x) + 150*log(x)) + 3*x/4 + 75*x/(8*x*log(x) - 100)
Time = 0.10 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.15 \[ \int \frac {-75 e^{4 x} x-150 e^{4 x} x \log (x)+\left (6 e^{4 x} x^2+e^{2 x} \left (-18 x-18 x^2\right )\right ) \log ^2(x)}{1250 e^{4 x}+\left (300 e^{2 x}-200 e^{4 x} x\right ) \log (x)+\left (18-24 e^{2 x} x+8 e^{4 x} x^2\right ) \log ^2(x)} \, dx=\frac {3 \, x^{2} e^{\left (2 \, x\right )} \log \left (x\right )}{2 \, {\left ({\left (2 \, x \log \left (x\right ) - 25\right )} e^{\left (2 \, x\right )} - 3 \, \log \left (x\right )\right )}} \] Input:
integrate(((6*x^2*exp(x)^4+(-18*x^2-18*x)*exp(x)^2)*log(x)^2-150*x*exp(x)^ 4*log(x)-75*x*exp(x)^4)/((8*x^2*exp(x)^4-24*x*exp(x)^2+18)*log(x)^2+(-200* x*exp(x)^4+300*exp(x)^2)*log(x)+1250*exp(x)^4),x, algorithm="maxima")
Output:
3/2*x^2*e^(2*x)*log(x)/((2*x*log(x) - 25)*e^(2*x) - 3*log(x))
Time = 0.14 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.27 \[ \int \frac {-75 e^{4 x} x-150 e^{4 x} x \log (x)+\left (6 e^{4 x} x^2+e^{2 x} \left (-18 x-18 x^2\right )\right ) \log ^2(x)}{1250 e^{4 x}+\left (300 e^{2 x}-200 e^{4 x} x\right ) \log (x)+\left (18-24 e^{2 x} x+8 e^{4 x} x^2\right ) \log ^2(x)} \, dx=\frac {3 \, x^{2} e^{\left (2 \, x\right )} \log \left (x\right )}{2 \, {\left (2 \, x e^{\left (2 \, x\right )} \log \left (x\right ) - 25 \, e^{\left (2 \, x\right )} - 3 \, \log \left (x\right )\right )}} \] Input:
integrate(((6*x^2*exp(x)^4+(-18*x^2-18*x)*exp(x)^2)*log(x)^2-150*x*exp(x)^ 4*log(x)-75*x*exp(x)^4)/((8*x^2*exp(x)^4-24*x*exp(x)^2+18)*log(x)^2+(-200* x*exp(x)^4+300*exp(x)^2)*log(x)+1250*exp(x)^4),x, algorithm="giac")
Output:
3/2*x^2*e^(2*x)*log(x)/(2*x*e^(2*x)*log(x) - 25*e^(2*x) - 3*log(x))
Time = 4.10 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.27 \[ \int \frac {-75 e^{4 x} x-150 e^{4 x} x \log (x)+\left (6 e^{4 x} x^2+e^{2 x} \left (-18 x-18 x^2\right )\right ) \log ^2(x)}{1250 e^{4 x}+\left (300 e^{2 x}-200 e^{4 x} x\right ) \log (x)+\left (18-24 e^{2 x} x+8 e^{4 x} x^2\right ) \log ^2(x)} \, dx=-\frac {3\,x^2\,{\mathrm {e}}^{2\,x}\,\ln \left (x\right )}{2\,\left (25\,{\mathrm {e}}^{2\,x}+3\,\ln \left (x\right )-2\,x\,{\mathrm {e}}^{2\,x}\,\ln \left (x\right )\right )} \] Input:
int(-(75*x*exp(4*x) + log(x)^2*(exp(2*x)*(18*x + 18*x^2) - 6*x^2*exp(4*x)) + 150*x*exp(4*x)*log(x))/(1250*exp(4*x) + log(x)*(300*exp(2*x) - 200*x*ex p(4*x)) + log(x)^2*(8*x^2*exp(4*x) - 24*x*exp(2*x) + 18)),x)
Output:
-(3*x^2*exp(2*x)*log(x))/(2*(25*exp(2*x) + 3*log(x) - 2*x*exp(2*x)*log(x)) )
Time = 0.18 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.38 \[ \int \frac {-75 e^{4 x} x-150 e^{4 x} x \log (x)+\left (6 e^{4 x} x^2+e^{2 x} \left (-18 x-18 x^2\right )\right ) \log ^2(x)}{1250 e^{4 x}+\left (300 e^{2 x}-200 e^{4 x} x\right ) \log (x)+\left (18-24 e^{2 x} x+8 e^{4 x} x^2\right ) \log ^2(x)} \, dx=\frac {3 e^{2 x} \mathrm {log}\left (x \right ) x^{2}}{4 e^{2 x} \mathrm {log}\left (x \right ) x -50 e^{2 x}-6 \,\mathrm {log}\left (x \right )} \] Input:
int(((6*x^2*exp(x)^4+(-18*x^2-18*x)*exp(x)^2)*log(x)^2-150*x*exp(x)^4*log( x)-75*x*exp(x)^4)/((8*x^2*exp(x)^4-24*x*exp(x)^2+18)*log(x)^2+(-200*x*exp( x)^4+300*exp(x)^2)*log(x)+1250*exp(x)^4),x)
Output:
(3*e**(2*x)*log(x)*x**2)/(2*(2*e**(2*x)*log(x)*x - 25*e**(2*x) - 3*log(x)) )