Integrand size = 102, antiderivative size = 25 \[ \int \frac {\left (\frac {x^2}{e^3+x}\right )^{\frac {1}{36} \left (900+132 x+x^2\right )} \left (900 x+132 x^2+x^3+e^3 \left (1800+264 x+2 x^2\right )+\left (132 x^2+2 x^3+e^3 \left (132 x+2 x^2\right )\right ) \log \left (\frac {x^2}{e^3+x}\right )\right )}{36 e^3 x+36 x^2} \, dx=\left (\frac {x^2}{e^3+x}\right )^{\left (5+\frac {x}{6}\right )^2+2 x} \]
Time = 5.16 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.24 \[ \int \frac {\left (\frac {x^2}{e^3+x}\right )^{\frac {1}{36} \left (900+132 x+x^2\right )} \left (900 x+132 x^2+x^3+e^3 \left (1800+264 x+2 x^2\right )+\left (132 x^2+2 x^3+e^3 \left (132 x+2 x^2\right )\right ) \log \left (\frac {x^2}{e^3+x}\right )\right )}{36 e^3 x+36 x^2} \, dx=\frac {x^{50} \left (\frac {x^2}{e^3+x}\right )^{\frac {1}{36} x (132+x)}}{\left (e^3+x\right )^{25}} \]
Integrate[((x^2/(E^3 + x))^((900 + 132*x + x^2)/36)*(900*x + 132*x^2 + x^3 + E^3*(1800 + 264*x + 2*x^2) + (132*x^2 + 2*x^3 + E^3*(132*x + 2*x^2))*Lo g[x^2/(E^3 + x)]))/(36*E^3*x + 36*x^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.
| \(\displaystyle \int \frac {\left (\frac {x^2}{x+e^3}\right )^{\frac {1}{36} \left (x^2+132 x+900\right )} \left (x^3+132 x^2+e^3 \left (2 x^2+264 x+1800\right )+\left (2 x^3+132 x^2+e^3 \left (2 x^2+132 x\right )\right ) \log \left (\frac {x^2}{x+e^3}\right )+900 x\right )}{36 x^2+36 e^3 x} \, dx\) |
| \(\Big \downarrow \) 2026 |
| \(\displaystyle \int \frac {\left (\frac {x^2}{x+e^3}\right )^{\frac {1}{36} \left (x^2+132 x+900\right )} \left (x^3+132 x^2+e^3 \left (2 x^2+264 x+1800\right )+\left (2 x^3+132 x^2+e^3 \left (2 x^2+132 x\right )\right ) \log \left (\frac {x^2}{x+e^3}\right )+900 x\right )}{x \left (36 x+36 e^3\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {\left (x+2 e^3\right ) \left (x^2+132 x+900\right ) \left (\frac {x^2}{x+e^3}\right )^{\frac {1}{36} \left (x^2+132 x+900\right )}}{36 x \left (x+e^3\right )}+\frac {1}{18} (x+66) \left (\frac {x^2}{x+e^3}\right )^{\frac {1}{36} \left (x^2+132 x+900\right )} \log \left (\frac {x^2}{x+e^3}\right )\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{36} \left (132+e^3\right ) \int \left (\frac {x^2}{x+e^3}\right )^{\frac {1}{36} \left (x^2+132 x+900\right )}dx+50 \int \frac {\left (\frac {x^2}{x+e^3}\right )^{\frac {1}{36} \left (x^2+132 x+900\right )}}{x}dx+\frac {1}{36} \int x \left (\frac {x^2}{x+e^3}\right )^{\frac {1}{36} \left (x^2+132 x+900\right )}dx-\frac {1}{36} \left (900-132 e^3+e^6\right ) \int \frac {\left (\frac {x^2}{x+e^3}\right )^{\frac {1}{36} \left (x^2+132 x+900\right )}}{x+e^3}dx-\frac {22}{3} \int \frac {\int \left (\frac {x^2}{x+e^3}\right )^{\frac {1}{36} \left (x^2+132 x+900\right )}dx}{x}dx+\frac {11}{3} \int \frac {\int \left (\frac {x^2}{x+e^3}\right )^{\frac {1}{36} \left (x^2+132 x+900\right )}dx}{x+e^3}dx-\frac {1}{9} \int \frac {\int x \left (\frac {x^2}{x+e^3}\right )^{\frac {1}{36} \left (x^2+132 x+900\right )}dx}{x}dx+\frac {1}{18} \int \frac {\int x \left (\frac {x^2}{x+e^3}\right )^{\frac {1}{36} \left (x^2+132 x+900\right )}dx}{x+e^3}dx+\frac {11}{3} \log \left (\frac {x^2}{x+e^3}\right ) \int \left (\frac {x^2}{x+e^3}\right )^{\frac {1}{36} \left (x^2+132 x+900\right )}dx+\frac {1}{18} \log \left (\frac {x^2}{x+e^3}\right ) \int x \left (\frac {x^2}{x+e^3}\right )^{\frac {1}{36} \left (x^2+132 x+900\right )}dx\) |
Int[((x^2/(E^3 + x))^((900 + 132*x + x^2)/36)*(900*x + 132*x^2 + x^3 + E^3 *(1800 + 264*x + 2*x^2) + (132*x^2 + 2*x^3 + E^3*(132*x + 2*x^2))*Log[x^2/ (E^3 + x)]))/(36*E^3*x + 36*x^2),x]
3.11.22.3.1 Defintions of rubi rules used
Int[(Fx_.)*(Px_)^(p_.), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Int[x^(p *r)*ExpandToSum[Px/x^r, x]^p*Fx, x] /; IGtQ[r, 0]] /; PolyQ[Px, x] && Integ erQ[p] && !MonomialQ[Px, x] && (ILtQ[p, 0] || !PolyQ[u, x])
Time = 1.20 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.88
| method | result | size |
| risch | \(\left (\frac {x^{2}}{{\mathrm e}^{3}+x}\right )^{\frac {1}{36} x^{2}+\frac {11}{3} x +25}\) | \(22\) |
| norman | \({\mathrm e}^{\frac {\left (x^{2}+132 x +900\right ) \ln \left (\frac {x^{2}}{{\mathrm e}^{3}+x}\right )}{36}}\) | \(23\) |
| parallelrisch | \({\mathrm e}^{\frac {\left (x^{2}+132 x +900\right ) \ln \left (\frac {x^{2}}{{\mathrm e}^{3}+x}\right )}{36}}\) | \(23\) |
int((((2*x^2+132*x)*exp(3)+2*x^3+132*x^2)*ln(x^2/(exp(3)+x))+(2*x^2+264*x+ 1800)*exp(3)+x^3+132*x^2+900*x)*exp(1/36*(x^2+132*x+900)*ln(x^2/(exp(3)+x) ))/(36*x*exp(3)+36*x^2),x,method=_RETURNVERBOSE)
Time = 0.25 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.84 \[ \int \frac {\left (\frac {x^2}{e^3+x}\right )^{\frac {1}{36} \left (900+132 x+x^2\right )} \left (900 x+132 x^2+x^3+e^3 \left (1800+264 x+2 x^2\right )+\left (132 x^2+2 x^3+e^3 \left (132 x+2 x^2\right )\right ) \log \left (\frac {x^2}{e^3+x}\right )\right )}{36 e^3 x+36 x^2} \, dx=\left (\frac {x^{2}}{x + e^{3}}\right )^{\frac {1}{36} \, x^{2} + \frac {11}{3} \, x + 25} \]
integrate((((2*x^2+132*x)*exp(3)+2*x^3+132*x^2)*log(x^2/(exp(3)+x))+(2*x^2 +264*x+1800)*exp(3)+x^3+132*x^2+900*x)*exp(1/36*(x^2+132*x+900)*log(x^2/(e xp(3)+x)))/(36*x*exp(3)+36*x^2),x, algorithm=\
Time = 0.35 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.88 \[ \int \frac {\left (\frac {x^2}{e^3+x}\right )^{\frac {1}{36} \left (900+132 x+x^2\right )} \left (900 x+132 x^2+x^3+e^3 \left (1800+264 x+2 x^2\right )+\left (132 x^2+2 x^3+e^3 \left (132 x+2 x^2\right )\right ) \log \left (\frac {x^2}{e^3+x}\right )\right )}{36 e^3 x+36 x^2} \, dx=e^{\left (\frac {x^{2}}{36} + \frac {11 x}{3} + 25\right ) \log {\left (\frac {x^{2}}{x + e^{3}} \right )}} \]
integrate((((2*x**2+132*x)*exp(3)+2*x**3+132*x**2)*ln(x**2/(exp(3)+x))+(2* x**2+264*x+1800)*exp(3)+x**3+132*x**2+900*x)*exp(1/36*(x**2+132*x+900)*ln( x**2/(exp(3)+x)))/(36*x*exp(3)+36*x**2),x)
Leaf count of result is larger than twice the leaf count of optimal. 210 vs. \(2 (22) = 44\).
Time = 0.41 (sec) , antiderivative size = 210, normalized size of antiderivative = 8.40 \[ \int \frac {\left (\frac {x^2}{e^3+x}\right )^{\frac {1}{36} \left (900+132 x+x^2\right )} \left (900 x+132 x^2+x^3+e^3 \left (1800+264 x+2 x^2\right )+\left (132 x^2+2 x^3+e^3 \left (132 x+2 x^2\right )\right ) \log \left (\frac {x^2}{e^3+x}\right )\right )}{36 e^3 x+36 x^2} \, dx=\frac {x^{50} e^{\left (-\frac {1}{36} \, x^{2} \log \left (x + e^{3}\right ) + \frac {1}{18} \, x^{2} \log \left (x\right ) - \frac {11}{3} \, x \log \left (x + e^{3}\right ) + \frac {22}{3} \, x \log \left (x\right )\right )}}{x^{25} + 25 \, x^{24} e^{3} + 300 \, x^{23} e^{6} + 2300 \, x^{22} e^{9} + 12650 \, x^{21} e^{12} + 53130 \, x^{20} e^{15} + 177100 \, x^{19} e^{18} + 480700 \, x^{18} e^{21} + 1081575 \, x^{17} e^{24} + 2042975 \, x^{16} e^{27} + 3268760 \, x^{15} e^{30} + 4457400 \, x^{14} e^{33} + 5200300 \, x^{13} e^{36} + 5200300 \, x^{12} e^{39} + 4457400 \, x^{11} e^{42} + 3268760 \, x^{10} e^{45} + 2042975 \, x^{9} e^{48} + 1081575 \, x^{8} e^{51} + 480700 \, x^{7} e^{54} + 177100 \, x^{6} e^{57} + 53130 \, x^{5} e^{60} + 12650 \, x^{4} e^{63} + 2300 \, x^{3} e^{66} + 300 \, x^{2} e^{69} + 25 \, x e^{72} + e^{75}} \]
integrate((((2*x^2+132*x)*exp(3)+2*x^3+132*x^2)*log(x^2/(exp(3)+x))+(2*x^2 +264*x+1800)*exp(3)+x^3+132*x^2+900*x)*exp(1/36*(x^2+132*x+900)*log(x^2/(e xp(3)+x)))/(36*x*exp(3)+36*x^2),x, algorithm=\
x^50*e^(-1/36*x^2*log(x + e^3) + 1/18*x^2*log(x) - 11/3*x*log(x + e^3) + 2 2/3*x*log(x))/(x^25 + 25*x^24*e^3 + 300*x^23*e^6 + 2300*x^22*e^9 + 12650*x ^21*e^12 + 53130*x^20*e^15 + 177100*x^19*e^18 + 480700*x^18*e^21 + 1081575 *x^17*e^24 + 2042975*x^16*e^27 + 3268760*x^15*e^30 + 4457400*x^14*e^33 + 5 200300*x^13*e^36 + 5200300*x^12*e^39 + 4457400*x^11*e^42 + 3268760*x^10*e^ 45 + 2042975*x^9*e^48 + 1081575*x^8*e^51 + 480700*x^7*e^54 + 177100*x^6*e^ 57 + 53130*x^5*e^60 + 12650*x^4*e^63 + 2300*x^3*e^66 + 300*x^2*e^69 + 25*x *e^72 + e^75)
\[ \int \frac {\left (\frac {x^2}{e^3+x}\right )^{\frac {1}{36} \left (900+132 x+x^2\right )} \left (900 x+132 x^2+x^3+e^3 \left (1800+264 x+2 x^2\right )+\left (132 x^2+2 x^3+e^3 \left (132 x+2 x^2\right )\right ) \log \left (\frac {x^2}{e^3+x}\right )\right )}{36 e^3 x+36 x^2} \, dx=\int { \frac {{\left (x^{3} + 132 \, x^{2} + 2 \, {\left (x^{2} + 132 \, x + 900\right )} e^{3} + 2 \, {\left (x^{3} + 66 \, x^{2} + {\left (x^{2} + 66 \, x\right )} e^{3}\right )} \log \left (\frac {x^{2}}{x + e^{3}}\right ) + 900 \, x\right )} \left (\frac {x^{2}}{x + e^{3}}\right )^{\frac {1}{36} \, x^{2} + \frac {11}{3} \, x + 25}}{36 \, {\left (x^{2} + x e^{3}\right )}} \,d x } \]
integrate((((2*x^2+132*x)*exp(3)+2*x^3+132*x^2)*log(x^2/(exp(3)+x))+(2*x^2 +264*x+1800)*exp(3)+x^3+132*x^2+900*x)*exp(1/36*(x^2+132*x+900)*log(x^2/(e xp(3)+x)))/(36*x*exp(3)+36*x^2),x, algorithm=\
integrate(1/36*(x^3 + 132*x^2 + 2*(x^2 + 132*x + 900)*e^3 + 2*(x^3 + 66*x^ 2 + (x^2 + 66*x)*e^3)*log(x^2/(x + e^3)) + 900*x)*(x^2/(x + e^3))^(1/36*x^ 2 + 11/3*x + 25)/(x^2 + x*e^3), x)
Time = 13.64 (sec) , antiderivative size = 217, normalized size of antiderivative = 8.68 \[ \int \frac {\left (\frac {x^2}{e^3+x}\right )^{\frac {1}{36} \left (900+132 x+x^2\right )} \left (900 x+132 x^2+x^3+e^3 \left (1800+264 x+2 x^2\right )+\left (132 x^2+2 x^3+e^3 \left (132 x+2 x^2\right )\right ) \log \left (\frac {x^2}{e^3+x}\right )\right )}{36 e^3 x+36 x^2} \, dx=\frac {x^{50}\,{\mathrm {e}}^{\frac {11\,x\,\ln \left (\frac {1}{x+{\mathrm {e}}^3}\right )}{3}}\,{\left (\frac {1}{x+{\mathrm {e}}^3}\right )}^{\frac {x^2}{36}}\,{\left (x^2\right )}^{\frac {11\,x}{3}}\,{\left (x^2\right )}^{\frac {x^2}{36}}}{x^{25}+25\,{\mathrm {e}}^3\,x^{24}+300\,{\mathrm {e}}^6\,x^{23}+2300\,{\mathrm {e}}^9\,x^{22}+12650\,{\mathrm {e}}^{12}\,x^{21}+53130\,{\mathrm {e}}^{15}\,x^{20}+177100\,{\mathrm {e}}^{18}\,x^{19}+480700\,{\mathrm {e}}^{21}\,x^{18}+1081575\,{\mathrm {e}}^{24}\,x^{17}+2042975\,{\mathrm {e}}^{27}\,x^{16}+3268760\,{\mathrm {e}}^{30}\,x^{15}+4457400\,{\mathrm {e}}^{33}\,x^{14}+5200300\,{\mathrm {e}}^{36}\,x^{13}+5200300\,{\mathrm {e}}^{39}\,x^{12}+4457400\,{\mathrm {e}}^{42}\,x^{11}+3268760\,{\mathrm {e}}^{45}\,x^{10}+2042975\,{\mathrm {e}}^{48}\,x^9+1081575\,{\mathrm {e}}^{51}\,x^8+480700\,{\mathrm {e}}^{54}\,x^7+177100\,{\mathrm {e}}^{57}\,x^6+53130\,{\mathrm {e}}^{60}\,x^5+12650\,{\mathrm {e}}^{63}\,x^4+2300\,{\mathrm {e}}^{66}\,x^3+300\,{\mathrm {e}}^{69}\,x^2+25\,{\mathrm {e}}^{72}\,x+{\mathrm {e}}^{75}} \]
int((exp((log(x^2/(x + exp(3)))*(132*x + x^2 + 900))/36)*(900*x + exp(3)*( 264*x + 2*x^2 + 1800) + log(x^2/(x + exp(3)))*(exp(3)*(132*x + 2*x^2) + 13 2*x^2 + 2*x^3) + 132*x^2 + x^3))/(36*x*exp(3) + 36*x^2),x)
(x^50*exp((11*x*log(1/(x + exp(3))))/3)*(1/(x + exp(3)))^(x^2/36)*(x^2)^(( 11*x)/3)*(x^2)^(x^2/36))/(exp(75) + 25*x*exp(72) + 25*x^24*exp(3) + 300*x^ 23*exp(6) + 2300*x^22*exp(9) + 12650*x^21*exp(12) + 53130*x^20*exp(15) + 1 77100*x^19*exp(18) + 480700*x^18*exp(21) + 1081575*x^17*exp(24) + 2042975* x^16*exp(27) + 3268760*x^15*exp(30) + 4457400*x^14*exp(33) + 5200300*x^13* exp(36) + 5200300*x^12*exp(39) + 4457400*x^11*exp(42) + 3268760*x^10*exp(4 5) + 2042975*x^9*exp(48) + 1081575*x^8*exp(51) + 480700*x^7*exp(54) + 1771 00*x^6*exp(57) + 53130*x^5*exp(60) + 12650*x^4*exp(63) + 2300*x^3*exp(66) + 300*x^2*exp(69) + x^25)