Integrand size = 213, antiderivative size = 34 \[ \int \frac {-540+e^2 (45-9 x)-81 x+81 x^2+(108-27 x) \log (5)}{-3375 x^3-2025 x^4-405 x^5-27 x^6+e^6 \left (1+3 x+3 x^2+x^3\right )+e^4 \left (-45 x-99 x^2-63 x^3-9 x^4\right )+e^2 \left (675 x^2+945 x^3+297 x^4+27 x^5\right )+\left (2025 x^3+810 x^4+81 x^5+e^4 \left (9 x+18 x^2+9 x^3\right )+e^2 \left (-270 x^2-324 x^3-54 x^4\right )\right ) \log (5)+\left (-405 x^3-81 x^4+e^2 \left (27 x^2+27 x^3\right )\right ) \log ^2(5)+27 x^3 \log ^3(5)} \, dx=\frac {-2+x}{x^2 \left (5+\frac {e^2 \left (-1+\frac {2-x}{3}\right )}{x}+x-\log (5)\right )^2} \]
Time = 0.04 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.79 \[ \int \frac {-540+e^2 (45-9 x)-81 x+81 x^2+(108-27 x) \log (5)}{-3375 x^3-2025 x^4-405 x^5-27 x^6+e^6 \left (1+3 x+3 x^2+x^3\right )+e^4 \left (-45 x-99 x^2-63 x^3-9 x^4\right )+e^2 \left (675 x^2+945 x^3+297 x^4+27 x^5\right )+\left (2025 x^3+810 x^4+81 x^5+e^4 \left (9 x+18 x^2+9 x^3\right )+e^2 \left (-270 x^2-324 x^3-54 x^4\right )\right ) \log (5)+\left (-405 x^3-81 x^4+e^2 \left (27 x^2+27 x^3\right )\right ) \log ^2(5)+27 x^3 \log ^3(5)} \, dx=-\frac {9 (2-x)}{\left (e^2 (1+x)-3 x (5+x-\log (5))\right )^2} \]
Integrate[(-540 + E^2*(45 - 9*x) - 81*x + 81*x^2 + (108 - 27*x)*Log[5])/(- 3375*x^3 - 2025*x^4 - 405*x^5 - 27*x^6 + E^6*(1 + 3*x + 3*x^2 + x^3) + E^4 *(-45*x - 99*x^2 - 63*x^3 - 9*x^4) + E^2*(675*x^2 + 945*x^3 + 297*x^4 + 27 *x^5) + (2025*x^3 + 810*x^4 + 81*x^5 + E^4*(9*x + 18*x^2 + 9*x^3) + E^2*(- 270*x^2 - 324*x^3 - 54*x^4))*Log[5] + (-405*x^3 - 81*x^4 + E^2*(27*x^2 + 2 7*x^3))*Log[5]^2 + 27*x^3*Log[5]^3),x]
Leaf count is larger than twice the leaf count of optimal. \(273\) vs. \(2(34)=68\).
Time = 0.78 (sec) , antiderivative size = 273, normalized size of antiderivative = 8.03, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.014, Rules used = {6, 2462, 2009}
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.
| \(\displaystyle \int \frac {81 x^2-81 x+e^2 (45-9 x)+(108-27 x) \log (5)-540}{-27 x^6-405 x^5-2025 x^4-3375 x^3+27 x^3 \log ^3(5)+e^6 \left (x^3+3 x^2+3 x+1\right )+e^4 \left (-9 x^4-63 x^3-99 x^2-45 x\right )+\left (-81 x^4-405 x^3+e^2 \left (27 x^3+27 x^2\right )\right ) \log ^2(5)+e^2 \left (27 x^5+297 x^4+945 x^3+675 x^2\right )+\left (81 x^5+810 x^4+2025 x^3+e^4 \left (9 x^3+18 x^2+9 x\right )+e^2 \left (-54 x^4-324 x^3-270 x^2\right )\right ) \log (5)} \, dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {81 x^2-81 x+e^2 (45-9 x)+(108-27 x) \log (5)-540}{-27 x^6-405 x^5-2025 x^4+x^3 \left (27 \log ^3(5)-3375\right )+e^6 \left (x^3+3 x^2+3 x+1\right )+e^4 \left (-9 x^4-63 x^3-99 x^2-45 x\right )+\left (-81 x^4-405 x^3+e^2 \left (27 x^3+27 x^2\right )\right ) \log ^2(5)+e^2 \left (27 x^5+297 x^4+945 x^3+675 x^2\right )+\left (81 x^5+810 x^4+2025 x^3+e^4 \left (9 x^3+18 x^2+9 x\right )+e^2 \left (-54 x^4-324 x^3-270 x^2\right )\right ) \log (5)}dx\) |
| \(\Big \downarrow \) 2462 |
| \(\displaystyle \int \left (\frac {18 \left (-\left (x \left (27-e^2-3 \log (5)\right )\right )+4 e^2-30+3 \log (25)\right )}{\left (-3 x^2-x \left (15-e^2-3 \log (5)\right )+e^2\right )^3}-\frac {27}{\left (-3 x^2-x \left (15-e^2-3 \log (5)\right )+e^2\right )^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {27 \left (e^4+9 \left (25+\log ^2(5)-14 \log (5)+\log (625)\right )-6 e^2 (3-\log (5))\right ) \left (6 x-e^2+15-3 \log (5)\right )}{\left (e^4-6 e^2 (3-\log (5))+9 (5-\log (5))^2\right )^2 \left (-3 x^2-x \left (15-e^2-3 \log (5)\right )+e^2\right )}-\frac {9 \left (-\left (x \left (e^4+9 \left (25+\log ^2(5)-14 \log (5)+\log (625)\right )-6 e^2 (3-\log (5))\right )\right )+2 e^4-3 e^2 (12-\log (625))+9 (5-\log (5)) (10-\log (25))\right )}{\left (12 e^2+\left (e^2-3 (5-\log (5))\right )^2\right ) \left (-3 x^2-x \left (15-e^2-3 \log (5)\right )+e^2\right )^2}-\frac {27 \left (6 x-e^2+15-3 \log (5)\right )}{\left (12 e^2+\left (e^2-3 (5-\log (5))\right )^2\right ) \left (-3 x^2-x \left (15-e^2-3 \log (5)\right )+e^2\right )}\) |
Int[(-540 + E^2*(45 - 9*x) - 81*x + 81*x^2 + (108 - 27*x)*Log[5])/(-3375*x ^3 - 2025*x^4 - 405*x^5 - 27*x^6 + E^6*(1 + 3*x + 3*x^2 + x^3) + E^4*(-45* x - 99*x^2 - 63*x^3 - 9*x^4) + E^2*(675*x^2 + 945*x^3 + 297*x^4 + 27*x^5) + (2025*x^3 + 810*x^4 + 81*x^5 + E^4*(9*x + 18*x^2 + 9*x^3) + E^2*(-270*x^ 2 - 324*x^3 - 54*x^4))*Log[5] + (-405*x^3 - 81*x^4 + E^2*(27*x^2 + 27*x^3) )*Log[5]^2 + 27*x^3*Log[5]^3),x]
(-27*(15 - E^2 + 6*x - 3*Log[5]))/((12*E^2 + (E^2 - 3*(5 - Log[5]))^2)*(E^ 2 - 3*x^2 - x*(15 - E^2 - 3*Log[5]))) + (27*(15 - E^2 + 6*x - 3*Log[5])*(E ^4 - 6*E^2*(3 - Log[5]) + 9*(25 - 14*Log[5] + Log[5]^2 + Log[625])))/((E^2 - 3*x^2 - x*(15 - E^2 - 3*Log[5]))*(E^4 - 6*E^2*(3 - Log[5]) + 9*(5 - Log [5])^2)^2) - (9*(2*E^4 + 9*(5 - Log[5])*(10 - Log[25]) - 3*E^2*(12 - Log[6 25]) - x*(E^4 - 6*E^2*(3 - Log[5]) + 9*(25 - 14*Log[5] + Log[5]^2 + Log[62 5]))))/((12*E^2 + (E^2 - 3*(5 - Log[5]))^2)*(E^2 - 3*x^2 - x*(15 - E^2 - 3 *Log[5]))^2)
3.20.20.3.1 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[{Qx = Factor[Px]}, Int[ExpandIntegr and[u*Qx^p, x], x] /; !SumQ[NonfreeFactors[Qx, x]]] /; PolyQ[Px, x] && GtQ [Expon[Px, x], 2] && !BinomialQ[Px, x] && !TrinomialQ[Px, x] && ILtQ[p, 0 ] && RationalFunctionQ[u, x]
Time = 0.67 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.85
| method | result | size |
| norman | \(\frac {9 x -18}{\left ({\mathrm e}^{2} x +3 x \ln \left (5\right )-3 x^{2}+{\mathrm e}^{2}-15 x \right )^{2}}\) | \(29\) |
| risch | \(\frac {9 x -18}{x^{2} {\mathrm e}^{4}+6 \,{\mathrm e}^{2} \ln \left (5\right ) x^{2}-6 x^{3} {\mathrm e}^{2}+9 x^{2} \ln \left (5\right )^{2}-18 x^{3} \ln \left (5\right )+9 x^{4}+2 x \,{\mathrm e}^{4}+6 x \,{\mathrm e}^{2} \ln \left (5\right )-36 x^{2} {\mathrm e}^{2}-90 x^{2} \ln \left (5\right )+90 x^{3}+{\mathrm e}^{4}-30 \,{\mathrm e}^{2} x +225 x^{2}}\) | \(96\) |
| gosper | \(\frac {9 x -18}{x^{2} {\mathrm e}^{4}+6 \,{\mathrm e}^{2} \ln \left (5\right ) x^{2}-6 x^{3} {\mathrm e}^{2}+9 x^{2} \ln \left (5\right )^{2}-18 x^{3} \ln \left (5\right )+9 x^{4}+2 x \,{\mathrm e}^{4}+6 x \,{\mathrm e}^{2} \ln \left (5\right )-36 x^{2} {\mathrm e}^{2}-90 x^{2} \ln \left (5\right )+90 x^{3}+{\mathrm e}^{4}-30 \,{\mathrm e}^{2} x +225 x^{2}}\) | \(101\) |
| parallelrisch | \(\frac {-162+81 x}{9 x^{2} {\mathrm e}^{4}+54 \,{\mathrm e}^{2} \ln \left (5\right ) x^{2}-54 x^{3} {\mathrm e}^{2}+81 x^{2} \ln \left (5\right )^{2}-162 x^{3} \ln \left (5\right )+81 x^{4}+18 x \,{\mathrm e}^{4}+54 x \,{\mathrm e}^{2} \ln \left (5\right )-324 x^{2} {\mathrm e}^{2}-810 x^{2} \ln \left (5\right )+810 x^{3}+9 \,{\mathrm e}^{4}-270 \,{\mathrm e}^{2} x +2025 x^{2}}\) | \(103\) |
| default | \(-3 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (27 \textit {\_Z}^{6}+\left (-27 \,{\mathrm e}^{2}-81 \ln \left (5\right )+405\right ) \textit {\_Z}^{5}+\left (54 \,{\mathrm e}^{2} \ln \left (5\right )+81 \ln \left (5\right )^{2}-297 \,{\mathrm e}^{2}-810 \ln \left (5\right )+9 \,{\mathrm e}^{4}+2025\right ) \textit {\_Z}^{4}+\left (-27 \,{\mathrm e}^{2} \ln \left (5\right )^{2}-27 \ln \left (5\right )^{3}+324 \,{\mathrm e}^{2} \ln \left (5\right )+405 \ln \left (5\right )^{2}-9 \,{\mathrm e}^{4} \ln \left (5\right )-945 \,{\mathrm e}^{2}-2025 \ln \left (5\right )+63 \,{\mathrm e}^{4}-{\mathrm e}^{6}+3375\right ) \textit {\_Z}^{3}+\left (-27 \,{\mathrm e}^{2} \ln \left (5\right )^{2}+270 \,{\mathrm e}^{2} \ln \left (5\right )-18 \,{\mathrm e}^{4} \ln \left (5\right )-675 \,{\mathrm e}^{2}+99 \,{\mathrm e}^{4}-3 \,{\mathrm e}^{6}\right ) \textit {\_Z}^{2}+\left (-9 \,{\mathrm e}^{4} \ln \left (5\right )+45 \,{\mathrm e}^{4}-3 \,{\mathrm e}^{6}\right ) \textit {\_Z} -{\mathrm e}^{6}\right )}{\sum }\frac {\left (-9 \textit {\_R}^{2}+\left ({\mathrm e}^{2}+3 \ln \left (5\right )+9\right ) \textit {\_R} -5 \,{\mathrm e}^{2}-12 \ln \left (5\right )+60\right ) \ln \left (x -\textit {\_R} \right )}{3 \,{\mathrm e}^{4} \ln \left (5\right )+\textit {\_R}^{2} {\mathrm e}^{6}+2 \textit {\_R} \,{\mathrm e}^{6}+{\mathrm e}^{6}-15 \,{\mathrm e}^{4}-63 \textit {\_R}^{2} {\mathrm e}^{4}-12 \textit {\_R}^{3} {\mathrm e}^{4}-66 \textit {\_R} \,{\mathrm e}^{4}+450 \,{\mathrm e}^{2} \textit {\_R} +45 \textit {\_R}^{4} {\mathrm e}^{2}+396 \textit {\_R}^{3} {\mathrm e}^{2}-54 \textit {\_R}^{5}-675 \textit {\_R}^{4}-2700 \textit {\_R}^{3}-3375 \textit {\_R}^{2}-180 \textit {\_R} \,{\mathrm e}^{2} \ln \left (5\right )-324 \,{\mathrm e}^{2} \ln \left (5\right ) \textit {\_R}^{2}+27 \,{\mathrm e}^{2} \ln \left (5\right )^{2} \textit {\_R}^{2}-72 \,{\mathrm e}^{2} \ln \left (5\right ) \textit {\_R}^{3}+18 \,{\mathrm e}^{2} \ln \left (5\right )^{2} \textit {\_R} +9 \ln \left (5\right ) {\mathrm e}^{4} \textit {\_R}^{2}+12 \ln \left (5\right ) {\mathrm e}^{4} \textit {\_R} +945 \textit {\_R}^{2} {\mathrm e}^{2}+2025 \textit {\_R}^{2} \ln \left (5\right )-108 \textit {\_R}^{3} \ln \left (5\right )^{2}-405 \textit {\_R}^{2} \ln \left (5\right )^{2}+1080 \textit {\_R}^{3} \ln \left (5\right )+135 \textit {\_R}^{4} \ln \left (5\right )+27 \textit {\_R}^{2} \ln \left (5\right )^{3}}\right )\) | \(401\) |
int(((-27*x+108)*ln(5)+(-9*x+45)*exp(2)+81*x^2-81*x-540)/(27*x^3*ln(5)^3+( (27*x^3+27*x^2)*exp(2)-81*x^4-405*x^3)*ln(5)^2+((9*x^3+18*x^2+9*x)*exp(2)^ 2+(-54*x^4-324*x^3-270*x^2)*exp(2)+81*x^5+810*x^4+2025*x^3)*ln(5)+(x^3+3*x ^2+3*x+1)*exp(2)^3+(-9*x^4-63*x^3-99*x^2-45*x)*exp(2)^2+(27*x^5+297*x^4+94 5*x^3+675*x^2)*exp(2)-27*x^6-405*x^5-2025*x^4-3375*x^3),x,method=_RETURNVE RBOSE)
Leaf count of result is larger than twice the leaf count of optimal. 83 vs. \(2 (29) = 58\).
Time = 0.25 (sec) , antiderivative size = 83, normalized size of antiderivative = 2.44 \[ \int \frac {-540+e^2 (45-9 x)-81 x+81 x^2+(108-27 x) \log (5)}{-3375 x^3-2025 x^4-405 x^5-27 x^6+e^6 \left (1+3 x+3 x^2+x^3\right )+e^4 \left (-45 x-99 x^2-63 x^3-9 x^4\right )+e^2 \left (675 x^2+945 x^3+297 x^4+27 x^5\right )+\left (2025 x^3+810 x^4+81 x^5+e^4 \left (9 x+18 x^2+9 x^3\right )+e^2 \left (-270 x^2-324 x^3-54 x^4\right )\right ) \log (5)+\left (-405 x^3-81 x^4+e^2 \left (27 x^2+27 x^3\right )\right ) \log ^2(5)+27 x^3 \log ^3(5)} \, dx=\frac {9 \, {\left (x - 2\right )}}{9 \, x^{4} + 9 \, x^{2} \log \left (5\right )^{2} + 90 \, x^{3} + 225 \, x^{2} + {\left (x^{2} + 2 \, x + 1\right )} e^{4} - 6 \, {\left (x^{3} + 6 \, x^{2} + 5 \, x\right )} e^{2} - 6 \, {\left (3 \, x^{3} + 15 \, x^{2} - {\left (x^{2} + x\right )} e^{2}\right )} \log \left (5\right )} \]
integrate(((-27*x+108)*log(5)+(-9*x+45)*exp(2)+81*x^2-81*x-540)/(27*x^3*lo g(5)^3+((27*x^3+27*x^2)*exp(2)-81*x^4-405*x^3)*log(5)^2+((9*x^3+18*x^2+9*x )*exp(2)^2+(-54*x^4-324*x^3-270*x^2)*exp(2)+81*x^5+810*x^4+2025*x^3)*log(5 )+(x^3+3*x^2+3*x+1)*exp(2)^3+(-9*x^4-63*x^3-99*x^2-45*x)*exp(2)^2+(27*x^5+ 297*x^4+945*x^3+675*x^2)*exp(2)-27*x^6-405*x^5-2025*x^4-3375*x^3),x, algor ithm=\
9*(x - 2)/(9*x^4 + 9*x^2*log(5)^2 + 90*x^3 + 225*x^2 + (x^2 + 2*x + 1)*e^4 - 6*(x^3 + 6*x^2 + 5*x)*e^2 - 6*(3*x^3 + 15*x^2 - (x^2 + x)*e^2)*log(5))
Leaf count of result is larger than twice the leaf count of optimal. 83 vs. \(2 (27) = 54\).
Time = 2.69 (sec) , antiderivative size = 83, normalized size of antiderivative = 2.44 \[ \int \frac {-540+e^2 (45-9 x)-81 x+81 x^2+(108-27 x) \log (5)}{-3375 x^3-2025 x^4-405 x^5-27 x^6+e^6 \left (1+3 x+3 x^2+x^3\right )+e^4 \left (-45 x-99 x^2-63 x^3-9 x^4\right )+e^2 \left (675 x^2+945 x^3+297 x^4+27 x^5\right )+\left (2025 x^3+810 x^4+81 x^5+e^4 \left (9 x+18 x^2+9 x^3\right )+e^2 \left (-270 x^2-324 x^3-54 x^4\right )\right ) \log (5)+\left (-405 x^3-81 x^4+e^2 \left (27 x^2+27 x^3\right )\right ) \log ^2(5)+27 x^3 \log ^3(5)} \, dx=- \frac {18 - 9 x}{9 x^{4} + x^{3} \left (- 6 e^{2} - 18 \log {\left (5 \right )} + 90\right ) + x^{2} \left (- 36 e^{2} - 90 \log {\left (5 \right )} + 9 \log {\left (5 \right )}^{2} + e^{4} + 6 e^{2} \log {\left (5 \right )} + 225\right ) + x \left (- 30 e^{2} + 6 e^{2} \log {\left (5 \right )} + 2 e^{4}\right ) + e^{4}} \]
integrate(((-27*x+108)*ln(5)+(-9*x+45)*exp(2)+81*x**2-81*x-540)/(27*x**3*l n(5)**3+((27*x**3+27*x**2)*exp(2)-81*x**4-405*x**3)*ln(5)**2+((9*x**3+18*x **2+9*x)*exp(2)**2+(-54*x**4-324*x**3-270*x**2)*exp(2)+81*x**5+810*x**4+20 25*x**3)*ln(5)+(x**3+3*x**2+3*x+1)*exp(2)**3+(-9*x**4-63*x**3-99*x**2-45*x )*exp(2)**2+(27*x**5+297*x**4+945*x**3+675*x**2)*exp(2)-27*x**6-405*x**5-2 025*x**4-3375*x**3),x)
-(18 - 9*x)/(9*x**4 + x**3*(-6*exp(2) - 18*log(5) + 90) + x**2*(-36*exp(2) - 90*log(5) + 9*log(5)**2 + exp(4) + 6*exp(2)*log(5) + 225) + x*(-30*exp( 2) + 6*exp(2)*log(5) + 2*exp(4)) + exp(4))
Leaf count of result is larger than twice the leaf count of optimal. 70 vs. \(2 (29) = 58\).
Time = 0.22 (sec) , antiderivative size = 70, normalized size of antiderivative = 2.06 \[ \int \frac {-540+e^2 (45-9 x)-81 x+81 x^2+(108-27 x) \log (5)}{-3375 x^3-2025 x^4-405 x^5-27 x^6+e^6 \left (1+3 x+3 x^2+x^3\right )+e^4 \left (-45 x-99 x^2-63 x^3-9 x^4\right )+e^2 \left (675 x^2+945 x^3+297 x^4+27 x^5\right )+\left (2025 x^3+810 x^4+81 x^5+e^4 \left (9 x+18 x^2+9 x^3\right )+e^2 \left (-270 x^2-324 x^3-54 x^4\right )\right ) \log (5)+\left (-405 x^3-81 x^4+e^2 \left (27 x^2+27 x^3\right )\right ) \log ^2(5)+27 x^3 \log ^3(5)} \, dx=\frac {9 \, {\left (x - 2\right )}}{9 \, x^{4} - 6 \, x^{3} {\left (e^{2} + 3 \, \log \left (5\right ) - 15\right )} + {\left (6 \, {\left (e^{2} - 15\right )} \log \left (5\right ) + 9 \, \log \left (5\right )^{2} + e^{4} - 36 \, e^{2} + 225\right )} x^{2} + 2 \, {\left (3 \, e^{2} \log \left (5\right ) + e^{4} - 15 \, e^{2}\right )} x + e^{4}} \]
integrate(((-27*x+108)*log(5)+(-9*x+45)*exp(2)+81*x^2-81*x-540)/(27*x^3*lo g(5)^3+((27*x^3+27*x^2)*exp(2)-81*x^4-405*x^3)*log(5)^2+((9*x^3+18*x^2+9*x )*exp(2)^2+(-54*x^4-324*x^3-270*x^2)*exp(2)+81*x^5+810*x^4+2025*x^3)*log(5 )+(x^3+3*x^2+3*x+1)*exp(2)^3+(-9*x^4-63*x^3-99*x^2-45*x)*exp(2)^2+(27*x^5+ 297*x^4+945*x^3+675*x^2)*exp(2)-27*x^6-405*x^5-2025*x^4-3375*x^3),x, algor ithm=\
9*(x - 2)/(9*x^4 - 6*x^3*(e^2 + 3*log(5) - 15) + (6*(e^2 - 15)*log(5) + 9* log(5)^2 + e^4 - 36*e^2 + 225)*x^2 + 2*(3*e^2*log(5) + e^4 - 15*e^2)*x + e ^4)
Time = 0.28 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.88 \[ \int \frac {-540+e^2 (45-9 x)-81 x+81 x^2+(108-27 x) \log (5)}{-3375 x^3-2025 x^4-405 x^5-27 x^6+e^6 \left (1+3 x+3 x^2+x^3\right )+e^4 \left (-45 x-99 x^2-63 x^3-9 x^4\right )+e^2 \left (675 x^2+945 x^3+297 x^4+27 x^5\right )+\left (2025 x^3+810 x^4+81 x^5+e^4 \left (9 x+18 x^2+9 x^3\right )+e^2 \left (-270 x^2-324 x^3-54 x^4\right )\right ) \log (5)+\left (-405 x^3-81 x^4+e^2 \left (27 x^2+27 x^3\right )\right ) \log ^2(5)+27 x^3 \log ^3(5)} \, dx=\frac {9 \, {\left (x - 2\right )}}{{\left (3 \, x^{2} - x e^{2} - 3 \, x \log \left (5\right ) + 15 \, x - e^{2}\right )}^{2}} \]
integrate(((-27*x+108)*log(5)+(-9*x+45)*exp(2)+81*x^2-81*x-540)/(27*x^3*lo g(5)^3+((27*x^3+27*x^2)*exp(2)-81*x^4-405*x^3)*log(5)^2+((9*x^3+18*x^2+9*x )*exp(2)^2+(-54*x^4-324*x^3-270*x^2)*exp(2)+81*x^5+810*x^4+2025*x^3)*log(5 )+(x^3+3*x^2+3*x+1)*exp(2)^3+(-9*x^4-63*x^3-99*x^2-45*x)*exp(2)^2+(27*x^5+ 297*x^4+945*x^3+675*x^2)*exp(2)-27*x^6-405*x^5-2025*x^4-3375*x^3),x, algor ithm=\
Time = 10.57 (sec) , antiderivative size = 525, normalized size of antiderivative = 15.44 \[ \int \frac {-540+e^2 (45-9 x)-81 x+81 x^2+(108-27 x) \log (5)}{-3375 x^3-2025 x^4-405 x^5-27 x^6+e^6 \left (1+3 x+3 x^2+x^3\right )+e^4 \left (-45 x-99 x^2-63 x^3-9 x^4\right )+e^2 \left (675 x^2+945 x^3+297 x^4+27 x^5\right )+\left (2025 x^3+810 x^4+81 x^5+e^4 \left (9 x+18 x^2+9 x^3\right )+e^2 \left (-270 x^2-324 x^3-54 x^4\right )\right ) \log (5)+\left (-405 x^3-81 x^4+e^2 \left (27 x^2+27 x^3\right )\right ) \log ^2(5)+27 x^3 \log ^3(5)} \, dx =\text {Too large to display} \]
int((81*x + log(5)*(27*x - 108) - 81*x^2 + exp(2)*(9*x - 45) + 540)/(log(5 )^2*(405*x^3 - exp(2)*(27*x^2 + 27*x^3) + 81*x^4) - 27*x^3*log(5)^3 - exp( 6)*(3*x + 3*x^2 + x^3 + 1) - log(5)*(exp(4)*(9*x + 18*x^2 + 9*x^3) - exp(2 )*(270*x^2 + 324*x^3 + 54*x^4) + 2025*x^3 + 810*x^4 + 81*x^5) + exp(4)*(45 *x + 99*x^2 + 63*x^3 + 9*x^4) + 3375*x^3 + 2025*x^4 + 405*x^5 + 27*x^6 - e xp(2)*(675*x^2 + 945*x^3 + 297*x^4 + 27*x^5)),x)
((x^2*(2187*log(125)*log(625) - 26244*log(5)^2))/(774*exp(4) - 8100*exp(2) - 36*exp(6) + exp(8) - 40500*log(5) + log(5^(12*exp(6) - 396*exp(4))) + 5 940*exp(2)*log(5) - 1404*exp(2)*log(5)^2 + 108*exp(2)*log(5)^3 + 54*exp(4) *log(5)^2 + 12150*log(5)^2 - 1620*log(5)^3 + 81*log(5)^4 + 50625) - ((9*lo g(5^(2700*exp(2) - 180*exp(4) + 48*exp(6) + 324*exp(2)*log(5)^2 + 324*log( 5)^3)))/2 - 145800*exp(2) + 13932*exp(4) - 648*exp(6) + 18*exp(8) - 546750 *log(5) - (91125*log(625))/2 + 94770*exp(2)*log(5) - 6318*exp(4)*log(5) + (54675*log(5)*log(625))/2 + (9*log(625)*log(5^(27*exp(4) - 540*exp(2))))/2 - 15552*exp(2)*log(5)^2 + 486*exp(2)*log(5)^3 + 486*exp(4)*log(5)^2 - (10 935*log(5)^2*log(625))/2 + 109350*log(5)^2 - 7290*log(5)^3 + 911250)/(log( 15625^exp(2)/8077935669463160887416100508495730991853633895516395568847656 25) - 18*exp(2) + exp(4) + 9*log(5)^2 + 225)^2 + (9*x*(774*exp(4) - 8100*e xp(2) - 36*exp(6) + exp(8) - 24300*log(5) - 4050*log(625) + log(5^(3240*ex p(2) - 396*exp(4) + 12*exp(6) - 648*log(5)^2)) + 2700*exp(2)*log(5) + 1620 *log(5)*log(625) + log(625)*log(1/5^(108*exp(2))) - 972*exp(2)*log(5)^2 + 108*exp(2)*log(5)^3 + 54*exp(4)*log(5)^2 + 5670*log(5)^2 - 972*log(5)^3 + 81*log(5)^4 + 50625))/(774*exp(4) - 8100*exp(2) - 36*exp(6) + exp(8) - 405 00*log(5) + log(5^(12*exp(6) - 396*exp(4))) + 5940*exp(2)*log(5) - 1404*ex p(2)*log(5)^2 + 108*exp(2)*log(5)^3 + 54*exp(4)*log(5)^2 + 12150*log(5)^2 - 1620*log(5)^3 + 81*log(5)^4 + 50625))/(exp(4) + x*(2*exp(4) - 30*exp(...