Integrand size = 150, antiderivative size = 27 \[ \int \frac {1800 x^7+2880 x^8+1152 x^9+\left (1200 x^7+960 x^8\right ) \log (5)+200 x^7 \log ^2(5)+e^{e^{\frac {-3+15 x+12 x^2+5 x \log (5)}{15+12 x+5 \log (5)}}+\frac {-3+15 x+12 x^2+5 x \log (5)}{15+12 x+5 \log (5)}} \left (-783-1080 x-432 x^2+(-450-360 x) \log (5)-75 \log ^2(5)\right )}{225+360 x+144 x^2+(150+120 x) \log (5)+25 \log ^2(5)} \, dx=-3 e^{e^{x+\frac {1}{x-5 \left (1+x+\frac {\log (5)}{3}\right )}}}+x^8 \]
Time = 0.42 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.93 \[ \int \frac {1800 x^7+2880 x^8+1152 x^9+\left (1200 x^7+960 x^8\right ) \log (5)+200 x^7 \log ^2(5)+e^{e^{\frac {-3+15 x+12 x^2+5 x \log (5)}{15+12 x+5 \log (5)}}+\frac {-3+15 x+12 x^2+5 x \log (5)}{15+12 x+5 \log (5)}} \left (-783-1080 x-432 x^2+(-450-360 x) \log (5)-75 \log ^2(5)\right )}{225+360 x+144 x^2+(150+120 x) \log (5)+25 \log ^2(5)} \, dx=-3 e^{e^{x-\frac {3}{15+12 x+5 \log (5)}}}+x^8 \]
Integrate[(1800*x^7 + 2880*x^8 + 1152*x^9 + (1200*x^7 + 960*x^8)*Log[5] + 200*x^7*Log[5]^2 + E^(E^((-3 + 15*x + 12*x^2 + 5*x*Log[5])/(15 + 12*x + 5* Log[5])) + (-3 + 15*x + 12*x^2 + 5*x*Log[5])/(15 + 12*x + 5*Log[5]))*(-783 - 1080*x - 432*x^2 + (-450 - 360*x)*Log[5] - 75*Log[5]^2))/(225 + 360*x + 144*x^2 + (150 + 120*x)*Log[5] + 25*Log[5]^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.
| \(\displaystyle \int \frac {\left (-432 x^2-1080 x+(-360 x-450) \log (5)-783-75 \log ^2(5)\right ) \exp \left (\exp \left (\frac {12 x^2+15 x+5 x \log (5)-3}{12 x+15+5 \log (5)}\right )+\frac {12 x^2+15 x+5 x \log (5)-3}{12 x+15+5 \log (5)}\right )+1152 x^9+2880 x^8+1800 x^7+200 x^7 \log ^2(5)+\left (960 x^8+1200 x^7\right ) \log (5)}{144 x^2+360 x+(120 x+150) \log (5)+225+25 \log ^2(5)} \, dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {\left (-432 x^2-1080 x+(-360 x-450) \log (5)-783-75 \log ^2(5)\right ) \exp \left (\exp \left (\frac {12 x^2+15 x+5 x \log (5)-3}{12 x+15+5 \log (5)}\right )+\frac {12 x^2+15 x+5 x \log (5)-3}{12 x+15+5 \log (5)}\right )+1152 x^9+2880 x^8+x^7 \left (1800+200 \log ^2(5)\right )+\left (960 x^8+1200 x^7\right ) \log (5)}{144 x^2+360 x+(120 x+150) \log (5)+225+25 \log ^2(5)}dx\) |
| \(\Big \downarrow \) 2007 |
| \(\displaystyle \int \frac {\left (-432 x^2-1080 x+(-360 x-450) \log (5)-783-75 \log ^2(5)\right ) \exp \left (\exp \left (\frac {12 x^2+15 x+5 x \log (5)-3}{12 x+15+5 \log (5)}\right )+\frac {12 x^2+15 x+5 x \log (5)-3}{12 x+15+5 \log (5)}\right )+1152 x^9+2880 x^8+x^7 \left (1800+200 \log ^2(5)\right )+\left (960 x^8+1200 x^7\right ) \log (5)}{(12 x+5 (3+\log (5)))^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {3\ 5^{\frac {5 x}{12 x+5 (3+\log (5))}} \left (-144 x^2-120 x (3+\log (5))-261-25 \log ^2(5)-150 \log (5)\right ) \exp \left (5^{\frac {5 x}{12 x+5 (3+\log (5))}} \exp \left (\frac {12 x^2}{12 x+5 (3+\log (5))}+\frac {15 x}{12 x+5 (3+\log (5))}-\frac {3}{12 x+5 (3+\log (5))}\right )+\frac {12 x^2}{12 x+5 (3+\log (5))}+\frac {15 x}{12 x+5 (3+\log (5))}-\frac {3}{12 x+5 (3+\log (5))}\right )}{(12 x+15+5 \log (5))^2}+\frac {1152 x^9}{(12 x+15+5 \log (5))^2}+\frac {2880 x^8}{(12 x+15+5 \log (5))^2}+\frac {200 x^7 \left (9+\log ^2(5)\right )}{(12 x+15+5 \log (5))^2}+\frac {240 (4 x+5) x^7 \log (5)}{(12 x+15+5 \log (5))^2}\right )dx\) |
| \(\Big \downarrow \) 7299 |
| \(\displaystyle \int \left (\frac {3\ 5^{\frac {5 x}{12 x+5 (3+\log (5))}} \left (-144 x^2-120 x (3+\log (5))-261-25 \log ^2(5)-150 \log (5)\right ) \exp \left (5^{\frac {5 x}{12 x+5 (3+\log (5))}} \exp \left (\frac {12 x^2}{12 x+5 (3+\log (5))}+\frac {15 x}{12 x+5 (3+\log (5))}-\frac {3}{12 x+5 (3+\log (5))}\right )+\frac {12 x^2}{12 x+5 (3+\log (5))}+\frac {15 x}{12 x+5 (3+\log (5))}-\frac {3}{12 x+5 (3+\log (5))}\right )}{(12 x+15+5 \log (5))^2}+\frac {1152 x^9}{(12 x+15+5 \log (5))^2}+\frac {2880 x^8}{(12 x+15+5 \log (5))^2}+\frac {200 x^7 \left (9+\log ^2(5)\right )}{(12 x+15+5 \log (5))^2}+\frac {240 (4 x+5) x^7 \log (5)}{(12 x+15+5 \log (5))^2}\right )dx\) |
Int[(1800*x^7 + 2880*x^8 + 1152*x^9 + (1200*x^7 + 960*x^8)*Log[5] + 200*x^ 7*Log[5]^2 + E^(E^((-3 + 15*x + 12*x^2 + 5*x*Log[5])/(15 + 12*x + 5*Log[5] )) + (-3 + 15*x + 12*x^2 + 5*x*Log[5])/(15 + 12*x + 5*Log[5]))*(-783 - 108 0*x - 432*x^2 + (-450 - 360*x)*Log[5] - 75*Log[5]^2))/(225 + 360*x + 144*x ^2 + (150 + 120*x)*Log[5] + 25*Log[5]^2),x]
3.3.47.3.1 Defintions of rubi rules used
Int[(u_.)*((v_.) + (a_.)*(Fx_) + (b_.)*(Fx_))^(p_.), x_Symbol] :> Int[u*(v + (a + b)*Fx)^p, x] /; FreeQ[{a, b}, x] && !FreeQ[Fx, x]
Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{a = Rt[Coeff[Px, x, 0], Expon[Px, x]], b = Rt[Coeff[Px, x, Expon[Px, x]], Expon[Px, x]]}, Int[u*(a + b*x)^(Ex pon[Px, x]*p), x] /; EqQ[Px, (a + b*x)^Expon[Px, x]]] /; IntegerQ[p] && Pol yQ[Px, x] && GtQ[Expon[Px, x], 1] && NeQ[Coeff[Px, x, 0], 0]
Time = 5.49 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.33
| method | result | size |
| risch | \(x^{8}-3 \,{\mathrm e}^{{\mathrm e}^{\frac {5 x \ln \left (5\right )+12 x^{2}+15 x -3}{5 \ln \left (5\right )+12 x +15}}}\) | \(36\) |
| parallelrisch | \(x^{8}-3 \,{\mathrm e}^{{\mathrm e}^{\frac {5 x \ln \left (5\right )+12 x^{2}+15 x -3}{5 \ln \left (5\right )+12 x +15}}}\) | \(36\) |
| parts | \(x^{8}+\frac {\left (-45-15 \ln \left (5\right )\right ) {\mathrm e}^{{\mathrm e}^{\frac {5 x \ln \left (5\right )+12 x^{2}+15 x -3}{5 \ln \left (5\right )+12 x +15}}}-36 x \,{\mathrm e}^{{\mathrm e}^{\frac {5 x \ln \left (5\right )+12 x^{2}+15 x -3}{5 \ln \left (5\right )+12 x +15}}}}{5 \ln \left (5\right )+12 x +15}\) | \(86\) |
int(((-75*ln(5)^2+(-360*x-450)*ln(5)-432*x^2-1080*x-783)*exp((5*x*ln(5)+12 *x^2+15*x-3)/(5*ln(5)+12*x+15))*exp(exp((5*x*ln(5)+12*x^2+15*x-3)/(5*ln(5) +12*x+15)))+200*x^7*ln(5)^2+(960*x^8+1200*x^7)*ln(5)+1152*x^9+2880*x^8+180 0*x^7)/(25*ln(5)^2+(120*x+150)*ln(5)+144*x^2+360*x+225),x,method=_RETURNVE RBOSE)
Leaf count of result is larger than twice the leaf count of optimal. 131 vs. \(2 (23) = 46\).
Time = 0.28 (sec) , antiderivative size = 131, normalized size of antiderivative = 4.85 \[ \int \frac {1800 x^7+2880 x^8+1152 x^9+\left (1200 x^7+960 x^8\right ) \log (5)+200 x^7 \log ^2(5)+e^{e^{\frac {-3+15 x+12 x^2+5 x \log (5)}{15+12 x+5 \log (5)}}+\frac {-3+15 x+12 x^2+5 x \log (5)}{15+12 x+5 \log (5)}} \left (-783-1080 x-432 x^2+(-450-360 x) \log (5)-75 \log ^2(5)\right )}{225+360 x+144 x^2+(150+120 x) \log (5)+25 \log ^2(5)} \, dx={\left (x^{8} e^{\left (\frac {12 \, x^{2} + 5 \, x \log \left (5\right ) + 15 \, x - 3}{12 \, x + 5 \, \log \left (5\right ) + 15}\right )} - 3 \, e^{\left (\frac {12 \, x^{2} + {\left (12 \, x + 5 \, \log \left (5\right ) + 15\right )} e^{\left (\frac {12 \, x^{2} + 5 \, x \log \left (5\right ) + 15 \, x - 3}{12 \, x + 5 \, \log \left (5\right ) + 15}\right )} + 5 \, x \log \left (5\right ) + 15 \, x - 3}{12 \, x + 5 \, \log \left (5\right ) + 15}\right )}\right )} e^{\left (-\frac {12 \, x^{2} + 5 \, x \log \left (5\right ) + 15 \, x - 3}{12 \, x + 5 \, \log \left (5\right ) + 15}\right )} \]
integrate(((-75*log(5)^2+(-360*x-450)*log(5)-432*x^2-1080*x-783)*exp((5*x* log(5)+12*x^2+15*x-3)/(5*log(5)+12*x+15))*exp(exp((5*x*log(5)+12*x^2+15*x- 3)/(5*log(5)+12*x+15)))+200*x^7*log(5)^2+(960*x^8+1200*x^7)*log(5)+1152*x^ 9+2880*x^8+1800*x^7)/(25*log(5)^2+(120*x+150)*log(5)+144*x^2+360*x+225),x, algorithm=\
(x^8*e^((12*x^2 + 5*x*log(5) + 15*x - 3)/(12*x + 5*log(5) + 15)) - 3*e^((1 2*x^2 + (12*x + 5*log(5) + 15)*e^((12*x^2 + 5*x*log(5) + 15*x - 3)/(12*x + 5*log(5) + 15)) + 5*x*log(5) + 15*x - 3)/(12*x + 5*log(5) + 15)))*e^(-(12 *x^2 + 5*x*log(5) + 15*x - 3)/(12*x + 5*log(5) + 15))
Time = 0.41 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.26 \[ \int \frac {1800 x^7+2880 x^8+1152 x^9+\left (1200 x^7+960 x^8\right ) \log (5)+200 x^7 \log ^2(5)+e^{e^{\frac {-3+15 x+12 x^2+5 x \log (5)}{15+12 x+5 \log (5)}}+\frac {-3+15 x+12 x^2+5 x \log (5)}{15+12 x+5 \log (5)}} \left (-783-1080 x-432 x^2+(-450-360 x) \log (5)-75 \log ^2(5)\right )}{225+360 x+144 x^2+(150+120 x) \log (5)+25 \log ^2(5)} \, dx=x^{8} - 3 e^{e^{\frac {12 x^{2} + 5 x \log {\left (5 \right )} + 15 x - 3}{12 x + 5 \log {\left (5 \right )} + 15}}} \]
integrate(((-75*ln(5)**2+(-360*x-450)*ln(5)-432*x**2-1080*x-783)*exp((5*x* ln(5)+12*x**2+15*x-3)/(5*ln(5)+12*x+15))*exp(exp((5*x*ln(5)+12*x**2+15*x-3 )/(5*ln(5)+12*x+15)))+200*x**7*ln(5)**2+(960*x**8+1200*x**7)*ln(5)+1152*x* *9+2880*x**8+1800*x**7)/(25*ln(5)**2+(120*x+150)*ln(5)+144*x**2+360*x+225) ,x)
\[ \int \frac {1800 x^7+2880 x^8+1152 x^9+\left (1200 x^7+960 x^8\right ) \log (5)+200 x^7 \log ^2(5)+e^{e^{\frac {-3+15 x+12 x^2+5 x \log (5)}{15+12 x+5 \log (5)}}+\frac {-3+15 x+12 x^2+5 x \log (5)}{15+12 x+5 \log (5)}} \left (-783-1080 x-432 x^2+(-450-360 x) \log (5)-75 \log ^2(5)\right )}{225+360 x+144 x^2+(150+120 x) \log (5)+25 \log ^2(5)} \, dx=\int { \frac {1152 \, x^{9} + 200 \, x^{7} \log \left (5\right )^{2} + 2880 \, x^{8} + 1800 \, x^{7} - 3 \, {\left (144 \, x^{2} + 30 \, {\left (4 \, x + 5\right )} \log \left (5\right ) + 25 \, \log \left (5\right )^{2} + 360 \, x + 261\right )} e^{\left (\frac {12 \, x^{2} + 5 \, x \log \left (5\right ) + 15 \, x - 3}{12 \, x + 5 \, \log \left (5\right ) + 15} + e^{\left (\frac {12 \, x^{2} + 5 \, x \log \left (5\right ) + 15 \, x - 3}{12 \, x + 5 \, \log \left (5\right ) + 15}\right )}\right )} + 240 \, {\left (4 \, x^{8} + 5 \, x^{7}\right )} \log \left (5\right )}{144 \, x^{2} + 30 \, {\left (4 \, x + 5\right )} \log \left (5\right ) + 25 \, \log \left (5\right )^{2} + 360 \, x + 225} \,d x } \]
integrate(((-75*log(5)^2+(-360*x-450)*log(5)-432*x^2-1080*x-783)*exp((5*x* log(5)+12*x^2+15*x-3)/(5*log(5)+12*x+15))*exp(exp((5*x*log(5)+12*x^2+15*x- 3)/(5*log(5)+12*x+15)))+200*x^7*log(5)^2+(960*x^8+1200*x^7)*log(5)+1152*x^ 9+2880*x^8+1800*x^7)/(25*log(5)^2+(120*x+150)*log(5)+144*x^2+360*x+225),x, algorithm=\
x^8 - 20/21*x^7*(log(5) + 3) + 25/36*(log(5)^2 + 6*log(5) + 9)*x^6 + 20/7* x^7 - 25/9*x^6*(log(5) + 3) - 25/54*(log(5)^3 + 9*log(5)^2 + 27*log(5) + 2 7)*x^5 + 25/12*(log(5)^2 + 6*log(5) + 9)*x^5 + 25/12*x^6 - 25/12*x^5*(log( 5) + 3) + 3125/10368*(log(5)^4 + 12*log(5)^3 + 54*log(5)^2 + 108*log(5) + 81)*x^4 - 625/432*(log(5)^3 + 9*log(5)^2 + 27*log(5) + 27)*x^4 + 625/384*( log(5)^2 + 6*log(5) + 9)*x^4 - 3125/15552*(log(5)^5 + 15*log(5)^4 + 90*log (5)^3 + 270*log(5)^2 + 405*log(5) + 243)*x^3 + 15625/15552*(log(5)^4 + 12* log(5)^3 + 54*log(5)^2 + 108*log(5) + 81)*x^3 - 3125/2592*(log(5)^3 + 9*lo g(5)^2 + 27*log(5) + 27)*x^3 + 109375/746496*(log(5)^6 + 18*log(5)^5 + 135 *log(5)^4 + 540*log(5)^3 + 1215*log(5)^2 + 1458*log(5) + 729)*x^2 - 15625/ 20736*(log(5)^5 + 15*log(5)^4 + 90*log(5)^3 + 270*log(5)^2 + 405*log(5) + 243)*x^2 + 78125/82944*(log(5)^4 + 12*log(5)^3 + 54*log(5)^2 + 108*log(5) + 81)*x^2 + 25/53747712*(497664*x^6 - 497664*x^5*(log(5) + 3) + 388800*(lo g(5)^2 + 6*log(5) + 9)*x^4 - 288000*(log(5)^3 + 9*log(5)^2 + 27*log(5) + 2 7)*x^3 + 225000*(log(5)^4 + 12*log(5)^3 + 54*log(5)^2 + 108*log(5) + 81)*x ^2 - 225000*(log(5)^5 + 15*log(5)^4 + 90*log(5)^3 + 270*log(5)^2 + 405*log (5) + 243)*x + 109375*(log(5)^6 + 18*log(5)^5 + 135*log(5)^4 + 540*log(5)^ 3 + 1215*log(5)^2 + 1458*log(5) + 729)*log(12*x + 5*log(5) + 15) + 78125*( log(5)^7 + 21*log(5)^6 + 189*log(5)^5 + 945*log(5)^4 + 2835*log(5)^3 + 510 3*log(5)^2 + 5103*log(5) + 2187)/(12*x + 5*log(5) + 15))*log(5)^2 - 781...
\[ \int \frac {1800 x^7+2880 x^8+1152 x^9+\left (1200 x^7+960 x^8\right ) \log (5)+200 x^7 \log ^2(5)+e^{e^{\frac {-3+15 x+12 x^2+5 x \log (5)}{15+12 x+5 \log (5)}}+\frac {-3+15 x+12 x^2+5 x \log (5)}{15+12 x+5 \log (5)}} \left (-783-1080 x-432 x^2+(-450-360 x) \log (5)-75 \log ^2(5)\right )}{225+360 x+144 x^2+(150+120 x) \log (5)+25 \log ^2(5)} \, dx=\int { \frac {1152 \, x^{9} + 200 \, x^{7} \log \left (5\right )^{2} + 2880 \, x^{8} + 1800 \, x^{7} - 3 \, {\left (144 \, x^{2} + 30 \, {\left (4 \, x + 5\right )} \log \left (5\right ) + 25 \, \log \left (5\right )^{2} + 360 \, x + 261\right )} e^{\left (\frac {12 \, x^{2} + 5 \, x \log \left (5\right ) + 15 \, x - 3}{12 \, x + 5 \, \log \left (5\right ) + 15} + e^{\left (\frac {12 \, x^{2} + 5 \, x \log \left (5\right ) + 15 \, x - 3}{12 \, x + 5 \, \log \left (5\right ) + 15}\right )}\right )} + 240 \, {\left (4 \, x^{8} + 5 \, x^{7}\right )} \log \left (5\right )}{144 \, x^{2} + 30 \, {\left (4 \, x + 5\right )} \log \left (5\right ) + 25 \, \log \left (5\right )^{2} + 360 \, x + 225} \,d x } \]
integrate(((-75*log(5)^2+(-360*x-450)*log(5)-432*x^2-1080*x-783)*exp((5*x* log(5)+12*x^2+15*x-3)/(5*log(5)+12*x+15))*exp(exp((5*x*log(5)+12*x^2+15*x- 3)/(5*log(5)+12*x+15)))+200*x^7*log(5)^2+(960*x^8+1200*x^7)*log(5)+1152*x^ 9+2880*x^8+1800*x^7)/(25*log(5)^2+(120*x+150)*log(5)+144*x^2+360*x+225),x, algorithm=\
integrate((1152*x^9 + 200*x^7*log(5)^2 + 2880*x^8 + 1800*x^7 - 3*(144*x^2 + 30*(4*x + 5)*log(5) + 25*log(5)^2 + 360*x + 261)*e^((12*x^2 + 5*x*log(5) + 15*x - 3)/(12*x + 5*log(5) + 15) + e^((12*x^2 + 5*x*log(5) + 15*x - 3)/ (12*x + 5*log(5) + 15))) + 240*(4*x^8 + 5*x^7)*log(5))/(144*x^2 + 30*(4*x + 5)*log(5) + 25*log(5)^2 + 360*x + 225), x)
Time = 12.85 (sec) , antiderivative size = 70, normalized size of antiderivative = 2.59 \[ \int \frac {1800 x^7+2880 x^8+1152 x^9+\left (1200 x^7+960 x^8\right ) \log (5)+200 x^7 \log ^2(5)+e^{e^{\frac {-3+15 x+12 x^2+5 x \log (5)}{15+12 x+5 \log (5)}}+\frac {-3+15 x+12 x^2+5 x \log (5)}{15+12 x+5 \log (5)}} \left (-783-1080 x-432 x^2+(-450-360 x) \log (5)-75 \log ^2(5)\right )}{225+360 x+144 x^2+(150+120 x) \log (5)+25 \log ^2(5)} \, dx=x^8-3\,{\mathrm {e}}^{5^{\frac {5\,x}{12\,x+5\,\ln \left (5\right )+15}}\,{\mathrm {e}}^{-\frac {3}{12\,x+5\,\ln \left (5\right )+15}}\,{\mathrm {e}}^{\frac {15\,x}{12\,x+5\,\ln \left (5\right )+15}}\,{\mathrm {e}}^{\frac {12\,x^2}{12\,x+5\,\ln \left (5\right )+15}}} \]
int((200*x^7*log(5)^2 + log(5)*(1200*x^7 + 960*x^8) + 1800*x^7 + 2880*x^8 + 1152*x^9 - exp(exp((15*x + 5*x*log(5) + 12*x^2 - 3)/(12*x + 5*log(5) + 1 5)))*exp((15*x + 5*x*log(5) + 12*x^2 - 3)/(12*x + 5*log(5) + 15))*(1080*x + log(5)*(360*x + 450) + 75*log(5)^2 + 432*x^2 + 783))/(360*x + log(5)*(12 0*x + 150) + 25*log(5)^2 + 144*x^2 + 225),x)