Integrand size = 144, antiderivative size = 31 \[ \int \frac {-15+29 x+4 x^2-4 x^3+\left (-3+7 x-2 x^2\right ) \log (5)+e^{2 x^2} \left (-25 x+55 x^2-128 x^3+12 x^4+8 x^5+\left (-5 x+12 x^2-28 x^3+8 x^4\right ) \log (5)\right )+\left (25 x+5 x^2+5 x \log (5)\right ) \log \left (5 x+x^2+x \log (5)\right )}{45 x-21 x^2-x^3+x^4+\left (9 x-6 x^2+x^3\right ) \log (5)} \, dx=\frac {(-1+2 x) \left (-e^{2 x^2}+\log (x (5+x+\log (5)))\right )}{3-x} \]
Leaf count is larger than twice the leaf count of optimal. \(76\) vs. \(2(31)=62\).
Time = 1.57 (sec) , antiderivative size = 76, normalized size of antiderivative = 2.45 \[ \int \frac {-15+29 x+4 x^2-4 x^3+\left (-3+7 x-2 x^2\right ) \log (5)+e^{2 x^2} \left (-25 x+55 x^2-128 x^3+12 x^4+8 x^5+\left (-5 x+12 x^2-28 x^3+8 x^4\right ) \log (5)\right )+\left (25 x+5 x^2+5 x \log (5)\right ) \log \left (5 x+x^2+x \log (5)\right )}{45 x-21 x^2-x^3+x^4+\left (9 x-6 x^2+x^3\right ) \log (5)} \, dx=\frac {e^{2 x^2} \left (-1+2 x+8 \log ^3(5)-4 \log ^2(5) (-14+\log (25))-76 \log (25)-4 \log (5) (-38+7 \log (25))\right )}{-3+x}-2 \log (x)-2 \log (5+x+\log (5))-\frac {5 \log (x (5+x+\log (5)))}{-3+x} \]
Integrate[(-15 + 29*x + 4*x^2 - 4*x^3 + (-3 + 7*x - 2*x^2)*Log[5] + E^(2*x ^2)*(-25*x + 55*x^2 - 128*x^3 + 12*x^4 + 8*x^5 + (-5*x + 12*x^2 - 28*x^3 + 8*x^4)*Log[5]) + (25*x + 5*x^2 + 5*x*Log[5])*Log[5*x + x^2 + x*Log[5]])/( 45*x - 21*x^2 - x^3 + x^4 + (9*x - 6*x^2 + x^3)*Log[5]),x]
(E^(2*x^2)*(-1 + 2*x + 8*Log[5]^3 - 4*Log[5]^2*(-14 + Log[25]) - 76*Log[25 ] - 4*Log[5]*(-38 + 7*Log[25])))/(-3 + x) - 2*Log[x] - 2*Log[5 + x + Log[5 ]] - (5*Log[x*(5 + x + Log[5])])/(-3 + 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.
| \(\displaystyle \int \frac {-4 x^3+4 x^2+\left (5 x^2+25 x+5 x \log (5)\right ) \log \left (x^2+5 x+x \log (5)\right )+\left (-2 x^2+7 x-3\right ) \log (5)+e^{2 x^2} \left (8 x^5+12 x^4-128 x^3+55 x^2+\left (8 x^4-28 x^3+12 x^2-5 x\right ) \log (5)-25 x\right )+29 x-15}{x^4-x^3-21 x^2+\left (x^3-6 x^2+9 x\right ) \log (5)+45 x} \, dx\) |
| \(\Big \downarrow \) 2026 |
| \(\displaystyle \int \frac {-4 x^3+4 x^2+\left (5 x^2+25 x+5 x \log (5)\right ) \log \left (x^2+5 x+x \log (5)\right )+\left (-2 x^2+7 x-3\right ) \log (5)+e^{2 x^2} \left (8 x^5+12 x^4-128 x^3+55 x^2+\left (8 x^4-28 x^3+12 x^2-5 x\right ) \log (5)-25 x\right )+29 x-15}{x \left (x^3-x^2 (1-\log (5))-3 x (7+\log (25))+9 (5+\log (5))\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {4 x^2}{-x^3+x^2 (1-\log (5))+3 x (7+\log (25))-9 (5+\log (5))}+\frac {4 x}{x^3-x^2 (1-\log (5))-3 x (7+\log (25))+9 (5+\log (5))}+\frac {5 (x+5+\log (5)) \log (x (x+5+\log (5)))}{x^3-x^2 (1-\log (5))-3 x (7+\log (25))+9 (5+\log (5))}+\frac {e^{2 x^2} \left (-8 x^3+28 x^2-12 x+5\right ) (-x-5-\log (5))}{x^3-x^2 (1-\log (5))-3 x (7+\log (25))+9 (5+\log (5))}+\frac {29}{x^3-x^2 (1-\log (5))-3 x (7+\log (25))+9 (5+\log (5))}+\frac {(1-2 x) (x-3) \log (5)}{x \left (x^3-x^2 (1-\log (5))-3 x (7+\log (25))+9 (5+\log (5))\right )}+\frac {15}{x \left (-x^3+x^2 (1-\log (5))+3 x (7+\log (25))-9 (5+\log (5))\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\log (5) (58+\log (48828125)) \text {arctanh}\left (\frac {2 x+\log (5)+2}{\sqrt {64+\log ^2(5)+4 \log (125)+\log (625)}}\right )}{3 (5+\log (5)) \sqrt {64+\log ^2(5)+4 \log (125)+\log (625)}}+2 e^{2 x^2}-\frac {20 \left (64-2 \log (5)+\log ^2(5)+9 \log (25)\right )^3 \log (x)}{3 \left (320-3 \log ^2(5)+\log ^3(5)-12 \log (25)+12 \log (5) (14+\log (25))\right ) (128-\log (25)+\log (5) (34+\log (25)))^2}-\frac {\log (5) \log (x)}{3 (5+\log (5))}-\frac {5 (\log (x)+\log (x+\log (5)+5)-\log (x (x+\log (5)+5)))}{3-x}-\frac {16 \left (320-3 \log ^2(5)+\log ^3(5)-12 \log (25)+12 \log (5) (14+\log (25))\right )^2 \log \left (\left (64-2 \log (5)+\log ^2(5)+9 \log (25)\right ) x+12 \log (5) (14+\log (25))-12 \log (25)+\log ^3(5)-3 \log ^2(5)+320\right )}{\left (1024-6 \log ^2(5)+2 \log ^3(5)-27 \log (25)+3 \log (5) (146+9 \log (25))\right )^2}-\frac {16 \left (64-2 \log (5)+\log ^2(5)+9 \log (25)\right ) \left (320-3 \log ^2(5)+\log ^3(5)-12 \log (25)+12 \log (5) (14+\log (25))\right ) \log \left (\left (64-2 \log (5)+\log ^2(5)+9 \log (25)\right ) x+12 \log (5) (14+\log (25))-12 \log (25)+\log ^3(5)-3 \log ^2(5)+320\right )}{\left (1024-6 \log ^2(5)+2 \log ^3(5)-27 \log (25)+3 \log (5) (146+9 \log (25))\right )^2}+\frac {60 \left (64-2 \log (5)+\log ^2(5)+9 \log (25)\right )^3 \log \left (\left (64-2 \log (5)+\log ^2(5)+9 \log (25)\right ) x+12 \log (5) (14+\log (25))-12 \log (25)+\log ^3(5)-3 \log ^2(5)+320\right )}{\left (320-3 \log ^2(5)+\log ^3(5)-12 \log (25)+12 \log (5) (14+\log (25))\right ) \left (1024-6 \log ^2(5)+2 \log ^3(5)-27 \log (25)+3 \log (5) (146+9 \log (25))\right )^2}+\frac {116 \left (64-2 \log (5)+\log ^2(5)+9 \log (25)\right )^2 \log \left (\left (64-2 \log (5)+\log ^2(5)+9 \log (25)\right ) x+12 \log (5) (14+\log (25))-12 \log (25)+\log ^3(5)-3 \log ^2(5)+320\right )}{\left (1024-6 \log ^2(5)+2 \log ^3(5)-27 \log (25)+3 \log (5) (146+9 \log (25))\right )^2}-\frac {12 \left (212992-8192 \log (25)+51 \log ^2(25)+4 \log ^4(5) (34+\log (25))+\log (5) \left (155648+5684 \log (25)-102 \log ^2(25)\right )+3 \log ^2(5) \left (8260+840 \log (25)+17 \log ^2(25)\right )+8 \log ^3(5) (13-\log (625))\right ) \log \left (2 x \left (64-2 \log (5)+\log ^2(5)+9 \log (25)\right )-3 (128-\log (25)+\log (5) (34+\log (25)))\right )}{\left (1024-6 \log ^2(5)+2 \log ^3(5)-27 \log (25)+3 \log (5) (146+9 \log (25))\right )^2}+\frac {80 \left (64-2 \log (5)+\log ^2(5)+9 \log (25)\right )^3 \left (704-3 \log ^2(5)+\log ^3(5)-15 \log (25)+15 \log (5) (18+\log (25))\right ) \log \left (2 x \left (64-2 \log (5)+\log ^2(5)+9 \log (25)\right )-3 (128-\log (25)+\log (5) (34+\log (25)))\right )}{3 (128-\log (25)+\log (5) (34+\log (25)))^2 \left (1024-6 \log ^2(5)+2 \log ^3(5)-27 \log (25)+3 \log (5) (146+9 \log (25))\right )^2}+\frac {16 \left (64-2 \log (5)+\log ^2(5)+9 \log (25)\right ) \left (320-3 \log ^2(5)+\log ^3(5)-12 \log (25)+12 \log (5) (14+\log (25))\right ) \log \left (2 x \left (64-2 \log (5)+\log ^2(5)+9 \log (25)\right )-3 (128-\log (25)+\log (5) (34+\log (25)))\right )}{\left (1024-6 \log ^2(5)+2 \log ^3(5)-27 \log (25)+3 \log (5) (146+9 \log (25))\right )^2}-\frac {116 \left (64-2 \log (5)+\log ^2(5)+9 \log (25)\right )^2 \log \left (2 x \left (64-2 \log (5)+\log ^2(5)+9 \log (25)\right )-3 (128-\log (25)+\log (5) (34+\log (25)))\right )}{\left (1024-6 \log ^2(5)+2 \log ^3(5)-27 \log (25)+3 \log (5) (146+9 \log (25))\right )^2}+\frac {\log (5) \log \left (-x^2-(2+\log (5)) x+\log (125)+15\right )}{6 (5+\log (5))}+5 \int \frac {\log (x)}{x^2-6 x-3 \log (25)+6 \log (5)+9}dx+5 \int \frac {\log (x+\log (5)+5)}{x^2-6 x-3 \log (25)+6 \log (5)+9}dx-\frac {5 e^{2 x^2}}{3-x}+\frac {36 (128-\log (25)+\log (5) (34+\log (25)))^2}{\left (1024-6 \log ^2(5)+2 \log ^3(5)-27 \log (25)+3 \log (5) (146+9 \log (25))\right ) \left (2 x \left (64-2 \log (5)+\log ^2(5)+9 \log (25)\right )-3 (128-\log (25)+\log (5) (34+\log (25)))\right )}-\frac {24 \left (64-2 \log (5)+\log ^2(5)+9 \log (25)\right ) (128-\log (25)+\log (5) (34+\log (25)))}{\left (1024-6 \log ^2(5)+2 \log ^3(5)-27 \log (25)+3 \log (5) (146+9 \log (25))\right ) \left (2 x \left (64-2 \log (5)+\log ^2(5)+9 \log (25)\right )-3 (128-\log (25)+\log (5) (34+\log (25)))\right )}+\frac {40 \left (64-2 \log (5)+\log ^2(5)+9 \log (25)\right )^3}{(128-\log (25)+\log (5) (34+\log (25))) \left (1024-6 \log ^2(5)+2 \log ^3(5)-27 \log (25)+3 \log (5) (146+9 \log (25))\right ) \left (2 x \left (64-2 \log (5)+\log ^2(5)+9 \log (25)\right )-3 (128-\log (25)+\log (5) (34+\log (25)))\right )}-\frac {116 \left (64-2 \log (5)+\log ^2(5)+9 \log (25)\right )^2}{\left (1024-6 \log ^2(5)+2 \log ^3(5)-27 \log (25)+3 \log (5) (146+9 \log (25))\right ) \left (2 x \left (64-2 \log (5)+\log ^2(5)+9 \log (25)\right )-3 (128-\log (25)+\log (5) (34+\log (25)))\right )}\) |
Int[(-15 + 29*x + 4*x^2 - 4*x^3 + (-3 + 7*x - 2*x^2)*Log[5] + E^(2*x^2)*(- 25*x + 55*x^2 - 128*x^3 + 12*x^4 + 8*x^5 + (-5*x + 12*x^2 - 28*x^3 + 8*x^4 )*Log[5]) + (25*x + 5*x^2 + 5*x*Log[5])*Log[5*x + x^2 + x*Log[5]])/(45*x - 21*x^2 - x^3 + x^4 + (9*x - 6*x^2 + x^3)*Log[5]),x]
3.8.10.3.1 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 = 2.08 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.42
| method | result | size |
| parallelrisch | \(\frac {2 x \,{\mathrm e}^{2 x^{2}}-2 \ln \left (x \left (x +\ln \left (5\right )+5\right )\right ) x -{\mathrm e}^{2 x^{2}}+\ln \left (x \left (x +\ln \left (5\right )+5\right )\right )}{-3+x}\) | \(44\) |
| risch | \(-\frac {5 \ln \left (x +\ln \left (5\right )+5\right )}{-3+x}-\frac {-5 i \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i \left (x +\ln \left (5\right )+5\right )\right ) \operatorname {csgn}\left (i x \left (x +\ln \left (5\right )+5\right )\right )+5 i \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x \left (x +\ln \left (5\right )+5\right )\right )^{2}+5 i \pi \,\operatorname {csgn}\left (i \left (x +\ln \left (5\right )+5\right )\right ) \operatorname {csgn}\left (i x \left (x +\ln \left (5\right )+5\right )\right )^{2}-5 i \pi \operatorname {csgn}\left (i x \left (x +\ln \left (5\right )+5\right )\right )^{3}-4 x \,{\mathrm e}^{2 x^{2}}+4 \ln \left (x^{2}+\left (5+\ln \left (5\right )\right ) x \right ) x +2 \,{\mathrm e}^{2 x^{2}}-12 \ln \left (x^{2}+\left (5+\ln \left (5\right )\right ) x \right )+10 \ln \left (x \right )}{2 \left (-3+x \right )}\) | \(161\) |
int(((5*x*ln(5)+5*x^2+25*x)*ln(x*ln(5)+x^2+5*x)+((8*x^4-28*x^3+12*x^2-5*x) *ln(5)+8*x^5+12*x^4-128*x^3+55*x^2-25*x)*exp(x^2)^2+(-2*x^2+7*x-3)*ln(5)-4 *x^3+4*x^2+29*x-15)/((x^3-6*x^2+9*x)*ln(5)+x^4-x^3-21*x^2+45*x),x,method=_ RETURNVERBOSE)
Time = 0.25 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.23 \[ \int \frac {-15+29 x+4 x^2-4 x^3+\left (-3+7 x-2 x^2\right ) \log (5)+e^{2 x^2} \left (-25 x+55 x^2-128 x^3+12 x^4+8 x^5+\left (-5 x+12 x^2-28 x^3+8 x^4\right ) \log (5)\right )+\left (25 x+5 x^2+5 x \log (5)\right ) \log \left (5 x+x^2+x \log (5)\right )}{45 x-21 x^2-x^3+x^4+\left (9 x-6 x^2+x^3\right ) \log (5)} \, dx=\frac {{\left (2 \, x - 1\right )} e^{\left (2 \, x^{2}\right )} - {\left (2 \, x - 1\right )} \log \left (x^{2} + x \log \left (5\right ) + 5 \, x\right )}{x - 3} \]
integrate(((5*x*log(5)+5*x^2+25*x)*log(x*log(5)+x^2+5*x)+((8*x^4-28*x^3+12 *x^2-5*x)*log(5)+8*x^5+12*x^4-128*x^3+55*x^2-25*x)*exp(x^2)^2+(-2*x^2+7*x- 3)*log(5)-4*x^3+4*x^2+29*x-15)/((x^3-6*x^2+9*x)*log(5)+x^4-x^3-21*x^2+45*x ),x, algorithm=\
Time = 0.31 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.48 \[ \int \frac {-15+29 x+4 x^2-4 x^3+\left (-3+7 x-2 x^2\right ) \log (5)+e^{2 x^2} \left (-25 x+55 x^2-128 x^3+12 x^4+8 x^5+\left (-5 x+12 x^2-28 x^3+8 x^4\right ) \log (5)\right )+\left (25 x+5 x^2+5 x \log (5)\right ) \log \left (5 x+x^2+x \log (5)\right )}{45 x-21 x^2-x^3+x^4+\left (9 x-6 x^2+x^3\right ) \log (5)} \, dx=- 2 \log {\left (x^{2} + x \left (\log {\left (5 \right )} + 5\right ) \right )} + \frac {\left (2 x - 1\right ) e^{2 x^{2}}}{x - 3} - \frac {5 \log {\left (x^{2} + x \log {\left (5 \right )} + 5 x \right )}}{x - 3} \]
integrate(((5*x*ln(5)+5*x**2+25*x)*ln(x*ln(5)+x**2+5*x)+((8*x**4-28*x**3+1 2*x**2-5*x)*ln(5)+8*x**5+12*x**4-128*x**3+55*x**2-25*x)*exp(x**2)**2+(-2*x **2+7*x-3)*ln(5)-4*x**3+4*x**2+29*x-15)/((x**3-6*x**2+9*x)*ln(5)+x**4-x**3 -21*x**2+45*x),x)
-2*log(x**2 + x*(log(5) + 5)) + (2*x - 1)*exp(2*x**2)/(x - 3) - 5*log(x**2 + x*log(5) + 5*x)/(x - 3)
Leaf count of result is larger than twice the leaf count of optimal. 464 vs. \(2 (28) = 56\).
Time = 0.32 (sec) , antiderivative size = 464, normalized size of antiderivative = 14.97 \[ \int \frac {-15+29 x+4 x^2-4 x^3+\left (-3+7 x-2 x^2\right ) \log (5)+e^{2 x^2} \left (-25 x+55 x^2-128 x^3+12 x^4+8 x^5+\left (-5 x+12 x^2-28 x^3+8 x^4\right ) \log (5)\right )+\left (25 x+5 x^2+5 x \log (5)\right ) \log \left (5 x+x^2+x \log (5)\right )}{45 x-21 x^2-x^3+x^4+\left (9 x-6 x^2+x^3\right ) \log (5)} \, dx =\text {Too large to display} \]
integrate(((5*x*log(5)+5*x^2+25*x)*log(x*log(5)+x^2+5*x)+((8*x^4-28*x^3+12 *x^2-5*x)*log(5)+8*x^5+12*x^4-128*x^3+55*x^2-25*x)*exp(x^2)^2+(-2*x^2+7*x- 3)*log(5)-4*x^3+4*x^2+29*x-15)/((x^3-6*x^2+9*x)*log(5)+x^4-x^3-21*x^2+45*x ),x, algorithm=\
2*((log(5) + 5)*log(x + log(5) + 5)/(log(5)^2 + 16*log(5) + 64) - (log(5) + 5)*log(x - 3)/(log(5)^2 + 16*log(5) + 64) + 3/(x*(log(5) + 8) - 3*log(5) - 24))*log(5) + 1/3*((log(5) + 11)*log(x - 3)/(log(5)^2 + 16*log(5) + 64) + 9*log(x + log(5) + 5)/(log(5)^3 + 21*log(5)^2 + 144*log(5) + 320) - log (x)/(log(5) + 5) + 3/(x*(log(5) + 8) - 3*log(5) - 24))*log(5) + 7*(log(x + log(5) + 5)/(log(5)^2 + 16*log(5) + 64) - log(x - 3)/(log(5)^2 + 16*log(5 ) + 64) - 1/(x*(log(5) + 8) - 3*log(5) - 24))*log(5) - 4*(log(5)^2 + 10*lo g(5) + 25)*log(x + log(5) + 5)/(log(5)^2 + 16*log(5) + 64) - 4*(log(5) + 5 )*log(x + log(5) + 5)/(log(5)^2 + 16*log(5) + 64) - 12*(2*log(5) + 13)*log (x - 3)/(log(5)^2 + 16*log(5) + 64) + 5/3*(log(5) + 11)*log(x - 3)/(log(5) ^2 + 16*log(5) + 64) + 5/3*(log(5) + 11)*log(x - 3)/(log(5) + 8) + 4*(log( 5) + 5)*log(x - 3)/(log(5)^2 + 16*log(5) + 64) + ((2*x - 1)*e^(2*x^2) - 5* log(x + log(5) + 5) - 5*log(x))/(x - 3) + 15*log(x + log(5) + 5)/(log(5)^3 + 21*log(5)^2 + 144*log(5) + 320) + 29*log(x + log(5) + 5)/(log(5)^2 + 16 *log(5) + 64) - 5*log(x + log(5) + 5)/(log(5) + 8) - 29*log(x - 3)/(log(5) ^2 + 16*log(5) + 64) - 5/3*log(x)/(log(5) + 5) - 5/3*log(x)
Leaf count of result is larger than twice the leaf count of optimal. 64 vs. \(2 (28) = 56\).
Time = 0.29 (sec) , antiderivative size = 64, normalized size of antiderivative = 2.06 \[ \int \frac {-15+29 x+4 x^2-4 x^3+\left (-3+7 x-2 x^2\right ) \log (5)+e^{2 x^2} \left (-25 x+55 x^2-128 x^3+12 x^4+8 x^5+\left (-5 x+12 x^2-28 x^3+8 x^4\right ) \log (5)\right )+\left (25 x+5 x^2+5 x \log (5)\right ) \log \left (5 x+x^2+x \log (5)\right )}{45 x-21 x^2-x^3+x^4+\left (9 x-6 x^2+x^3\right ) \log (5)} \, dx=\frac {2 \, x e^{\left (2 \, x^{2}\right )} - 2 \, x \log \left (x + \log \left (5\right ) + 5\right ) - 2 \, x \log \left (x\right ) - e^{\left (2 \, x^{2}\right )} - 5 \, \log \left (x^{2} + x \log \left (5\right ) + 5 \, x\right ) + 6 \, \log \left (x + \log \left (5\right ) + 5\right ) + 6 \, \log \left (x\right )}{x - 3} \]
integrate(((5*x*log(5)+5*x^2+25*x)*log(x*log(5)+x^2+5*x)+((8*x^4-28*x^3+12 *x^2-5*x)*log(5)+8*x^5+12*x^4-128*x^3+55*x^2-25*x)*exp(x^2)^2+(-2*x^2+7*x- 3)*log(5)-4*x^3+4*x^2+29*x-15)/((x^3-6*x^2+9*x)*log(5)+x^4-x^3-21*x^2+45*x ),x, algorithm=\
(2*x*e^(2*x^2) - 2*x*log(x + log(5) + 5) - 2*x*log(x) - e^(2*x^2) - 5*log( x^2 + x*log(5) + 5*x) + 6*log(x + log(5) + 5) + 6*log(x))/(x - 3)
Time = 8.91 (sec) , antiderivative size = 71, normalized size of antiderivative = 2.29 \[ \int \frac {-15+29 x+4 x^2-4 x^3+\left (-3+7 x-2 x^2\right ) \log (5)+e^{2 x^2} \left (-25 x+55 x^2-128 x^3+12 x^4+8 x^5+\left (-5 x+12 x^2-28 x^3+8 x^4\right ) \log (5)\right )+\left (25 x+5 x^2+5 x \log (5)\right ) \log \left (5 x+x^2+x \log (5)\right )}{45 x-21 x^2-x^3+x^4+\left (9 x-6 x^2+x^3\right ) \log (5)} \, dx=\ln \left (x+\ln \left (5\right )+5\right )\,\left (\frac {\ln \left (25\right )+10}{\ln \left (5\right )+5}-4\right )-\frac {5\,\ln \left (5\,x+x\,\ln \left (5\right )+x^2\right )}{x-3}-\frac {\ln \left (x\right )\,\left (\ln \left (25\right )+10\right )}{\ln \left (5\right )+5}+\frac {{\mathrm {e}}^{2\,x^2}\,\left (2\,x-1\right )}{x-3} \]
int(-(log(5)*(2*x^2 - 7*x + 3) - 29*x + exp(2*x^2)*(25*x + log(5)*(5*x - 1 2*x^2 + 28*x^3 - 8*x^4) - 55*x^2 + 128*x^3 - 12*x^4 - 8*x^5) - log(5*x + x *log(5) + x^2)*(25*x + 5*x*log(5) + 5*x^2) - 4*x^2 + 4*x^3 + 15)/(45*x + l og(5)*(9*x - 6*x^2 + x^3) - 21*x^2 - x^3 + x^4),x)