Integrand size = 252, antiderivative size = 31 \[ \int \frac {180+90 x^2+360 e x^2-90 e^6 x^2+30 x^3-90 e^2 x^3}{144+1152 x+2160 x^2+9 e^{20} x^2-600 x^3-60 x^4+12 x^5-144 e^3 x^5+x^6+9 e^4 x^6+e^{15} \left (-144 x^2+36 e x^3\right )+e^{10} \left (72 x+864 x^2-36 x^3-432 e x^3-6 x^4+54 e^2 x^4\right )+e \left (-576 x^2-2304 x^3+288 x^4+48 x^5\right )+e^2 \left (72 x^3+864 x^4-36 x^5-6 x^6\right )+e^5 \left (-576 x-2304 x^2+288 x^3+48 x^4-432 e^2 x^4+36 e^3 x^5+e \left (144 x^2+1728 x^3-72 x^4-12 x^5\right )\right )} \, dx=\frac {5}{-2 x+\left (-4+e^5+e x\right )^2+\frac {4-\frac {x^3}{3}}{x}} \]
Time = 0.07 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.32 \[ \int \frac {180+90 x^2+360 e x^2-90 e^6 x^2+30 x^3-90 e^2 x^3}{144+1152 x+2160 x^2+9 e^{20} x^2-600 x^3-60 x^4+12 x^5-144 e^3 x^5+x^6+9 e^4 x^6+e^{15} \left (-144 x^2+36 e x^3\right )+e^{10} \left (72 x+864 x^2-36 x^3-432 e x^3-6 x^4+54 e^2 x^4\right )+e \left (-576 x^2-2304 x^3+288 x^4+48 x^5\right )+e^2 \left (72 x^3+864 x^4-36 x^5-6 x^6\right )+e^5 \left (-576 x-2304 x^2+288 x^3+48 x^4-432 e^2 x^4+36 e^3 x^5+e \left (144 x^2+1728 x^3-72 x^4-12 x^5\right )\right )} \, dx=\frac {15 x}{12+3 \left (-4+e^5\right )^2 x+6 \left (-1-4 e+e^6\right ) x^2+\left (-1+3 e^2\right ) x^3} \]
Integrate[(180 + 90*x^2 + 360*E*x^2 - 90*E^6*x^2 + 30*x^3 - 90*E^2*x^3)/(1 44 + 1152*x + 2160*x^2 + 9*E^20*x^2 - 600*x^3 - 60*x^4 + 12*x^5 - 144*E^3* x^5 + x^6 + 9*E^4*x^6 + E^15*(-144*x^2 + 36*E*x^3) + E^10*(72*x + 864*x^2 - 36*x^3 - 432*E*x^3 - 6*x^4 + 54*E^2*x^4) + E*(-576*x^2 - 2304*x^3 + 288* x^4 + 48*x^5) + E^2*(72*x^3 + 864*x^4 - 36*x^5 - 6*x^6) + E^5*(-576*x - 23 04*x^2 + 288*x^3 + 48*x^4 - 432*E^2*x^4 + 36*E^3*x^5 + E*(144*x^2 + 1728*x ^3 - 72*x^4 - 12*x^5))),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.
| \(\displaystyle \int \frac {-90 e^2 x^3+30 x^3-90 e^6 x^2+360 e x^2+90 x^2+180}{9 e^4 x^6+x^6-144 e^3 x^5+12 x^5-60 x^4-600 x^3+9 e^{20} x^2+2160 x^2+e^{15} \left (36 e x^3-144 x^2\right )+e^{10} \left (54 e^2 x^4-6 x^4-432 e x^3-36 x^3+864 x^2+72 x\right )+e^2 \left (-6 x^6-36 x^5+864 x^4+72 x^3\right )+e \left (48 x^5+288 x^4-2304 x^3-576 x^2\right )+e^5 \left (36 e^3 x^5-432 e^2 x^4+48 x^4+288 x^3-2304 x^2+e \left (-12 x^5-72 x^4+1728 x^3+144 x^2\right )-576 x\right )+1152 x+144} \, dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {-90 e^2 x^3+30 x^3+(90+360 e) x^2-90 e^6 x^2+180}{9 e^4 x^6+x^6-144 e^3 x^5+12 x^5-60 x^4-600 x^3+9 e^{20} x^2+2160 x^2+e^{15} \left (36 e x^3-144 x^2\right )+e^{10} \left (54 e^2 x^4-6 x^4-432 e x^3-36 x^3+864 x^2+72 x\right )+e^2 \left (-6 x^6-36 x^5+864 x^4+72 x^3\right )+e \left (48 x^5+288 x^4-2304 x^3-576 x^2\right )+e^5 \left (36 e^3 x^5-432 e^2 x^4+48 x^4+288 x^3-2304 x^2+e \left (-12 x^5-72 x^4+1728 x^3+144 x^2\right )-576 x\right )+1152 x+144}dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {-90 e^2 x^3+30 x^3+\left (90+360 e-90 e^6\right ) x^2+180}{9 e^4 x^6+x^6-144 e^3 x^5+12 x^5-60 x^4-600 x^3+9 e^{20} x^2+2160 x^2+e^{15} \left (36 e x^3-144 x^2\right )+e^{10} \left (54 e^2 x^4-6 x^4-432 e x^3-36 x^3+864 x^2+72 x\right )+e^2 \left (-6 x^6-36 x^5+864 x^4+72 x^3\right )+e \left (48 x^5+288 x^4-2304 x^3-576 x^2\right )+e^5 \left (36 e^3 x^5-432 e^2 x^4+48 x^4+288 x^3-2304 x^2+e \left (-12 x^5-72 x^4+1728 x^3+144 x^2\right )-576 x\right )+1152 x+144}dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {\left (30-90 e^2\right ) x^3+\left (90+360 e-90 e^6\right ) x^2+180}{9 e^4 x^6+x^6-144 e^3 x^5+12 x^5-60 x^4-600 x^3+9 e^{20} x^2+2160 x^2+e^{15} \left (36 e x^3-144 x^2\right )+e^{10} \left (54 e^2 x^4-6 x^4-432 e x^3-36 x^3+864 x^2+72 x\right )+e^2 \left (-6 x^6-36 x^5+864 x^4+72 x^3\right )+e \left (48 x^5+288 x^4-2304 x^3-576 x^2\right )+e^5 \left (36 e^3 x^5-432 e^2 x^4+48 x^4+288 x^3-2304 x^2+e \left (-12 x^5-72 x^4+1728 x^3+144 x^2\right )-576 x\right )+1152 x+144}dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {\left (30-90 e^2\right ) x^3+\left (90+360 e-90 e^6\right ) x^2+180}{9 e^4 x^6+x^6-144 e^3 x^5+12 x^5-60 x^4-600 x^3+\left (2160+9 e^{20}\right ) x^2+e^{15} \left (36 e x^3-144 x^2\right )+e^{10} \left (54 e^2 x^4-6 x^4-432 e x^3-36 x^3+864 x^2+72 x\right )+e^2 \left (-6 x^6-36 x^5+864 x^4+72 x^3\right )+e \left (48 x^5+288 x^4-2304 x^3-576 x^2\right )+e^5 \left (36 e^3 x^5-432 e^2 x^4+48 x^4+288 x^3-2304 x^2+e \left (-12 x^5-72 x^4+1728 x^3+144 x^2\right )-576 x\right )+1152 x+144}dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {\left (30-90 e^2\right ) x^3+\left (90+360 e-90 e^6\right ) x^2+180}{9 e^4 x^6+x^6+\left (12-144 e^3\right ) x^5-60 x^4-600 x^3+\left (2160+9 e^{20}\right ) x^2+e^{15} \left (36 e x^3-144 x^2\right )+e^{10} \left (54 e^2 x^4-6 x^4-432 e x^3-36 x^3+864 x^2+72 x\right )+e^2 \left (-6 x^6-36 x^5+864 x^4+72 x^3\right )+e \left (48 x^5+288 x^4-2304 x^3-576 x^2\right )+e^5 \left (36 e^3 x^5-432 e^2 x^4+48 x^4+288 x^3-2304 x^2+e \left (-12 x^5-72 x^4+1728 x^3+144 x^2\right )-576 x\right )+1152 x+144}dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {\left (30-90 e^2\right ) x^3+\left (90+360 e-90 e^6\right ) x^2+180}{\left (1+9 e^4\right ) x^6+\left (12-144 e^3\right ) x^5-60 x^4-600 x^3+\left (2160+9 e^{20}\right ) x^2+e^{15} \left (36 e x^3-144 x^2\right )+e^{10} \left (54 e^2 x^4-6 x^4-432 e x^3-36 x^3+864 x^2+72 x\right )+e^2 \left (-6 x^6-36 x^5+864 x^4+72 x^3\right )+e \left (48 x^5+288 x^4-2304 x^3-576 x^2\right )+e^5 \left (36 e^3 x^5-432 e^2 x^4+48 x^4+288 x^3-2304 x^2+e \left (-12 x^5-72 x^4+1728 x^3+144 x^2\right )-576 x\right )+1152 x+144}dx\) |
| \(\Big \downarrow \) 2462 |
| \(\displaystyle \int \left (\frac {90 \left (-\left (\left (1+4 e-e^6\right ) x^2\right )+\left (4-e^5\right )^2 x+6\right )}{\left (-\left (\left (1-3 e^2\right ) x^3\right )-6 \left (1+4 e-e^6\right ) x^2+3 \left (4-e^5\right )^2 x+12\right )^2}+\frac {30}{\left (1-3 e^2\right ) x^3+6 \left (1+4 e-e^6\right ) x^2-3 \left (4-e^5\right )^2 x-12}\right )dx\) |
| \(\Big \downarrow \) 7299 |
| \(\displaystyle \int \left (\frac {90 \left (-\left (\left (1+4 e-e^6\right ) x^2\right )+\left (4-e^5\right )^2 x+6\right )}{\left (-\left (\left (1-3 e^2\right ) x^3\right )-6 \left (1+4 e-e^6\right ) x^2+3 \left (4-e^5\right )^2 x+12\right )^2}+\frac {30}{\left (1-3 e^2\right ) x^3+6 \left (1+4 e-e^6\right ) x^2-3 \left (4-e^5\right )^2 x-12}\right )dx\) |
Int[(180 + 90*x^2 + 360*E*x^2 - 90*E^6*x^2 + 30*x^3 - 90*E^2*x^3)/(144 + 1 152*x + 2160*x^2 + 9*E^20*x^2 - 600*x^3 - 60*x^4 + 12*x^5 - 144*E^3*x^5 + x^6 + 9*E^4*x^6 + E^15*(-144*x^2 + 36*E*x^3) + E^10*(72*x + 864*x^2 - 36*x ^3 - 432*E*x^3 - 6*x^4 + 54*E^2*x^4) + E*(-576*x^2 - 2304*x^3 + 288*x^4 + 48*x^5) + E^2*(72*x^3 + 864*x^4 - 36*x^5 - 6*x^6) + E^5*(-576*x - 2304*x^2 + 288*x^3 + 48*x^4 - 432*E^2*x^4 + 36*E^3*x^5 + E*(144*x^2 + 1728*x^3 - 7 2*x^4 - 12*x^5))),x]
3.3.16.3.1 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 = 1.00 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.61
| method | result | size |
| risch | \(\frac {5 x}{x \,{\mathrm e}^{10}+2 x^{2} {\mathrm e}^{6}-8 x \,{\mathrm e}^{5}+x^{3} {\mathrm e}^{2}-8 x^{2} {\mathrm e}-\frac {x^{3}}{3}-2 x^{2}+16 x +4}\) | \(50\) |
| gosper | \(\frac {15 x}{3 x^{3} {\mathrm e}^{2}+6 x^{2} {\mathrm e} \,{\mathrm e}^{5}+3 x \,{\mathrm e}^{10}-24 x^{2} {\mathrm e}-x^{3}-24 x \,{\mathrm e}^{5}-6 x^{2}+48 x +12}\) | \(58\) |
| norman | \(\frac {15 x}{3 x^{3} {\mathrm e}^{2}+6 x^{2} {\mathrm e} \,{\mathrm e}^{5}+3 x \,{\mathrm e}^{10}-24 x^{2} {\mathrm e}-x^{3}-24 x \,{\mathrm e}^{5}-6 x^{2}+48 x +12}\) | \(58\) |
| parallelrisch | \(\frac {15 x}{3 x^{3} {\mathrm e}^{2}+6 x^{2} {\mathrm e} \,{\mathrm e}^{5}+3 x \,{\mathrm e}^{10}-24 x^{2} {\mathrm e}-x^{3}-24 x \,{\mathrm e}^{5}-6 x^{2}+48 x +12}\) | \(58\) |
| default | \(-5 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (-144+\left (6 \,{\mathrm e}^{2}-9 \,{\mathrm e}^{4}-1\right ) \textit {\_Z}^{6}+\left (144 \,{\mathrm e}^{3}+36 \,{\mathrm e}^{2}-48 \,{\mathrm e}+12 \,{\mathrm e}^{6}-36 \,{\mathrm e}^{8}-12\right ) \textit {\_Z}^{5}+\left (-48 \,{\mathrm e}^{5}-864 \,{\mathrm e}^{2}-288 \,{\mathrm e}+6 \,{\mathrm e}^{10}-54 \,{\mathrm e}^{12}+72 \,{\mathrm e}^{6}+432 \,{\mathrm e}^{7}+60\right ) \textit {\_Z}^{4}+\left (-288 \,{\mathrm e}^{5}-72 \,{\mathrm e}^{2}+2304 \,{\mathrm e}+36 \,{\mathrm e}^{10}-1728 \,{\mathrm e}^{6}-36 \,{\mathrm e}^{16}+432 \,{\mathrm e}^{11}+600\right ) \textit {\_Z}^{3}+\left (2304 \,{\mathrm e}^{5}+576 \,{\mathrm e}-864 \,{\mathrm e}^{10}-144 \,{\mathrm e}^{6}-9 \,{\mathrm e}^{20}+144 \,{\mathrm e}^{15}-2160\right ) \textit {\_Z}^{2}+\left (576 \,{\mathrm e}^{5}-72 \,{\mathrm e}^{10}-1152\right ) \textit {\_Z} \right )}{\sum }\frac {\left (6+\left (-3 \,{\mathrm e}^{2}+1\right ) \textit {\_R}^{3}+3 \left (1+4 \,{\mathrm e}-{\mathrm e}^{6}\right ) \textit {\_R}^{2}\right ) \ln \left (x -\textit {\_R} \right )}{-192+1152 \,{\mathrm e} \textit {\_R}^{2}+192 \textit {\_R} \,{\mathrm e}+10 \textit {\_R}^{4} {\mathrm e}^{6}+48 \textit {\_R}^{3} {\mathrm e}^{6}+120 \textit {\_R}^{4} {\mathrm e}^{3}+18 \textit {\_R}^{2} {\mathrm e}^{10}+216 \textit {\_R}^{2} {\mathrm e}^{11}+288 \textit {\_R}^{3} {\mathrm e}^{7}-18 \textit {\_R}^{2} {\mathrm e}^{16}+48 \textit {\_R} \,{\mathrm e}^{15}-192 \textit {\_R}^{3} {\mathrm e}-32 \textit {\_R}^{3} {\mathrm e}^{5}+30 \textit {\_R}^{4} {\mathrm e}^{2}-30 \textit {\_R}^{4} {\mathrm e}^{8}-3 \textit {\_R} \,{\mathrm e}^{20}+4 \textit {\_R}^{3} {\mathrm e}^{10}-9 \textit {\_R}^{5} {\mathrm e}^{4}+6 \,{\mathrm e}^{2} \textit {\_R}^{5}+40 \textit {\_R}^{3}+300 \textit {\_R}^{2}-\textit {\_R}^{5}-10 \textit {\_R}^{4}-144 \textit {\_R}^{2} {\mathrm e}^{5}-288 \textit {\_R} \,{\mathrm e}^{10}+768 \textit {\_R} \,{\mathrm e}^{5}-36 \,{\mathrm e}^{12} \textit {\_R}^{3}-720 \textit {\_R} -36 \textit {\_R}^{2} {\mathrm e}^{2}-12 \,{\mathrm e}^{10}+96 \,{\mathrm e}^{5}-576 \textit {\_R}^{3} {\mathrm e}^{2}-40 \textit {\_R}^{4} {\mathrm e}-864 \textit {\_R}^{2} {\mathrm e}^{6}-48 \textit {\_R} \,{\mathrm e}^{6}}\right )\) | \(405\) |
int((-90*x^2*exp(1)*exp(5)-90*x^3*exp(1)^2+360*x^2*exp(1)+30*x^3+90*x^2+18 0)/(9*x^2*exp(5)^4+(36*x^3*exp(1)-144*x^2)*exp(5)^3+(54*x^4*exp(1)^2-432*x ^3*exp(1)-6*x^4-36*x^3+864*x^2+72*x)*exp(5)^2+(36*x^5*exp(1)^3-432*x^4*exp (1)^2+(-12*x^5-72*x^4+1728*x^3+144*x^2)*exp(1)+48*x^4+288*x^3-2304*x^2-576 *x)*exp(5)+9*x^6*exp(1)^4-144*x^5*exp(1)^3+(-6*x^6-36*x^5+864*x^4+72*x^3)* exp(1)^2+(48*x^5+288*x^4-2304*x^3-576*x^2)*exp(1)+x^6+12*x^5-60*x^4-600*x^ 3+2160*x^2+1152*x+144),x,method=_RETURNVERBOSE)
Time = 0.26 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.65 \[ \int \frac {180+90 x^2+360 e x^2-90 e^6 x^2+30 x^3-90 e^2 x^3}{144+1152 x+2160 x^2+9 e^{20} x^2-600 x^3-60 x^4+12 x^5-144 e^3 x^5+x^6+9 e^4 x^6+e^{15} \left (-144 x^2+36 e x^3\right )+e^{10} \left (72 x+864 x^2-36 x^3-432 e x^3-6 x^4+54 e^2 x^4\right )+e \left (-576 x^2-2304 x^3+288 x^4+48 x^5\right )+e^2 \left (72 x^3+864 x^4-36 x^5-6 x^6\right )+e^5 \left (-576 x-2304 x^2+288 x^3+48 x^4-432 e^2 x^4+36 e^3 x^5+e \left (144 x^2+1728 x^3-72 x^4-12 x^5\right )\right )} \, dx=\frac {15 \, x}{3 \, x^{3} e^{2} - x^{3} + 6 \, x^{2} e^{6} - 24 \, x^{2} e - 6 \, x^{2} + 3 \, x e^{10} - 24 \, x e^{5} + 48 \, x + 12} \]
integrate((-90*x^2*exp(1)*exp(5)-90*x^3*exp(1)^2+360*x^2*exp(1)+30*x^3+90* x^2+180)/(9*x^2*exp(5)^4+(36*x^3*exp(1)-144*x^2)*exp(5)^3+(54*x^4*exp(1)^2 -432*x^3*exp(1)-6*x^4-36*x^3+864*x^2+72*x)*exp(5)^2+(36*x^5*exp(1)^3-432*x ^4*exp(1)^2+(-12*x^5-72*x^4+1728*x^3+144*x^2)*exp(1)+48*x^4+288*x^3-2304*x ^2-576*x)*exp(5)+9*x^6*exp(1)^4-144*x^5*exp(1)^3+(-6*x^6-36*x^5+864*x^4+72 *x^3)*exp(1)^2+(48*x^5+288*x^4-2304*x^3-576*x^2)*exp(1)+x^6+12*x^5-60*x^4- 600*x^3+2160*x^2+1152*x+144),x, algorithm=\
Time = 12.02 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.35 \[ \int \frac {180+90 x^2+360 e x^2-90 e^6 x^2+30 x^3-90 e^2 x^3}{144+1152 x+2160 x^2+9 e^{20} x^2-600 x^3-60 x^4+12 x^5-144 e^3 x^5+x^6+9 e^4 x^6+e^{15} \left (-144 x^2+36 e x^3\right )+e^{10} \left (72 x+864 x^2-36 x^3-432 e x^3-6 x^4+54 e^2 x^4\right )+e \left (-576 x^2-2304 x^3+288 x^4+48 x^5\right )+e^2 \left (72 x^3+864 x^4-36 x^5-6 x^6\right )+e^5 \left (-576 x-2304 x^2+288 x^3+48 x^4-432 e^2 x^4+36 e^3 x^5+e \left (144 x^2+1728 x^3-72 x^4-12 x^5\right )\right )} \, dx=\frac {15 x}{x^{3} \left (-1 + 3 e^{2}\right ) + x^{2} \left (- 24 e - 6 + 6 e^{6}\right ) + x \left (- 24 e^{5} + 48 + 3 e^{10}\right ) + 12} \]
integrate((-90*x**2*exp(1)*exp(5)-90*x**3*exp(1)**2+360*x**2*exp(1)+30*x** 3+90*x**2+180)/(9*x**2*exp(5)**4+(36*x**3*exp(1)-144*x**2)*exp(5)**3+(54*x **4*exp(1)**2-432*x**3*exp(1)-6*x**4-36*x**3+864*x**2+72*x)*exp(5)**2+(36* x**5*exp(1)**3-432*x**4*exp(1)**2+(-12*x**5-72*x**4+1728*x**3+144*x**2)*ex p(1)+48*x**4+288*x**3-2304*x**2-576*x)*exp(5)+9*x**6*exp(1)**4-144*x**5*ex p(1)**3+(-6*x**6-36*x**5+864*x**4+72*x**3)*exp(1)**2+(48*x**5+288*x**4-230 4*x**3-576*x**2)*exp(1)+x**6+12*x**5-60*x**4-600*x**3+2160*x**2+1152*x+144 ),x)
15*x/(x**3*(-1 + 3*exp(2)) + x**2*(-24*E - 6 + 6*exp(6)) + x*(-24*exp(5) + 48 + 3*exp(10)) + 12)
Time = 0.20 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.32 \[ \int \frac {180+90 x^2+360 e x^2-90 e^6 x^2+30 x^3-90 e^2 x^3}{144+1152 x+2160 x^2+9 e^{20} x^2-600 x^3-60 x^4+12 x^5-144 e^3 x^5+x^6+9 e^4 x^6+e^{15} \left (-144 x^2+36 e x^3\right )+e^{10} \left (72 x+864 x^2-36 x^3-432 e x^3-6 x^4+54 e^2 x^4\right )+e \left (-576 x^2-2304 x^3+288 x^4+48 x^5\right )+e^2 \left (72 x^3+864 x^4-36 x^5-6 x^6\right )+e^5 \left (-576 x-2304 x^2+288 x^3+48 x^4-432 e^2 x^4+36 e^3 x^5+e \left (144 x^2+1728 x^3-72 x^4-12 x^5\right )\right )} \, dx=\frac {15 \, x}{x^{3} {\left (3 \, e^{2} - 1\right )} + 6 \, x^{2} {\left (e^{6} - 4 \, e - 1\right )} + 3 \, x {\left (e^{10} - 8 \, e^{5} + 16\right )} + 12} \]
integrate((-90*x^2*exp(1)*exp(5)-90*x^3*exp(1)^2+360*x^2*exp(1)+30*x^3+90* x^2+180)/(9*x^2*exp(5)^4+(36*x^3*exp(1)-144*x^2)*exp(5)^3+(54*x^4*exp(1)^2 -432*x^3*exp(1)-6*x^4-36*x^3+864*x^2+72*x)*exp(5)^2+(36*x^5*exp(1)^3-432*x ^4*exp(1)^2+(-12*x^5-72*x^4+1728*x^3+144*x^2)*exp(1)+48*x^4+288*x^3-2304*x ^2-576*x)*exp(5)+9*x^6*exp(1)^4-144*x^5*exp(1)^3+(-6*x^6-36*x^5+864*x^4+72 *x^3)*exp(1)^2+(48*x^5+288*x^4-2304*x^3-576*x^2)*exp(1)+x^6+12*x^5-60*x^4- 600*x^3+2160*x^2+1152*x+144),x, algorithm=\
Time = 0.30 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.65 \[ \int \frac {180+90 x^2+360 e x^2-90 e^6 x^2+30 x^3-90 e^2 x^3}{144+1152 x+2160 x^2+9 e^{20} x^2-600 x^3-60 x^4+12 x^5-144 e^3 x^5+x^6+9 e^4 x^6+e^{15} \left (-144 x^2+36 e x^3\right )+e^{10} \left (72 x+864 x^2-36 x^3-432 e x^3-6 x^4+54 e^2 x^4\right )+e \left (-576 x^2-2304 x^3+288 x^4+48 x^5\right )+e^2 \left (72 x^3+864 x^4-36 x^5-6 x^6\right )+e^5 \left (-576 x-2304 x^2+288 x^3+48 x^4-432 e^2 x^4+36 e^3 x^5+e \left (144 x^2+1728 x^3-72 x^4-12 x^5\right )\right )} \, dx=\frac {15 \, x}{3 \, x^{3} e^{2} - x^{3} + 6 \, x^{2} e^{6} - 24 \, x^{2} e - 6 \, x^{2} + 3 \, x e^{10} - 24 \, x e^{5} + 48 \, x + 12} \]
integrate((-90*x^2*exp(1)*exp(5)-90*x^3*exp(1)^2+360*x^2*exp(1)+30*x^3+90* x^2+180)/(9*x^2*exp(5)^4+(36*x^3*exp(1)-144*x^2)*exp(5)^3+(54*x^4*exp(1)^2 -432*x^3*exp(1)-6*x^4-36*x^3+864*x^2+72*x)*exp(5)^2+(36*x^5*exp(1)^3-432*x ^4*exp(1)^2+(-12*x^5-72*x^4+1728*x^3+144*x^2)*exp(1)+48*x^4+288*x^3-2304*x ^2-576*x)*exp(5)+9*x^6*exp(1)^4-144*x^5*exp(1)^3+(-6*x^6-36*x^5+864*x^4+72 *x^3)*exp(1)^2+(48*x^5+288*x^4-2304*x^3-576*x^2)*exp(1)+x^6+12*x^5-60*x^4- 600*x^3+2160*x^2+1152*x+144),x, algorithm=\
Time = 0.60 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.42 \[ \int \frac {180+90 x^2+360 e x^2-90 e^6 x^2+30 x^3-90 e^2 x^3}{144+1152 x+2160 x^2+9 e^{20} x^2-600 x^3-60 x^4+12 x^5-144 e^3 x^5+x^6+9 e^4 x^6+e^{15} \left (-144 x^2+36 e x^3\right )+e^{10} \left (72 x+864 x^2-36 x^3-432 e x^3-6 x^4+54 e^2 x^4\right )+e \left (-576 x^2-2304 x^3+288 x^4+48 x^5\right )+e^2 \left (72 x^3+864 x^4-36 x^5-6 x^6\right )+e^5 \left (-576 x-2304 x^2+288 x^3+48 x^4-432 e^2 x^4+36 e^3 x^5+e \left (144 x^2+1728 x^3-72 x^4-12 x^5\right )\right )} \, dx=\frac {15\,x}{\left (3\,{\mathrm {e}}^2-1\right )\,x^3+\left (6\,{\mathrm {e}}^6-24\,\mathrm {e}-6\right )\,x^2+\left (3\,{\mathrm {e}}^{10}-24\,{\mathrm {e}}^5+48\right )\,x+12} \]
int((360*x^2*exp(1) - 90*x^3*exp(2) - 90*x^2*exp(6) + 90*x^2 + 30*x^3 + 18 0)/(1152*x - 144*x^5*exp(3) + 9*x^6*exp(4) + 9*x^2*exp(20) + 2160*x^2 - 60 0*x^3 - 60*x^4 + 12*x^5 + x^6 + exp(5)*(36*x^5*exp(3) - 432*x^4*exp(2) - 5 76*x - 2304*x^2 + 288*x^3 + 48*x^4 + exp(1)*(144*x^2 + 1728*x^3 - 72*x^4 - 12*x^5)) + exp(10)*(72*x - 432*x^3*exp(1) + 54*x^4*exp(2) + 864*x^2 - 36* x^3 - 6*x^4) + exp(15)*(36*x^3*exp(1) - 144*x^2) + exp(2)*(72*x^3 + 864*x^ 4 - 36*x^5 - 6*x^6) - exp(1)*(576*x^2 + 2304*x^3 - 288*x^4 - 48*x^5) + 144 ),x)