\(\int \frac {1800 x^7+2880 x^8+1152 x^9+(1200 x^7+960 x^8) \log (5)+200 x^7 \log ^2(5)+e^{e^{\frac {-3+15 x+12 x^2+5 x \log (5)}{15+12 x+5 \log (5)}}+\frac {-3+15 x+12 x^2+5 x \log (5)}{15+12 x+5 \log (5)}} (-783-1080 x-432 x^2+(-450-360 x) \log (5)-75 \log ^2(5))}{225+360 x+144 x^2+(150+120 x) \log (5)+25 \log ^2(5)} \, dx\) [6247]

Optimal result

Integrand size = 150, antiderivative size = 27 \[ \int \frac {1800 x^7+2880 x^8+1152 x^9+\left (1200 x^7+960 x^8\right ) \log (5)+200 x^7 \log ^2(5)+e^{e^{\frac {-3+15 x+12 x^2+5 x \log (5)}{15+12 x+5 \log (5)}}+\frac {-3+15 x+12 x^2+5 x \log (5)}{15+12 x+5 \log (5)}} \left (-783-1080 x-432 x^2+(-450-360 x) \log (5)-75 \log ^2(5)\right )}{225+360 x+144 x^2+(150+120 x) \log (5)+25 \log ^2(5)} \, dx=-3 e^{e^{x+\frac {1}{x-5 \left (1+x+\frac {\log (5)}{3}\right )}}}+x^8 \]

[Out]

x^8-3*exp(exp(x+1/(-4*x-5-5/3*ln(5))))

Rubi [F]

\[ \int \frac {1800 x^7+2880 x^8+1152 x^9+\left (1200 x^7+960 x^8\right ) \log (5)+200 x^7 \log ^2(5)+e^{e^{\frac {-3+15 x+12 x^2+5 x \log (5)}{15+12 x+5 \log (5)}}+\frac {-3+15 x+12 x^2+5 x \log (5)}{15+12 x+5 \log (5)}} \left (-783-1080 x-432 x^2+(-450-360 x) \log (5)-75 \log ^2(5)\right )}{225+360 x+144 x^2+(150+120 x) \log (5)+25 \log ^2(5)} \, dx=\int \frac {1800 x^7+2880 x^8+1152 x^9+\left (1200 x^7+960 x^8\right ) \log (5)+200 x^7 \log ^2(5)+\exp \left (\exp \left (\frac {-3+15 x+12 x^2+5 x \log (5)}{15+12 x+5 \log (5)}\right )+\frac {-3+15 x+12 x^2+5 x \log (5)}{15+12 x+5 \log (5)}\right ) \left (-783-1080 x-432 x^2+(-450-360 x) \log (5)-75 \log ^2(5)\right )}{225+360 x+144 x^2+(150+120 x) \log (5)+25 \log ^2(5)} \, dx \]

[In]

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 - 1
080*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]

[Out]

(20*x^7)/7 + x^8 + (20*x^7*Log[5])/21 - (25*x^6*(3 + Log[5]))/9 - (20*x^7*(3 + Log[5]))/21 + (25*x^5*(3 + Log[
5])^2)/12 + (25*x^6*(3 + Log[5])^2)/36 - (625*x^4*(3 + Log[5])^3)/432 - (25*x^5*(3 + Log[5])^3)/54 + (15625*x^
3*(3 + Log[5])^4)/15552 + (3125*x^4*(3 + Log[5])^4)/10368 - (15625*x^2*(3 + Log[5])^5)/20736 - (3125*x^3*(3 +
Log[5])^5)/15552 + (546875*x*(3 + Log[5])^6)/746496 + (109375*x^2*(3 + Log[5])^6)/746496 - (78125*x*(3 + Log[5
])^7)/559872 + (3125*x^3*Log[5]*(3 + Log[5])^3*(3 + 5*Log[5]))/46656 + (78125*x*Log[5]*(3 + Log[5])^5*(3 + 7*L
og[5]))/2239488 + (25*x^6*(9 + Log[5]^2))/108 - (25*x^5*(3 + Log[5])*(9 + Log[5]^2))/108 + (625*x^4*(3 + Log[5
])^2*(9 + Log[5]^2))/3456 - (3125*x^3*(3 + Log[5])^3*(9 + Log[5]^2))/23328 + (78125*x^2*(3 + Log[5])^4*(9 + Lo
g[5]^2))/746496 - (78125*x*(3 + Log[5])^5*(9 + Log[5]^2))/746496 - (1953125*Log[5]^2*(3 + Log[5])^7)/(26873856
*(12*x + 5*(3 + Log[5]))) - (1953125*(3 + Log[5])^8)/(8957952*(12*x + 5*(3 + Log[5]))) + (1953125*(3 + Log[5])
^9)/(53747712*(12*x + 5*(3 + Log[5]))) + (1953125*(3 + Log[5])^7*(9 + Log[5]^2))/(53747712*(12*x + 5*(3 + Log[
5]))) - (15625*x^2*Log[5]*(3 + Log[5])^4*(1 + Log[25]))/124416 - (25*x^6*Log[5]*(3 + Log[25]))/54 - (625*x^4*L
og[5]*(3 + Log[5])^2*(3 + Log[625]))/5184 + (25*x^5*Log[5]*(3 + Log[5]^2 + Log[625]))/36 - (390625*(3 + Log[5]
)^7*Log[12*x + 5*(3 + Log[5])])/1119744 + (390625*(3 + Log[5])^8*Log[12*x + 5*(3 + Log[5])])/5971968 - (390625
*Log[5]*(3 + Log[5])^6*(3 + 8*Log[5])*Log[12*x + 5*(3 + Log[5])])/26873856 + (2734375*(3 + Log[5])^6*(9 + Log[
5]^2)*Log[12*x + 5*(3 + Log[5])])/53747712 - 3*Defer[Int][5^((5*x)/(12*x + 5*(3 + Log[5])))*E^((-3 + 15*x + 12
*x^2 + 5^((5*x)/(12*x + 5*(3 + Log[5])))*E^((3*(-1 + 5*x + 4*x^2))/(12*x + 5*(3 + Log[5])))*(12*x + 5*(3 + Log
[5])))/(12*x + 5*(3 + Log[5]))), x] - 108*Defer[Int][(5^((5*x)/(12*x + 5*(3 + Log[5])))*E^((-3 + 15*x + 12*x^2
+ 5^((5*x)/(12*x + 5*(3 + Log[5])))*E^((3*(-1 + 5*x + 4*x^2))/(12*x + 5*(3 + Log[5])))*(12*x + 5*(3 + Log[5])
))/(12*x + 5*(3 + Log[5]))))/(15 + 12*x + 5*Log[5])^2, x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {2880 x^8+1152 x^9+\left (1200 x^7+960 x^8\right ) \log (5)+\exp \left (\exp \left (\frac {-3+15 x+12 x^2+5 x \log (5)}{15+12 x+5 \log (5)}\right )+\frac {-3+15 x+12 x^2+5 x \log (5)}{15+12 x+5 \log (5)}\right ) \left (-783-1080 x-432 x^2+(-450-360 x) \log (5)-75 \log ^2(5)\right )+x^7 \left (1800+200 \log ^2(5)\right )}{225+360 x+144 x^2+(150+120 x) \log (5)+25 \log ^2(5)} \, dx \\ & = \int \frac {2880 x^8+1152 x^9+\left (1200 x^7+960 x^8\right ) \log (5)+\exp \left (\exp \left (\frac {-3+15 x+12 x^2+5 x \log (5)}{15+12 x+5 \log (5)}\right )+\frac {-3+15 x+12 x^2+5 x \log (5)}{15+12 x+5 \log (5)}\right ) \left (-783-1080 x-432 x^2+(-450-360 x) \log (5)-75 \log ^2(5)\right )+x^7 \left (1800+200 \log ^2(5)\right )}{144 x^2+120 x (3+\log (5))+25 (3+\log (5))^2} \, dx \\ & = \int \frac {2880 x^8+1152 x^9+\left (1200 x^7+960 x^8\right ) \log (5)+\exp \left (\exp \left (\frac {-3+15 x+12 x^2+5 x \log (5)}{15+12 x+5 \log (5)}\right )+\frac {-3+15 x+12 x^2+5 x \log (5)}{15+12 x+5 \log (5)}\right ) \left (-783-1080 x-432 x^2+(-450-360 x) \log (5)-75 \log ^2(5)\right )+x^7 \left (1800+200 \log ^2(5)\right )}{(15+12 x+5 \log (5))^2} \, dx \\ & = \int \left (\frac {2880 x^8}{(15+12 x+5 \log (5))^2}+\frac {1152 x^9}{(15+12 x+5 \log (5))^2}+\frac {240 x^7 (5+4 x) \log (5)}{(15+12 x+5 \log (5))^2}+\frac {200 x^7 \left (9+\log ^2(5)\right )}{(15+12 x+5 \log (5))^2}+\frac {3\ 5^{\frac {5 x}{12 x+5 (3+\log (5))}} \exp \left (5^{\frac {5 x}{12 x+5 (3+\log (5))}} \exp \left (-\frac {3}{12 x+5 (3+\log (5))}+\frac {15 x}{12 x+5 (3+\log (5))}+\frac {12 x^2}{12 x+5 (3+\log (5))}\right )-\frac {3}{12 x+5 (3+\log (5))}+\frac {15 x}{12 x+5 (3+\log (5))}+\frac {12 x^2}{12 x+5 (3+\log (5))}\right ) \left (-261-144 x^2-150 \log (5)-25 \log ^2(5)-120 x (3+\log (5))\right )}{(15+12 x+5 \log (5))^2}\right ) \, dx \\ & = 3 \int \frac {5^{\frac {5 x}{12 x+5 (3+\log (5))}} \exp \left (5^{\frac {5 x}{12 x+5 (3+\log (5))}} \exp \left (-\frac {3}{12 x+5 (3+\log (5))}+\frac {15 x}{12 x+5 (3+\log (5))}+\frac {12 x^2}{12 x+5 (3+\log (5))}\right )-\frac {3}{12 x+5 (3+\log (5))}+\frac {15 x}{12 x+5 (3+\log (5))}+\frac {12 x^2}{12 x+5 (3+\log (5))}\right ) \left (-261-144 x^2-150 \log (5)-25 \log ^2(5)-120 x (3+\log (5))\right )}{(15+12 x+5 \log (5))^2} \, dx+1152 \int \frac {x^9}{(15+12 x+5 \log (5))^2} \, dx+2880 \int \frac {x^8}{(15+12 x+5 \log (5))^2} \, dx+(240 \log (5)) \int \frac {x^7 (5+4 x)}{(15+12 x+5 \log (5))^2} \, dx+\left (200 \left (9+\log ^2(5)\right )\right ) \int \frac {x^7}{(15+12 x+5 \log (5))^2} \, dx \\ & = 3 \int \frac {5^{\frac {5 x}{12 x+5 (3+\log (5))}} \exp \left (\frac {-3+15 x+12 x^2+5^{\frac {5 x}{12 x+5 (3+\log (5))}} \exp \left (\frac {3 \left (-1+5 x+4 x^2\right )}{12 x+5 (3+\log (5))}\right ) (12 x+5 (3+\log (5)))}{12 x+5 (3+\log (5))}\right ) \left (-261-144 x^2-150 \log (5)-25 \log ^2(5)-120 x (3+\log (5))\right )}{(12 x+5 (3+\log (5)))^2} \, dx+1152 \int \left (\frac {x^7}{144}-\frac {5}{864} x^6 (3+\log (5))+\frac {25 x^5 (3+\log (5))^2}{6912}-\frac {125 x^4 (3+\log (5))^3}{62208}+\frac {3125 x^3 (3+\log (5))^4}{2985984}-\frac {3125 x^2 (3+\log (5))^5}{5971968}+\frac {109375 x (3+\log (5))^6}{429981696}-\frac {78125 (3+\log (5))^7}{644972544}-\frac {1953125 (3+\log (5))^9}{5159780352 (15+12 x+5 \log (5))^2}+\frac {390625 (3+\log (5))^8}{573308928 (15+12 x+5 \log (5))}\right ) \, dx+2880 \int \left (\frac {x^6}{144}-\frac {5}{864} x^5 (3+\log (5))+\frac {25 x^4 (3+\log (5))^2}{6912}-\frac {125 x^3 (3+\log (5))^3}{62208}+\frac {3125 x^2 (3+\log (5))^4}{2985984}-\frac {3125 x (3+\log (5))^5}{5971968}+\frac {109375 (3+\log (5))^6}{429981696}+\frac {390625 (3+\log (5))^8}{429981696 (15+12 x+5 \log (5))^2}-\frac {78125 (3+\log (5))^7}{53747712 (15+12 x+5 \log (5))}\right ) \, dx+(240 \log (5)) \int \left (\frac {x^6}{36}+\frac {625 x^2 (3+\log (5))^3 (3+5 \log (5))}{746496}+\frac {390625 \log (5) (3+\log (5))^7}{107495424 (15+12 x+5 \log (5))^2}+\frac {15625 (3+\log (5))^5 (3+7 \log (5))}{107495424}-\frac {78125 (3+\log (5))^6 (3+8 \log (5))}{107495424 (15+12 x+5 \log (5))}-\frac {3125 x (3+\log (5))^4 (1+\log (25))}{2985984}-\frac {5}{432} x^5 (3+\log (25))-\frac {125 x^3 (3+\log (5))^2 (3+\log (625))}{62208}+\frac {25 x^4 \left (3+\log ^2(5)+\log (625)\right )}{1728}\right ) \, dx+\left (200 \left (9+\log ^2(5)\right )\right ) \int \left (\frac {x^5}{144}-\frac {5}{864} x^4 (3+\log (5))+\frac {25 x^3 (3+\log (5))^2}{6912}-\frac {125 x^2 (3+\log (5))^3}{62208}+\frac {3125 x (3+\log (5))^4}{2985984}-\frac {3125 (3+\log (5))^5}{5971968}-\frac {78125 (3+\log (5))^7}{35831808 (15+12 x+5 \log (5))^2}+\frac {109375 (3+\log (5))^6}{35831808 (15+12 x+5 \log (5))}\right ) \, dx \\ & = \frac {20 x^7}{7}+x^8+\frac {20}{21} x^7 \log (5)-\frac {25}{9} x^6 (3+\log (5))-\frac {20}{21} x^7 (3+\log (5))+\frac {25}{12} x^5 (3+\log (5))^2+\frac {25}{36} x^6 (3+\log (5))^2-\frac {625}{432} x^4 (3+\log (5))^3-\frac {25}{54} x^5 (3+\log (5))^3+\frac {15625 x^3 (3+\log (5))^4}{15552}+\frac {3125 x^4 (3+\log (5))^4}{10368}-\frac {15625 x^2 (3+\log (5))^5}{20736}-\frac {3125 x^3 (3+\log (5))^5}{15552}+\frac {546875 x (3+\log (5))^6}{746496}+\frac {109375 x^2 (3+\log (5))^6}{746496}-\frac {78125 x (3+\log (5))^7}{559872}+\frac {3125 x^3 \log (5) (3+\log (5))^3 (3+5 \log (5))}{46656}+\frac {78125 x \log (5) (3+\log (5))^5 (3+7 \log (5))}{2239488}+\frac {25}{108} x^6 \left (9+\log ^2(5)\right )-\frac {25}{108} x^5 (3+\log (5)) \left (9+\log ^2(5)\right )+\frac {625 x^4 (3+\log (5))^2 \left (9+\log ^2(5)\right )}{3456}-\frac {3125 x^3 (3+\log (5))^3 \left (9+\log ^2(5)\right )}{23328}+\frac {78125 x^2 (3+\log (5))^4 \left (9+\log ^2(5)\right )}{746496}-\frac {78125 x (3+\log (5))^5 \left (9+\log ^2(5)\right )}{746496}-\frac {1953125 \log ^2(5) (3+\log (5))^7}{26873856 (12 x+5 (3+\log (5)))}-\frac {1953125 (3+\log (5))^8}{8957952 (12 x+5 (3+\log (5)))}+\frac {1953125 (3+\log (5))^9}{53747712 (12 x+5 (3+\log (5)))}+\frac {1953125 (3+\log (5))^7 \left (9+\log ^2(5)\right )}{53747712 (12 x+5 (3+\log (5)))}-\frac {15625 x^2 \log (5) (3+\log (5))^4 (1+\log (25))}{124416}-\frac {25}{54} x^6 \log (5) (3+\log (25))-\frac {625 x^4 \log (5) (3+\log (5))^2 (3+\log (625))}{5184}+\frac {25}{36} x^5 \log (5) \left (3+\log ^2(5)+\log (625)\right )-\frac {390625 (3+\log (5))^7 \log (12 x+5 (3+\log (5)))}{1119744}+\frac {390625 (3+\log (5))^8 \log (12 x+5 (3+\log (5)))}{5971968}-\frac {390625 \log (5) (3+\log (5))^6 (3+8 \log (5)) \log (12 x+5 (3+\log (5)))}{26873856}+\frac {2734375 (3+\log (5))^6 \left (9+\log ^2(5)\right ) \log (12 x+5 (3+\log (5)))}{53747712}+3 \int \left (-5^{\frac {5 x}{12 x+5 (3+\log (5))}} e^{\frac {-3+15 x+12 x^2+5^{\frac {5 x}{12 x+5 (3+\log (5))}} e^{\frac {3 \left (-1+5 x+4 x^2\right )}{12 x+5 (3+\log (5))}} (12 x+5 (3+\log (5)))}{12 x+5 (3+\log (5))}}-\frac {36\ 5^{\frac {5 x}{12 x+5 (3+\log (5))}} e^{\frac {-3+15 x+12 x^2+5^{\frac {5 x}{12 x+5 (3+\log (5))}} e^{\frac {3 \left (-1+5 x+4 x^2\right )}{12 x+5 (3+\log (5))}} (12 x+5 (3+\log (5)))}{12 x+5 (3+\log (5))}}}{(15+12 x+5 \log (5))^2}\right ) \, dx \\ & = \frac {20 x^7}{7}+x^8+\frac {20}{21} x^7 \log (5)-\frac {25}{9} x^6 (3+\log (5))-\frac {20}{21} x^7 (3+\log (5))+\frac {25}{12} x^5 (3+\log (5))^2+\frac {25}{36} x^6 (3+\log (5))^2-\frac {625}{432} x^4 (3+\log (5))^3-\frac {25}{54} x^5 (3+\log (5))^3+\frac {15625 x^3 (3+\log (5))^4}{15552}+\frac {3125 x^4 (3+\log (5))^4}{10368}-\frac {15625 x^2 (3+\log (5))^5}{20736}-\frac {3125 x^3 (3+\log (5))^5}{15552}+\frac {546875 x (3+\log (5))^6}{746496}+\frac {109375 x^2 (3+\log (5))^6}{746496}-\frac {78125 x (3+\log (5))^7}{559872}+\frac {3125 x^3 \log (5) (3+\log (5))^3 (3+5 \log (5))}{46656}+\frac {78125 x \log (5) (3+\log (5))^5 (3+7 \log (5))}{2239488}+\frac {25}{108} x^6 \left (9+\log ^2(5)\right )-\frac {25}{108} x^5 (3+\log (5)) \left (9+\log ^2(5)\right )+\frac {625 x^4 (3+\log (5))^2 \left (9+\log ^2(5)\right )}{3456}-\frac {3125 x^3 (3+\log (5))^3 \left (9+\log ^2(5)\right )}{23328}+\frac {78125 x^2 (3+\log (5))^4 \left (9+\log ^2(5)\right )}{746496}-\frac {78125 x (3+\log (5))^5 \left (9+\log ^2(5)\right )}{746496}-\frac {1953125 \log ^2(5) (3+\log (5))^7}{26873856 (12 x+5 (3+\log (5)))}-\frac {1953125 (3+\log (5))^8}{8957952 (12 x+5 (3+\log (5)))}+\frac {1953125 (3+\log (5))^9}{53747712 (12 x+5 (3+\log (5)))}+\frac {1953125 (3+\log (5))^7 \left (9+\log ^2(5)\right )}{53747712 (12 x+5 (3+\log (5)))}-\frac {15625 x^2 \log (5) (3+\log (5))^4 (1+\log (25))}{124416}-\frac {25}{54} x^6 \log (5) (3+\log (25))-\frac {625 x^4 \log (5) (3+\log (5))^2 (3+\log (625))}{5184}+\frac {25}{36} x^5 \log (5) \left (3+\log ^2(5)+\log (625)\right )-\frac {390625 (3+\log (5))^7 \log (12 x+5 (3+\log (5)))}{1119744}+\frac {390625 (3+\log (5))^8 \log (12 x+5 (3+\log (5)))}{5971968}-\frac {390625 \log (5) (3+\log (5))^6 (3+8 \log (5)) \log (12 x+5 (3+\log (5)))}{26873856}+\frac {2734375 (3+\log (5))^6 \left (9+\log ^2(5)\right ) \log (12 x+5 (3+\log (5)))}{53747712}-3 \int 5^{\frac {5 x}{12 x+5 (3+\log (5))}} e^{\frac {-3+15 x+12 x^2+5^{\frac {5 x}{12 x+5 (3+\log (5))}} e^{\frac {3 \left (-1+5 x+4 x^2\right )}{12 x+5 (3+\log (5))}} (12 x+5 (3+\log (5)))}{12 x+5 (3+\log (5))}} \, dx-108 \int \frac {5^{\frac {5 x}{12 x+5 (3+\log (5))}} e^{\frac {-3+15 x+12 x^2+5^{\frac {5 x}{12 x+5 (3+\log (5))}} e^{\frac {3 \left (-1+5 x+4 x^2\right )}{12 x+5 (3+\log (5))}} (12 x+5 (3+\log (5)))}{12 x+5 (3+\log (5))}}}{(15+12 x+5 \log (5))^2} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.42 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.93 \[ \int \frac {1800 x^7+2880 x^8+1152 x^9+\left (1200 x^7+960 x^8\right ) \log (5)+200 x^7 \log ^2(5)+e^{e^{\frac {-3+15 x+12 x^2+5 x \log (5)}{15+12 x+5 \log (5)}}+\frac {-3+15 x+12 x^2+5 x \log (5)}{15+12 x+5 \log (5)}} \left (-783-1080 x-432 x^2+(-450-360 x) \log (5)-75 \log ^2(5)\right )}{225+360 x+144 x^2+(150+120 x) \log (5)+25 \log ^2(5)} \, dx=-3 e^{e^{x-\frac {3}{15+12 x+5 \log (5)}}}+x^8 \]

[In]

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]))*(-7
83 - 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]

[Out]

-3*E^E^(x - 3/(15 + 12*x + 5*Log[5])) + x^8

Maple [A] (verified)

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	\(x^{8}+\frac {\left (-45-15 \ln \left (5\right )\right ) {\mathrm e}^{{\mathrm e}^{\frac {5 x \ln \left (5\right )+12 x^{2}+15 x -3}{5 \ln \left (5\right )+12 x +15}}}-36 x \,{\mathrm e}^{{\mathrm e}^{\frac {5 x \ln \left (5\right )+12 x^{2}+15 x -3}{5 \ln \left (5\right )+12 x +15}}}}{5 \ln \left (5\right )+12 x +15}\)	\(86\)

[In]

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+1
800*x^7)/(25*ln(5)^2+(120*x+150)*ln(5)+144*x^2+360*x+225),x,method=_RETURNVERBOSE)

[Out]

x^8-3*exp(exp((5*x*ln(5)+12*x^2+15*x-3)/(5*ln(5)+12*x+15)))

Fricas [B] (verification not implemented)

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 \[ \int \frac {1800 x^7+2880 x^8+1152 x^9+\left (1200 x^7+960 x^8\right ) \log (5)+200 x^7 \log ^2(5)+e^{e^{\frac {-3+15 x+12 x^2+5 x \log (5)}{15+12 x+5 \log (5)}}+\frac {-3+15 x+12 x^2+5 x \log (5)}{15+12 x+5 \log (5)}} \left (-783-1080 x-432 x^2+(-450-360 x) \log (5)-75 \log ^2(5)\right )}{225+360 x+144 x^2+(150+120 x) \log (5)+25 \log ^2(5)} \, dx={\left (x^{8} e^{\left (\frac {12 \, x^{2} + 5 \, x \log \left (5\right ) + 15 \, x - 3}{12 \, x + 5 \, \log \left (5\right ) + 15}\right )} - 3 \, e^{\left (\frac {12 \, x^{2} + {\left (12 \, x + 5 \, \log \left (5\right ) + 15\right )} e^{\left (\frac {12 \, x^{2} + 5 \, x \log \left (5\right ) + 15 \, x - 3}{12 \, x + 5 \, \log \left (5\right ) + 15}\right )} + 5 \, x \log \left (5\right ) + 15 \, x - 3}{12 \, x + 5 \, \log \left (5\right ) + 15}\right )}\right )} e^{\left (-\frac {12 \, x^{2} + 5 \, x \log \left (5\right ) + 15 \, x - 3}{12 \, x + 5 \, \log \left (5\right ) + 15}\right )} \]

[In]

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="fricas")

[Out]

(x^8*e^((12*x^2 + 5*x*log(5) + 15*x - 3)/(12*x + 5*log(5) + 15)) - 3*e^((12*x^2 + (12*x + 5*log(5) + 15)*e^((1
2*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))

Sympy [A] (verification not implemented)

Time = 0.41 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.26 \[ \int \frac {1800 x^7+2880 x^8+1152 x^9+\left (1200 x^7+960 x^8\right ) \log (5)+200 x^7 \log ^2(5)+e^{e^{\frac {-3+15 x+12 x^2+5 x \log (5)}{15+12 x+5 \log (5)}}+\frac {-3+15 x+12 x^2+5 x \log (5)}{15+12 x+5 \log (5)}} \left (-783-1080 x-432 x^2+(-450-360 x) \log (5)-75 \log ^2(5)\right )}{225+360 x+144 x^2+(150+120 x) \log (5)+25 \log ^2(5)} \, dx=x^{8} - 3 e^{e^{\frac {12 x^{2} + 5 x \log {\left (5 \right )} + 15 x - 3}{12 x + 5 \log {\left (5 \right )} + 15}}} \]

[In]

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)

[Out]

x**8 - 3*exp(exp((12*x**2 + 5*x*log(5) + 15*x - 3)/(12*x + 5*log(5) + 15)))

Maxima [F]

\[ \int \frac {1800 x^7+2880 x^8+1152 x^9+\left (1200 x^7+960 x^8\right ) \log (5)+200 x^7 \log ^2(5)+e^{e^{\frac {-3+15 x+12 x^2+5 x \log (5)}{15+12 x+5 \log (5)}}+\frac {-3+15 x+12 x^2+5 x \log (5)}{15+12 x+5 \log (5)}} \left (-783-1080 x-432 x^2+(-450-360 x) \log (5)-75 \log ^2(5)\right )}{225+360 x+144 x^2+(150+120 x) \log (5)+25 \log ^2(5)} \, dx=\int { \frac {1152 \, x^{9} + 200 \, x^{7} \log \left (5\right )^{2} + 2880 \, x^{8} + 1800 \, x^{7} - 3 \, {\left (144 \, x^{2} + 30 \, {\left (4 \, x + 5\right )} \log \left (5\right ) + 25 \, \log \left (5\right )^{2} + 360 \, x + 261\right )} e^{\left (\frac {12 \, x^{2} + 5 \, x \log \left (5\right ) + 15 \, x - 3}{12 \, x + 5 \, \log \left (5\right ) + 15} + e^{\left (\frac {12 \, x^{2} + 5 \, x \log \left (5\right ) + 15 \, x - 3}{12 \, x + 5 \, \log \left (5\right ) + 15}\right )}\right )} + 240 \, {\left (4 \, x^{8} + 5 \, x^{7}\right )} \log \left (5\right )}{144 \, x^{2} + 30 \, {\left (4 \, x + 5\right )} \log \left (5\right ) + 25 \, \log \left (5\right )^{2} + 360 \, x + 225} \,d x } \]

[In]

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="maxima")

[Out]

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) + 27)*x^5 + 25/12*(log(5)^2 + 6*log(5) + 9)*x^5 + 25/12*x^6 - 25/12*x^5*(lo
g(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*lo
g(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*log(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*(log(5)^2 + 6*log(5) + 9)*x^4 -
288000*(log(5)^3 + 9*log(5)^2 + 27*log(5) + 27)*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) + 1
5) + 78125*(log(5)^7 + 21*log(5)^6 + 189*log(5)^5 + 945*log(5)^4 + 2835*log(5)^3 + 5103*log(5)^2 + 5103*log(5)
+ 2187)/(12*x + 5*log(5) + 15))*log(5)^2 - 78125/559872*(log(5)^7 + 21*log(5)^6 + 189*log(5)^5 + 945*log(5)^4
+ 2835*log(5)^3 + 5103*log(5)^2 + 5103*log(5) + 2187)*x + 546875/746496*(log(5)^6 + 18*log(5)^5 + 135*log(5)^
4 + 540*log(5)^3 + 1215*log(5)^2 + 1458*log(5) + 729)*x - 78125/82944*(log(5)^5 + 15*log(5)^4 + 90*log(5)^3 +
270*log(5)^2 + 405*log(5) + 243)*x + 5/188116992*(35831808*x^7 - 34836480*x^6*(log(5) + 3) + 26127360*(log(5)^
2 + 6*log(5) + 9)*x^5 - 18144000*(log(5)^3 + 9*log(5)^2 + 27*log(5) + 27)*x^4 + 12600000*(log(5)^4 + 12*log(5)
^3 + 54*log(5)^2 + 108*log(5) + 81)*x^3 - 9450000*(log(5)^5 + 15*log(5)^4 + 90*log(5)^3 + 270*log(5)^2 + 405*l
og(5) + 243)*x^2 + 9187500*(log(5)^6 + 18*log(5)^5 + 135*log(5)^4 + 540*log(5)^3 + 1215*log(5)^2 + 1458*log(5)
+ 729)*x - 4375000*(log(5)^7 + 21*log(5)^6 + 189*log(5)^5 + 945*log(5)^4 + 2835*log(5)^3 + 5103*log(5)^2 + 51
03*log(5) + 2187)*log(12*x + 5*log(5) + 15) - 2734375*(log(5)^8 + 24*log(5)^7 + 252*log(5)^6 + 1512*log(5)^5 +
5670*log(5)^4 + 13608*log(5)^3 + 20412*log(5)^2 + 17496*log(5) + 6561)/(12*x + 5*log(5) + 15))*log(5) + 25/89
57952*(497664*x^6 - 497664*x^5*(log(5) + 3) + 388800*(log(5)^2 + 6*log(5) + 9)*x^4 - 288000*(log(5)^3 + 9*log(
5)^2 + 27*log(5) + 27)*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 + 5103*log(5)^2 + 5103*log(5) + 2187)/(12*x + 5*log(5)
+ 15))*log(5) + 390625/5971968*(log(5)^8 + 24*log(5)^7 + 252*log(5)^6 + 1512*log(5)^5 + 5670*log(5)^4 + 13608*
log(5)^3 + 20412*log(5)^2 + 17496*log(5) + 6561)*log(12*x + 5*log(5) + 15) - 390625/1119744*(log(5)^7 + 21*log
(5)^6 + 189*log(5)^5 + 945*log(5)^4 + 2835*log(5)^3 + 5103*log(5)^2 + 5103*log(5) + 2187)*log(12*x + 5*log(5)
+ 15) + 2734375/5971968*(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) + 1953125/53747712*(log(5)^9 + 27*log(5)^8 + 324*log(5)^7 + 2268*log(5)^6 + 102
06*log(5)^5 + 30618*log(5)^4 + 61236*log(5)^3 + 78732*log(5)^2 + 59049*log(5) + 19683)/(12*x + 5*log(5) + 15)
- 1953125/8957952*(log(5)^8 + 24*log(5)^7 + 252*log(5)^6 + 1512*log(5)^5 + 5670*log(5)^4 + 13608*log(5)^3 + 20
412*log(5)^2 + 17496*log(5) + 6561)/(12*x + 5*log(5) + 15) + 1953125/5971968*(log(5)^7 + 21*log(5)^6 + 189*log
(5)^5 + 945*log(5)^4 + 2835*log(5)^3 + 5103*log(5)^2 + 5103*log(5) + 2187)/(12*x + 5*log(5) + 15) - integrate(
3*(144*x^2 + 120*x*(log(5) + 3) + 25*log(5)^2 + 150*log(5) + 261)*e^(x - 3/(12*x + 5*log(5) + 15) + e^(x - 3/(
12*x + 5*log(5) + 15)))/(144*x^2 + 120*x*(log(5) + 3) + 25*log(5)^2 + 150*log(5) + 225), x)

Giac [F]

\[ \int \frac {1800 x^7+2880 x^8+1152 x^9+\left (1200 x^7+960 x^8\right ) \log (5)+200 x^7 \log ^2(5)+e^{e^{\frac {-3+15 x+12 x^2+5 x \log (5)}{15+12 x+5 \log (5)}}+\frac {-3+15 x+12 x^2+5 x \log (5)}{15+12 x+5 \log (5)}} \left (-783-1080 x-432 x^2+(-450-360 x) \log (5)-75 \log ^2(5)\right )}{225+360 x+144 x^2+(150+120 x) \log (5)+25 \log ^2(5)} \, dx=\int { \frac {1152 \, x^{9} + 200 \, x^{7} \log \left (5\right )^{2} + 2880 \, x^{8} + 1800 \, x^{7} - 3 \, {\left (144 \, x^{2} + 30 \, {\left (4 \, x + 5\right )} \log \left (5\right ) + 25 \, \log \left (5\right )^{2} + 360 \, x + 261\right )} e^{\left (\frac {12 \, x^{2} + 5 \, x \log \left (5\right ) + 15 \, x - 3}{12 \, x + 5 \, \log \left (5\right ) + 15} + e^{\left (\frac {12 \, x^{2} + 5 \, x \log \left (5\right ) + 15 \, x - 3}{12 \, x + 5 \, \log \left (5\right ) + 15}\right )}\right )} + 240 \, {\left (4 \, x^{8} + 5 \, x^{7}\right )} \log \left (5\right )}{144 \, x^{2} + 30 \, {\left (4 \, x + 5\right )} \log \left (5\right ) + 25 \, \log \left (5\right )^{2} + 360 \, x + 225} \,d x } \]

[In]

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="giac")

[Out]

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)

Mupad [B] (verification not implemented)

Time = 12.85 (sec) , antiderivative size = 70, normalized size of antiderivative = 2.59 \[ \int \frac {1800 x^7+2880 x^8+1152 x^9+\left (1200 x^7+960 x^8\right ) \log (5)+200 x^7 \log ^2(5)+e^{e^{\frac {-3+15 x+12 x^2+5 x \log (5)}{15+12 x+5 \log (5)}}+\frac {-3+15 x+12 x^2+5 x \log (5)}{15+12 x+5 \log (5)}} \left (-783-1080 x-432 x^2+(-450-360 x) \log (5)-75 \log ^2(5)\right )}{225+360 x+144 x^2+(150+120 x) \log (5)+25 \log ^2(5)} \, dx=x^8-3\,{\mathrm {e}}^{5^{\frac {5\,x}{12\,x+5\,\ln \left (5\right )+15}}\,{\mathrm {e}}^{-\frac {3}{12\,x+5\,\ln \left (5\right )+15}}\,{\mathrm {e}}^{\frac {15\,x}{12\,x+5\,\ln \left (5\right )+15}}\,{\mathrm {e}}^{\frac {12\,x^2}{12\,x+5\,\ln \left (5\right )+15}}} \]

[In]

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) + 15)))*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)*(120*x + 150) + 25*log(5)^2 + 144*x^2
+ 225),x)

[Out]

x^8 - 3*exp(5^((5*x)/(12*x + 5*log(5) + 15))*exp(-3/(12*x + 5*log(5) + 15))*exp((15*x)/(12*x + 5*log(5) + 15))
*exp((12*x^2)/(12*x + 5*log(5) + 15)))