Integrand size = 275, antiderivative size = 32 \[ \int \frac {e^2 (-200-300 x)+48 x-104 x^2+40 x^3+24 x^4-12 x^5+e \left (-80-200 x-160 x^2+120 x^3\right )+\left (160+128 x+100 e^2 x+192 x^2-112 x^3-16 x^4+4 x^5+e \left (400+560 x+80 x^2-40 x^3\right )\right ) \log (x)+\left (-200-260 x-200 e x-80 x^2+40 x^3\right ) \log ^2(x)+100 x \log ^3(x)}{25 e^2 x^3+4 x^5-4 x^6+x^7+e \left (20 x^4-10 x^5\right )+\left (50 e^2 x^2-12 x^4+2 x^5+2 x^6+e \left (-10 x^3-20 x^4\right )\right ) \log (x)+\left (25 e^2 x-11 x^3+16 x^4+x^5+e \left (-80 x^2-10 x^3\right )\right ) \log ^2(x)+\left (-50 e x+30 x^2+10 x^3\right ) \log ^3(x)+25 x \log ^4(x)} \, dx=\frac {4 \left (2+x+\frac {2}{x-\frac {5 (-e+\log (x))}{2-x}}\right )}{x+\log (x)} \]
Time = 0.13 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.47 \[ \int \frac {e^2 (-200-300 x)+48 x-104 x^2+40 x^3+24 x^4-12 x^5+e \left (-80-200 x-160 x^2+120 x^3\right )+\left (160+128 x+100 e^2 x+192 x^2-112 x^3-16 x^4+4 x^5+e \left (400+560 x+80 x^2-40 x^3\right )\right ) \log (x)+\left (-200-260 x-200 e x-80 x^2+40 x^3\right ) \log ^2(x)+100 x \log ^3(x)}{25 e^2 x^3+4 x^5-4 x^6+x^7+e \left (20 x^4-10 x^5\right )+\left (50 e^2 x^2-12 x^4+2 x^5+2 x^6+e \left (-10 x^3-20 x^4\right )\right ) \log (x)+\left (25 e^2 x-11 x^3+16 x^4+x^5+e \left (-80 x^2-10 x^3\right )\right ) \log ^2(x)+\left (-50 e x+30 x^2+10 x^3\right ) \log ^3(x)+25 x \log ^4(x)} \, dx=\frac {4 \left (4+2 x-x^3+5 e (2+x)-5 (2+x) \log (x)\right )}{(5 e-(-2+x) x-5 \log (x)) (x+\log (x))} \]
Integrate[(E^2*(-200 - 300*x) + 48*x - 104*x^2 + 40*x^3 + 24*x^4 - 12*x^5 + E*(-80 - 200*x - 160*x^2 + 120*x^3) + (160 + 128*x + 100*E^2*x + 192*x^2 - 112*x^3 - 16*x^4 + 4*x^5 + E*(400 + 560*x + 80*x^2 - 40*x^3))*Log[x] + (-200 - 260*x - 200*E*x - 80*x^2 + 40*x^3)*Log[x]^2 + 100*x*Log[x]^3)/(25* E^2*x^3 + 4*x^5 - 4*x^6 + x^7 + E*(20*x^4 - 10*x^5) + (50*E^2*x^2 - 12*x^4 + 2*x^5 + 2*x^6 + E*(-10*x^3 - 20*x^4))*Log[x] + (25*E^2*x - 11*x^3 + 16* x^4 + x^5 + E*(-80*x^2 - 10*x^3))*Log[x]^2 + (-50*E*x + 30*x^2 + 10*x^3)*L og[x]^3 + 25*x*Log[x]^4),x]
(4*(4 + 2*x - x^3 + 5*E*(2 + x) - 5*(2 + x)*Log[x]))/((5*E - (-2 + x)*x - 5*Log[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.
| \(\displaystyle \int \frac {-12 x^5+24 x^4+40 x^3-104 x^2+e \left (120 x^3-160 x^2-200 x-80\right )+\left (40 x^3-80 x^2-200 e x-260 x-200\right ) \log ^2(x)+\left (4 x^5-16 x^4-112 x^3+192 x^2+e \left (-40 x^3+80 x^2+560 x+400\right )+100 e^2 x+128 x+160\right ) \log (x)+48 x+e^2 (-300 x-200)+100 x \log ^3(x)}{x^7-4 x^6+4 x^5+25 e^2 x^3+e \left (20 x^4-10 x^5\right )+\left (10 x^3+30 x^2-50 e x\right ) \log ^3(x)+\left (x^5+16 x^4-11 x^3+e \left (-10 x^3-80 x^2\right )+25 e^2 x\right ) \log ^2(x)+\left (2 x^6+2 x^5-12 x^4+50 e^2 x^2+e \left (-20 x^4-10 x^3\right )\right ) \log (x)+25 x \log ^4(x)} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {4 \left (10 e \left (3 x^3-4 x^2-5 x-2\right )-5 \left (-2 x^3+4 x^2+(13+10 e) x+10\right ) \log ^2(x)+x \left (-3 x^4+6 x^3+10 x^2-26 x+12\right )+\left (x^5-4 x^4-28 x^3+48 x^2-10 e \left (x^3-2 x^2-14 x-10\right )+25 e^2 x+32 x+40\right ) \log (x)-25 e^2 (3 x+2)+25 x \log ^3(x)\right )}{x (-((x-2) x)-5 \log (x)+5 e)^2 (x+\log (x))^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 4 \int -\frac {-25 x \log ^3(x)+5 \left (-2 x^3+4 x^2+(13+10 e) x+10\right ) \log ^2(x)-\left (x^5-4 x^4-28 x^3+48 x^2+25 e^2 x+32 x+10 e \left (-x^3+2 x^2+14 x+10\right )+40\right ) \log (x)+25 e^2 (3 x+2)+10 e \left (-3 x^3+4 x^2+5 x+2\right )-x \left (-3 x^4+6 x^3+10 x^2-26 x+12\right )}{x ((2-x) x-5 \log (x)+5 e)^2 (x+\log (x))^2}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -4 \int \frac {-25 x \log ^3(x)+5 \left (-2 x^3+4 x^2+(13+10 e) x+10\right ) \log ^2(x)-\left (x^5-4 x^4-28 x^3+48 x^2+25 e^2 x+32 x+10 e \left (-x^3+2 x^2+14 x+10\right )+40\right ) \log (x)+25 e^2 (3 x+2)+10 e \left (-3 x^3+4 x^2+5 x+2\right )-x \left (-3 x^4+6 x^3+10 x^2-26 x+12\right )}{x ((2-x) x-5 \log (x)+5 e)^2 (x+\log (x))^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -4 \int \left (\frac {10 \left (x^2-4 x+5 e+14\right )}{\left (-x^2+7 x+5 e\right )^2 \left (-x^2+2 x-5 \log (x)+5 e\right )}+\frac {-x^4+14 x^3-(47-10 e) x^2-2 (4+35 e) x-25 e^2+10 e+28}{\left (-x^2+7 x+5 e\right )^2 (x+\log (x))}+\frac {(x+1) \left (-x^3+5 x^2+(12+5 e) x+2 (2+5 e)\right )}{x \left (-x^2+7 x+5 e\right ) (x+\log (x))^2}-\frac {10 \left (2 x^3-6 x^2+9 x-10\right )}{x \left (x^2-7 x-5 e\right ) \left (x^2-2 x+5 \log (x)-5 e\right )^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -4 \left (-20 \int \frac {1}{\left (-x^2+2 x-5 \log (x)+5 e\right )^2}dx-\frac {20 (1-4 e) \left (1-\frac {7}{\sqrt {49+20 e}}\right ) \int \frac {1}{\left (-2 x-\sqrt {49+20 e}+7\right ) \left (-x^2+2 x-5 \log (x)+5 e\right )^2}dx}{e}-\frac {20 (1-4 e) \left (1+\frac {7}{\sqrt {49+20 e}}\right ) \int \frac {1}{\left (-2 x+\sqrt {49+20 e}+7\right ) \left (-x^2+2 x-5 \log (x)+5 e\right )^2}dx}{e}+\frac {20 (14+e (9+10 e)) \int \frac {1}{\left (-2 x+\sqrt {49+20 e}+7\right ) \left (-x^2+2 x-5 \log (x)+5 e\right )^2}dx}{e \sqrt {49+20 e}}-\frac {20 \int \frac {1}{x \left (-x^2+2 x-5 \log (x)+5 e\right )^2}dx}{e}+\frac {20 (14+e (9+10 e)) \int \frac {1}{\left (2 x+\sqrt {49+20 e}-7\right ) \left (-x^2+2 x-5 \log (x)+5 e\right )^2}dx}{e \sqrt {49+20 e}}-\frac {20 \int \frac {1}{\left (-2 x+\sqrt {49+20 e}+7\right ) \left (-x^2+2 x-5 \log (x)+5 e\right )}dx}{\sqrt {49+20 e}}-\frac {20 \int \frac {1}{\left (2 x+\sqrt {49+20 e}-7\right ) \left (-x^2+2 x-5 \log (x)+5 e\right )}dx}{\sqrt {49+20 e}}+20 (7+5 e) \int \frac {1}{\left (-x^2+7 x+5 e\right )^2 \left (-x^2+2 x-5 \log (x)+5 e\right )}dx+30 \int \frac {x}{\left (-x^2+7 x+5 e\right )^2 \left (-x^2+2 x-5 \log (x)+5 e\right )}dx+\int \frac {1}{-x-\log (x)}dx+3 \int \frac {1}{(x+\log (x))^2}dx+\frac {2 (2-5 e) \left (1-\frac {7}{\sqrt {49+20 e}}\right ) \int \frac {1}{\left (-2 x-\sqrt {49+20 e}+7\right ) (x+\log (x))^2}dx}{5 e}+\frac {2 (2-5 e) \left (1+\frac {7}{\sqrt {49+20 e}}\right ) \int \frac {1}{\left (-2 x+\sqrt {49+20 e}+7\right ) (x+\log (x))^2}dx}{5 e}-\frac {4 (14-5 e) \int \frac {1}{\left (-2 x+\sqrt {49+20 e}+7\right ) (x+\log (x))^2}dx}{5 e \sqrt {49+20 e}}+\frac {2 (2+5 e) \int \frac {1}{x (x+\log (x))^2}dx}{5 e}+\int \frac {x}{(x+\log (x))^2}dx-\frac {4 (14-5 e) \int \frac {1}{\left (2 x+\sqrt {49+20 e}-7\right ) (x+\log (x))^2}dx}{5 e \sqrt {49+20 e}}-\frac {4 \int \frac {1}{\left (-2 x+\sqrt {49+20 e}+7\right ) (x+\log (x))}dx}{\sqrt {49+20 e}}-\frac {4 \int \frac {1}{\left (2 x+\sqrt {49+20 e}-7\right ) (x+\log (x))}dx}{\sqrt {49+20 e}}+4 (7+5 e) \int \frac {1}{\left (-x^2+7 x+5 e\right )^2 (x+\log (x))}dx+6 \int \frac {x}{\left (-x^2+7 x+5 e\right )^2 (x+\log (x))}dx\right )\) |
Int[(E^2*(-200 - 300*x) + 48*x - 104*x^2 + 40*x^3 + 24*x^4 - 12*x^5 + E*(- 80 - 200*x - 160*x^2 + 120*x^3) + (160 + 128*x + 100*E^2*x + 192*x^2 - 112 *x^3 - 16*x^4 + 4*x^5 + E*(400 + 560*x + 80*x^2 - 40*x^3))*Log[x] + (-200 - 260*x - 200*E*x - 80*x^2 + 40*x^3)*Log[x]^2 + 100*x*Log[x]^3)/(25*E^2*x^ 3 + 4*x^5 - 4*x^6 + x^7 + E*(20*x^4 - 10*x^5) + (50*E^2*x^2 - 12*x^4 + 2*x ^5 + 2*x^6 + E*(-10*x^3 - 20*x^4))*Log[x] + (25*E^2*x - 11*x^3 + 16*x^4 + x^5 + E*(-80*x^2 - 10*x^3))*Log[x]^2 + (-50*E*x + 30*x^2 + 10*x^3)*Log[x]^ 3 + 25*x*Log[x]^4),x]
3.12.11.3.1 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]]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Leaf count of result is larger than twice the leaf count of optimal. \(71\) vs. \(2(33)=66\).
Time = 4.30 (sec) , antiderivative size = 72, normalized size of antiderivative = 2.25
| method | result | size |
| parallelrisch | \(\frac {16+8 x +40 \,{\mathrm e}+20 x \,{\mathrm e}-20 x \ln \left (x \right )-4 x^{3}-40 \ln \left (x \right )}{-x^{3}-x^{2} \ln \left (x \right )+5 x \,{\mathrm e}+5 \,{\mathrm e} \ln \left (x \right )+2 x^{2}-3 x \ln \left (x \right )-5 \ln \left (x \right )^{2}}\) | \(72\) |
| risch | \(\frac {16+8 x +40 \,{\mathrm e}+20 x \,{\mathrm e}-20 x \ln \left (x \right )-4 x^{3}-40 \ln \left (x \right )}{-x^{3}-x^{2} \ln \left (x \right )+5 x \,{\mathrm e}+5 \,{\mathrm e} \ln \left (x \right )+2 x^{2}-3 x \ln \left (x \right )-5 \ln \left (x \right )^{2}}\) | \(73\) |
| default | \(-\frac {4 \left (-x^{2} \ln \left (x \right )+5 \,{\mathrm e} \ln \left (x \right )-5 \ln \left (x \right )^{2}+2 x \ln \left (x \right )+2 x^{2}-10 \,{\mathrm e}+10 \ln \left (x \right )-2 x -4\right )}{-x^{3}-x^{2} \ln \left (x \right )+5 \,{\mathrm e}^{\ln \left (x \right )+1}+5 \,{\mathrm e} \ln \left (x \right )+2 x^{2}-3 x \ln \left (x \right )-5 \ln \left (x \right )^{2}}\) | \(89\) |
int((100*x*ln(x)^3+(-200*x*exp(1)+40*x^3-80*x^2-260*x-200)*ln(x)^2+(100*x* exp(1)^2+(-40*x^3+80*x^2+560*x+400)*exp(1)+4*x^5-16*x^4-112*x^3+192*x^2+12 8*x+160)*ln(x)+(-300*x-200)*exp(1)^2+(120*x^3-160*x^2-200*x-80)*exp(1)-12* x^5+24*x^4+40*x^3-104*x^2+48*x)/(25*x*ln(x)^4+(-50*x*exp(1)+10*x^3+30*x^2) *ln(x)^3+(25*x*exp(1)^2+(-10*x^3-80*x^2)*exp(1)+x^5+16*x^4-11*x^3)*ln(x)^2 +(50*x^2*exp(1)^2+(-20*x^4-10*x^3)*exp(1)+2*x^6+2*x^5-12*x^4)*ln(x)+25*x^3 *exp(1)^2+(-10*x^5+20*x^4)*exp(1)+x^7-4*x^6+4*x^5),x,method=_RETURNVERBOSE )
(16+8*x+40*exp(1)+20*x*exp(1)-20*x*ln(x)-4*x^3-40*ln(x))/(-x^3-x^2*ln(x)+5 *x*exp(1)+5*exp(1)*ln(x)+2*x^2-3*x*ln(x)-5*ln(x)^2)
Time = 0.27 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.88 \[ \int \frac {e^2 (-200-300 x)+48 x-104 x^2+40 x^3+24 x^4-12 x^5+e \left (-80-200 x-160 x^2+120 x^3\right )+\left (160+128 x+100 e^2 x+192 x^2-112 x^3-16 x^4+4 x^5+e \left (400+560 x+80 x^2-40 x^3\right )\right ) \log (x)+\left (-200-260 x-200 e x-80 x^2+40 x^3\right ) \log ^2(x)+100 x \log ^3(x)}{25 e^2 x^3+4 x^5-4 x^6+x^7+e \left (20 x^4-10 x^5\right )+\left (50 e^2 x^2-12 x^4+2 x^5+2 x^6+e \left (-10 x^3-20 x^4\right )\right ) \log (x)+\left (25 e^2 x-11 x^3+16 x^4+x^5+e \left (-80 x^2-10 x^3\right )\right ) \log ^2(x)+\left (-50 e x+30 x^2+10 x^3\right ) \log ^3(x)+25 x \log ^4(x)} \, dx=\frac {4 \, {\left (x^{3} - 5 \, {\left (x + 2\right )} e + 5 \, {\left (x + 2\right )} \log \left (x\right ) - 2 \, x - 4\right )}}{x^{3} - 2 \, x^{2} - 5 \, x e + {\left (x^{2} + 3 \, x - 5 \, e\right )} \log \left (x\right ) + 5 \, \log \left (x\right )^{2}} \]
integrate((100*x*log(x)^3+(-200*x*exp(1)+40*x^3-80*x^2-260*x-200)*log(x)^2 +(100*x*exp(1)^2+(-40*x^3+80*x^2+560*x+400)*exp(1)+4*x^5-16*x^4-112*x^3+19 2*x^2+128*x+160)*log(x)+(-300*x-200)*exp(1)^2+(120*x^3-160*x^2-200*x-80)*e xp(1)-12*x^5+24*x^4+40*x^3-104*x^2+48*x)/(25*x*log(x)^4+(-50*x*exp(1)+10*x ^3+30*x^2)*log(x)^3+(25*x*exp(1)^2+(-10*x^3-80*x^2)*exp(1)+x^5+16*x^4-11*x ^3)*log(x)^2+(50*x^2*exp(1)^2+(-20*x^4-10*x^3)*exp(1)+2*x^6+2*x^5-12*x^4)* log(x)+25*x^3*exp(1)^2+(-10*x^5+20*x^4)*exp(1)+x^7-4*x^6+4*x^5),x, algorit hm=\
4*(x^3 - 5*(x + 2)*e + 5*(x + 2)*log(x) - 2*x - 4)/(x^3 - 2*x^2 - 5*x*e + (x^2 + 3*x - 5*e)*log(x) + 5*log(x)^2)
Leaf count of result is larger than twice the leaf count of optimal. 66 vs. \(2 (26) = 52\).
Time = 0.16 (sec) , antiderivative size = 66, normalized size of antiderivative = 2.06 \[ \int \frac {e^2 (-200-300 x)+48 x-104 x^2+40 x^3+24 x^4-12 x^5+e \left (-80-200 x-160 x^2+120 x^3\right )+\left (160+128 x+100 e^2 x+192 x^2-112 x^3-16 x^4+4 x^5+e \left (400+560 x+80 x^2-40 x^3\right )\right ) \log (x)+\left (-200-260 x-200 e x-80 x^2+40 x^3\right ) \log ^2(x)+100 x \log ^3(x)}{25 e^2 x^3+4 x^5-4 x^6+x^7+e \left (20 x^4-10 x^5\right )+\left (50 e^2 x^2-12 x^4+2 x^5+2 x^6+e \left (-10 x^3-20 x^4\right )\right ) \log (x)+\left (25 e^2 x-11 x^3+16 x^4+x^5+e \left (-80 x^2-10 x^3\right )\right ) \log ^2(x)+\left (-50 e x+30 x^2+10 x^3\right ) \log ^3(x)+25 x \log ^4(x)} \, dx=\frac {4 x^{3} - 20 e x - 8 x + \left (20 x + 40\right ) \log {\left (x \right )} - 40 e - 16}{x^{3} - 2 x^{2} - 5 e x + \left (x^{2} + 3 x - 5 e\right ) \log {\left (x \right )} + 5 \log {\left (x \right )}^{2}} \]
integrate((100*x*ln(x)**3+(-200*x*exp(1)+40*x**3-80*x**2-260*x-200)*ln(x)* *2+(100*x*exp(1)**2+(-40*x**3+80*x**2+560*x+400)*exp(1)+4*x**5-16*x**4-112 *x**3+192*x**2+128*x+160)*ln(x)+(-300*x-200)*exp(1)**2+(120*x**3-160*x**2- 200*x-80)*exp(1)-12*x**5+24*x**4+40*x**3-104*x**2+48*x)/(25*x*ln(x)**4+(-5 0*x*exp(1)+10*x**3+30*x**2)*ln(x)**3+(25*x*exp(1)**2+(-10*x**3-80*x**2)*ex p(1)+x**5+16*x**4-11*x**3)*ln(x)**2+(50*x**2*exp(1)**2+(-20*x**4-10*x**3)* exp(1)+2*x**6+2*x**5-12*x**4)*ln(x)+25*x**3*exp(1)**2+(-10*x**5+20*x**4)*e xp(1)+x**7-4*x**6+4*x**5),x)
(4*x**3 - 20*E*x - 8*x + (20*x + 40)*log(x) - 40*E - 16)/(x**3 - 2*x**2 - 5*E*x + (x**2 + 3*x - 5*E)*log(x) + 5*log(x)**2)
Leaf count of result is larger than twice the leaf count of optimal. 63 vs. \(2 (31) = 62\).
Time = 0.26 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.97 \[ \int \frac {e^2 (-200-300 x)+48 x-104 x^2+40 x^3+24 x^4-12 x^5+e \left (-80-200 x-160 x^2+120 x^3\right )+\left (160+128 x+100 e^2 x+192 x^2-112 x^3-16 x^4+4 x^5+e \left (400+560 x+80 x^2-40 x^3\right )\right ) \log (x)+\left (-200-260 x-200 e x-80 x^2+40 x^3\right ) \log ^2(x)+100 x \log ^3(x)}{25 e^2 x^3+4 x^5-4 x^6+x^7+e \left (20 x^4-10 x^5\right )+\left (50 e^2 x^2-12 x^4+2 x^5+2 x^6+e \left (-10 x^3-20 x^4\right )\right ) \log (x)+\left (25 e^2 x-11 x^3+16 x^4+x^5+e \left (-80 x^2-10 x^3\right )\right ) \log ^2(x)+\left (-50 e x+30 x^2+10 x^3\right ) \log ^3(x)+25 x \log ^4(x)} \, dx=\frac {4 \, {\left (x^{3} - x {\left (5 \, e + 2\right )} + 5 \, {\left (x + 2\right )} \log \left (x\right ) - 10 \, e - 4\right )}}{x^{3} - 2 \, x^{2} - 5 \, x e + {\left (x^{2} + 3 \, x - 5 \, e\right )} \log \left (x\right ) + 5 \, \log \left (x\right )^{2}} \]
integrate((100*x*log(x)^3+(-200*x*exp(1)+40*x^3-80*x^2-260*x-200)*log(x)^2 +(100*x*exp(1)^2+(-40*x^3+80*x^2+560*x+400)*exp(1)+4*x^5-16*x^4-112*x^3+19 2*x^2+128*x+160)*log(x)+(-300*x-200)*exp(1)^2+(120*x^3-160*x^2-200*x-80)*e xp(1)-12*x^5+24*x^4+40*x^3-104*x^2+48*x)/(25*x*log(x)^4+(-50*x*exp(1)+10*x ^3+30*x^2)*log(x)^3+(25*x*exp(1)^2+(-10*x^3-80*x^2)*exp(1)+x^5+16*x^4-11*x ^3)*log(x)^2+(50*x^2*exp(1)^2+(-20*x^4-10*x^3)*exp(1)+2*x^6+2*x^5-12*x^4)* log(x)+25*x^3*exp(1)^2+(-10*x^5+20*x^4)*exp(1)+x^7-4*x^6+4*x^5),x, algorit hm=\
4*(x^3 - x*(5*e + 2) + 5*(x + 2)*log(x) - 10*e - 4)/(x^3 - 2*x^2 - 5*x*e + (x^2 + 3*x - 5*e)*log(x) + 5*log(x)^2)
Leaf count of result is larger than twice the leaf count of optimal. 67 vs. \(2 (31) = 62\).
Time = 4.16 (sec) , antiderivative size = 67, normalized size of antiderivative = 2.09 \[ \int \frac {e^2 (-200-300 x)+48 x-104 x^2+40 x^3+24 x^4-12 x^5+e \left (-80-200 x-160 x^2+120 x^3\right )+\left (160+128 x+100 e^2 x+192 x^2-112 x^3-16 x^4+4 x^5+e \left (400+560 x+80 x^2-40 x^3\right )\right ) \log (x)+\left (-200-260 x-200 e x-80 x^2+40 x^3\right ) \log ^2(x)+100 x \log ^3(x)}{25 e^2 x^3+4 x^5-4 x^6+x^7+e \left (20 x^4-10 x^5\right )+\left (50 e^2 x^2-12 x^4+2 x^5+2 x^6+e \left (-10 x^3-20 x^4\right )\right ) \log (x)+\left (25 e^2 x-11 x^3+16 x^4+x^5+e \left (-80 x^2-10 x^3\right )\right ) \log ^2(x)+\left (-50 e x+30 x^2+10 x^3\right ) \log ^3(x)+25 x \log ^4(x)} \, dx=\frac {4 \, {\left (x^{3} - 5 \, x e + 5 \, x \log \left (x\right ) - 2 \, x - 10 \, e + 10 \, \log \left (x\right ) - 4\right )}}{x^{3} + x^{2} \log \left (x\right ) - 2 \, x^{2} - 5 \, x e + 3 \, x \log \left (x\right ) - 5 \, e \log \left (x\right ) + 5 \, \log \left (x\right )^{2}} \]
integrate((100*x*log(x)^3+(-200*x*exp(1)+40*x^3-80*x^2-260*x-200)*log(x)^2 +(100*x*exp(1)^2+(-40*x^3+80*x^2+560*x+400)*exp(1)+4*x^5-16*x^4-112*x^3+19 2*x^2+128*x+160)*log(x)+(-300*x-200)*exp(1)^2+(120*x^3-160*x^2-200*x-80)*e xp(1)-12*x^5+24*x^4+40*x^3-104*x^2+48*x)/(25*x*log(x)^4+(-50*x*exp(1)+10*x ^3+30*x^2)*log(x)^3+(25*x*exp(1)^2+(-10*x^3-80*x^2)*exp(1)+x^5+16*x^4-11*x ^3)*log(x)^2+(50*x^2*exp(1)^2+(-20*x^4-10*x^3)*exp(1)+2*x^6+2*x^5-12*x^4)* log(x)+25*x^3*exp(1)^2+(-10*x^5+20*x^4)*exp(1)+x^7-4*x^6+4*x^5),x, algorit hm=\
4*(x^3 - 5*x*e + 5*x*log(x) - 2*x - 10*e + 10*log(x) - 4)/(x^3 + x^2*log(x ) - 2*x^2 - 5*x*e + 3*x*log(x) - 5*e*log(x) + 5*log(x)^2)
Timed out. \[ \int \frac {e^2 (-200-300 x)+48 x-104 x^2+40 x^3+24 x^4-12 x^5+e \left (-80-200 x-160 x^2+120 x^3\right )+\left (160+128 x+100 e^2 x+192 x^2-112 x^3-16 x^4+4 x^5+e \left (400+560 x+80 x^2-40 x^3\right )\right ) \log (x)+\left (-200-260 x-200 e x-80 x^2+40 x^3\right ) \log ^2(x)+100 x \log ^3(x)}{25 e^2 x^3+4 x^5-4 x^6+x^7+e \left (20 x^4-10 x^5\right )+\left (50 e^2 x^2-12 x^4+2 x^5+2 x^6+e \left (-10 x^3-20 x^4\right )\right ) \log (x)+\left (25 e^2 x-11 x^3+16 x^4+x^5+e \left (-80 x^2-10 x^3\right )\right ) \log ^2(x)+\left (-50 e x+30 x^2+10 x^3\right ) \log ^3(x)+25 x \log ^4(x)} \, dx=\int \frac {48\,x+100\,x\,{\ln \left (x\right )}^3-\mathrm {e}\,\left (-120\,x^3+160\,x^2+200\,x+80\right )-{\ln \left (x\right )}^2\,\left (260\,x+200\,x\,\mathrm {e}+80\,x^2-40\,x^3+200\right )-104\,x^2+40\,x^3+24\,x^4-12\,x^5-{\mathrm {e}}^2\,\left (300\,x+200\right )+\ln \left (x\right )\,\left (128\,x+100\,x\,{\mathrm {e}}^2+\mathrm {e}\,\left (-40\,x^3+80\,x^2+560\,x+400\right )+192\,x^2-112\,x^3-16\,x^4+4\,x^5+160\right )}{{\ln \left (x\right )}^2\,\left (25\,x\,{\mathrm {e}}^2-\mathrm {e}\,\left (10\,x^3+80\,x^2\right )-11\,x^3+16\,x^4+x^5\right )+25\,x\,{\ln \left (x\right )}^4+\ln \left (x\right )\,\left (50\,x^2\,{\mathrm {e}}^2-\mathrm {e}\,\left (20\,x^4+10\,x^3\right )-12\,x^4+2\,x^5+2\,x^6\right )+\mathrm {e}\,\left (20\,x^4-10\,x^5\right )+25\,x^3\,{\mathrm {e}}^2+{\ln \left (x\right )}^3\,\left (10\,x^3+30\,x^2-50\,\mathrm {e}\,x\right )+4\,x^5-4\,x^6+x^7} \,d x \]
int((48*x + 100*x*log(x)^3 - exp(1)*(200*x + 160*x^2 - 120*x^3 + 80) - log (x)^2*(260*x + 200*x*exp(1) + 80*x^2 - 40*x^3 + 200) - 104*x^2 + 40*x^3 + 24*x^4 - 12*x^5 - exp(2)*(300*x + 200) + log(x)*(128*x + 100*x*exp(2) + ex p(1)*(560*x + 80*x^2 - 40*x^3 + 400) + 192*x^2 - 112*x^3 - 16*x^4 + 4*x^5 + 160))/(log(x)^2*(25*x*exp(2) - exp(1)*(80*x^2 + 10*x^3) - 11*x^3 + 16*x^ 4 + x^5) + 25*x*log(x)^4 + log(x)*(50*x^2*exp(2) - exp(1)*(10*x^3 + 20*x^4 ) - 12*x^4 + 2*x^5 + 2*x^6) + exp(1)*(20*x^4 - 10*x^5) + 25*x^3*exp(2) + l og(x)^3*(30*x^2 - 50*x*exp(1) + 10*x^3) + 4*x^5 - 4*x^6 + x^7),x)
int((48*x + 100*x*log(x)^3 - exp(1)*(200*x + 160*x^2 - 120*x^3 + 80) - log (x)^2*(260*x + 200*x*exp(1) + 80*x^2 - 40*x^3 + 200) - 104*x^2 + 40*x^3 + 24*x^4 - 12*x^5 - exp(2)*(300*x + 200) + log(x)*(128*x + 100*x*exp(2) + ex p(1)*(560*x + 80*x^2 - 40*x^3 + 400) + 192*x^2 - 112*x^3 - 16*x^4 + 4*x^5 + 160))/(log(x)^2*(25*x*exp(2) - exp(1)*(80*x^2 + 10*x^3) - 11*x^3 + 16*x^ 4 + x^5) + 25*x*log(x)^4 + log(x)*(50*x^2*exp(2) - exp(1)*(10*x^3 + 20*x^4 ) - 12*x^4 + 2*x^5 + 2*x^6) + exp(1)*(20*x^4 - 10*x^5) + 25*x^3*exp(2) + l og(x)^3*(30*x^2 - 50*x*exp(1) + 10*x^3) + 4*x^5 - 4*x^6 + x^7), x)