Integrand size = 111, antiderivative size = 30 \[ \int \frac {e^{\frac {9+24 x^2+72 x^3+70 x^4+96 x^5+216 x^6+216 x^7+81 x^8}{3 x+x^2}} \left (-27-18 x+72 x^2+432 x^3+702 x^4+1292 x^5+3528 x^6+4752 x^7+2781 x^8+486 x^9\right )}{9 x^2+6 x^3+x^4} \, dx=-2+e^{\frac {\left (3+\left (-x+3 \left (x+x^2\right )\right )^2\right )^2}{x (3+x)}} \] Output:
exp(1/x/(3+x)*(3+(3*x^2+2*x)^2)^2)-2
Time = 0.03 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00 \[ \int \frac {e^{\frac {9+24 x^2+72 x^3+70 x^4+96 x^5+216 x^6+216 x^7+81 x^8}{3 x+x^2}} \left (-27-18 x+72 x^2+432 x^3+702 x^4+1292 x^5+3528 x^6+4752 x^7+2781 x^8+486 x^9\right )}{9 x^2+6 x^3+x^4} \, dx=e^{\frac {\left (3+4 x^2+12 x^3+9 x^4\right )^2}{x (3+x)}} \] Input:
Integrate[(E^((9 + 24*x^2 + 72*x^3 + 70*x^4 + 96*x^5 + 216*x^6 + 216*x^7 + 81*x^8)/(3*x + x^2))*(-27 - 18*x + 72*x^2 + 432*x^3 + 702*x^4 + 1292*x^5 + 3528*x^6 + 4752*x^7 + 2781*x^8 + 486*x^9))/(9*x^2 + 6*x^3 + x^4),x]
Output:
E^((3 + 4*x^2 + 12*x^3 + 9*x^4)^2/(x*(3 + 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.
| \(\displaystyle \int \frac {\left (486 x^9+2781 x^8+4752 x^7+3528 x^6+1292 x^5+702 x^4+432 x^3+72 x^2-18 x-27\right ) \exp \left (\frac {81 x^8+216 x^7+216 x^6+96 x^5+70 x^4+72 x^3+24 x^2+9}{x^2+3 x}\right )}{x^4+6 x^3+9 x^2} \, dx\) |
| \(\Big \downarrow \) 2026 |
| \(\displaystyle \int \frac {\left (486 x^9+2781 x^8+4752 x^7+3528 x^6+1292 x^5+702 x^4+432 x^3+72 x^2-18 x-27\right ) \exp \left (\frac {81 x^8+216 x^7+216 x^6+96 x^5+70 x^4+72 x^3+24 x^2+9}{x^2+3 x}\right )}{x^2 \left (x^2+6 x+9\right )}dx\) |
| \(\Big \downarrow \) 2007 |
| \(\displaystyle \int \frac {\left (486 x^9+2781 x^8+4752 x^7+3528 x^6+1292 x^5+702 x^4+432 x^3+72 x^2-18 x-27\right ) \exp \left (\frac {81 x^8+216 x^7+216 x^6+96 x^5+70 x^4+72 x^3+24 x^2+9}{x^2+3 x}\right )}{x^2 (x+3)^2}dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {e^{\frac {\left (9 x^4+12 x^3+4 x^2+3\right )^2}{x (x+3)}} \left (486 x^9+2781 x^8+4752 x^7+3528 x^6+1292 x^5+702 x^4+432 x^3+72 x^2-18 x-27\right )}{x^2 (x+3)^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-135 e^{\frac {\left (9 x^4+12 x^3+4 x^2+3\right )^2}{x (x+3)}} x^4+1188 e^{\frac {\left (9 x^4+12 x^3+4 x^2+3\right )^2}{x (x+3)}} x^3-2385 e^{\frac {\left (9 x^4+12 x^3+4 x^2+3\right )^2}{x (x+3)}} x^2+4910 e^{\frac {\left (9 x^4+12 x^3+4 x^2+3\right )^2}{x (x+3)}} x-7293 e^{\frac {\left (9 x^4+12 x^3+4 x^2+3\right )^2}{x (x+3)}}+\frac {65712 e^{\frac {\left (9 x^4+12 x^3+4 x^2+3\right )^2}{x (x+3)}}}{(x+3)^2}-\frac {3 e^{\frac {\left (9 x^4+12 x^3+4 x^2+3\right )^2}{x (x+3)}}}{x^2}+486 e^{\frac {\left (9 x^4+12 x^3+4 x^2+3\right )^2}{x (x+3)}} x^5\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -7293 \int e^{\frac {\left (9 x^4+12 x^3+4 x^2+3\right )^2}{x (x+3)}}dx-3 \int \frac {e^{\frac {\left (9 x^4+12 x^3+4 x^2+3\right )^2}{x (x+3)}}}{x^2}dx+4910 \int e^{\frac {\left (9 x^4+12 x^3+4 x^2+3\right )^2}{x (x+3)}} xdx-2385 \int e^{\frac {\left (9 x^4+12 x^3+4 x^2+3\right )^2}{x (x+3)}} x^2dx+1188 \int e^{\frac {\left (9 x^4+12 x^3+4 x^2+3\right )^2}{x (x+3)}} x^3dx-135 \int e^{\frac {\left (9 x^4+12 x^3+4 x^2+3\right )^2}{x (x+3)}} x^4dx+65712 \int \frac {e^{\frac {\left (9 x^4+12 x^3+4 x^2+3\right )^2}{x (x+3)}}}{(x+3)^2}dx+486 \int e^{\frac {\left (9 x^4+12 x^3+4 x^2+3\right )^2}{x (x+3)}} x^5dx\) |
Input:
Int[(E^((9 + 24*x^2 + 72*x^3 + 70*x^4 + 96*x^5 + 216*x^6 + 216*x^7 + 81*x^ 8)/(3*x + x^2))*(-27 - 18*x + 72*x^2 + 432*x^3 + 702*x^4 + 1292*x^5 + 3528 *x^6 + 4752*x^7 + 2781*x^8 + 486*x^9))/(9*x^2 + 6*x^3 + x^4),x]
Output:
$Aborted
Time = 0.33 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00
| method | result | size |
| risch | \({\mathrm e}^{\frac {\left (9 x^{4}+12 x^{3}+4 x^{2}+3\right )^{2}}{\left (3+x \right ) x}}\) | \(30\) |
| gosper | \({\mathrm e}^{\frac {81 x^{8}+216 x^{7}+216 x^{6}+96 x^{5}+70 x^{4}+72 x^{3}+24 x^{2}+9}{\left (3+x \right ) x}}\) | \(48\) |
| parallelrisch | \({\mathrm e}^{\frac {81 x^{8}+216 x^{7}+216 x^{6}+96 x^{5}+70 x^{4}+72 x^{3}+24 x^{2}+9}{\left (3+x \right ) x}}\) | \(48\) |
| norman | \(\frac {x^{2} {\mathrm e}^{\frac {81 x^{8}+216 x^{7}+216 x^{6}+96 x^{5}+70 x^{4}+72 x^{3}+24 x^{2}+9}{x^{2}+3 x}}+3 x \,{\mathrm e}^{\frac {81 x^{8}+216 x^{7}+216 x^{6}+96 x^{5}+70 x^{4}+72 x^{3}+24 x^{2}+9}{x^{2}+3 x}}}{\left (3+x \right ) x}\) | \(114\) |
| orering | \(\frac {\left (3+x \right )^{2} x^{2} \left (486 x^{9}+2781 x^{8}+4752 x^{7}+3528 x^{6}+1292 x^{5}+702 x^{4}+432 x^{3}+72 x^{2}-18 x -27\right ) {\mathrm e}^{\frac {81 x^{8}+216 x^{7}+216 x^{6}+96 x^{5}+70 x^{4}+72 x^{3}+24 x^{2}+9}{x^{2}+3 x}}}{\left (9 x^{4}+12 x^{3}+4 x^{2}+3\right ) \left (54 x^{5}+237 x^{4}+188 x^{3}+36 x^{2}-6 x -9\right ) \left (x^{4}+6 x^{3}+9 x^{2}\right )}\) | \(165\) |
Input:
int((486*x^9+2781*x^8+4752*x^7+3528*x^6+1292*x^5+702*x^4+432*x^3+72*x^2-18 *x-27)*exp((81*x^8+216*x^7+216*x^6+96*x^5+70*x^4+72*x^3+24*x^2+9)/(x^2+3*x ))/(x^4+6*x^3+9*x^2),x,method=_RETURNVERBOSE)
Output:
exp((9*x^4+12*x^3+4*x^2+3)^2/(3+x)/x)
Time = 0.07 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.60 \[ \int \frac {e^{\frac {9+24 x^2+72 x^3+70 x^4+96 x^5+216 x^6+216 x^7+81 x^8}{3 x+x^2}} \left (-27-18 x+72 x^2+432 x^3+702 x^4+1292 x^5+3528 x^6+4752 x^7+2781 x^8+486 x^9\right )}{9 x^2+6 x^3+x^4} \, dx=e^{\left (\frac {81 \, x^{8} + 216 \, x^{7} + 216 \, x^{6} + 96 \, x^{5} + 70 \, x^{4} + 72 \, x^{3} + 24 \, x^{2} + 9}{x^{2} + 3 \, x}\right )} \] Input:
integrate((486*x^9+2781*x^8+4752*x^7+3528*x^6+1292*x^5+702*x^4+432*x^3+72* x^2-18*x-27)*exp((81*x^8+216*x^7+216*x^6+96*x^5+70*x^4+72*x^3+24*x^2+9)/(x ^2+3*x))/(x^4+6*x^3+9*x^2),x, algorithm="fricas")
Output:
e^((81*x^8 + 216*x^7 + 216*x^6 + 96*x^5 + 70*x^4 + 72*x^3 + 24*x^2 + 9)/(x ^2 + 3*x))
Leaf count of result is larger than twice the leaf count of optimal. 44 vs. \(2 (20) = 40\).
Time = 0.11 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.47 \[ \int \frac {e^{\frac {9+24 x^2+72 x^3+70 x^4+96 x^5+216 x^6+216 x^7+81 x^8}{3 x+x^2}} \left (-27-18 x+72 x^2+432 x^3+702 x^4+1292 x^5+3528 x^6+4752 x^7+2781 x^8+486 x^9\right )}{9 x^2+6 x^3+x^4} \, dx=e^{\frac {81 x^{8} + 216 x^{7} + 216 x^{6} + 96 x^{5} + 70 x^{4} + 72 x^{3} + 24 x^{2} + 9}{x^{2} + 3 x}} \] Input:
integrate((486*x**9+2781*x**8+4752*x**7+3528*x**6+1292*x**5+702*x**4+432*x **3+72*x**2-18*x-27)*exp((81*x**8+216*x**7+216*x**6+96*x**5+70*x**4+72*x** 3+24*x**2+9)/(x**2+3*x))/(x**4+6*x**3+9*x**2),x)
Output:
exp((81*x**8 + 216*x**7 + 216*x**6 + 96*x**5 + 70*x**4 + 72*x**3 + 24*x**2 + 9)/(x**2 + 3*x))
Time = 0.53 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.43 \[ \int \frac {e^{\frac {9+24 x^2+72 x^3+70 x^4+96 x^5+216 x^6+216 x^7+81 x^8}{3 x+x^2}} \left (-27-18 x+72 x^2+432 x^3+702 x^4+1292 x^5+3528 x^6+4752 x^7+2781 x^8+486 x^9\right )}{9 x^2+6 x^3+x^4} \, dx=e^{\left (81 \, x^{6} - 27 \, x^{5} + 297 \, x^{4} - 795 \, x^{3} + 2455 \, x^{2} - 7293 \, x - \frac {65712}{x + 3} + \frac {3}{x} + 21903\right )} \] Input:
integrate((486*x^9+2781*x^8+4752*x^7+3528*x^6+1292*x^5+702*x^4+432*x^3+72* x^2-18*x-27)*exp((81*x^8+216*x^7+216*x^6+96*x^5+70*x^4+72*x^3+24*x^2+9)/(x ^2+3*x))/(x^4+6*x^3+9*x^2),x, algorithm="maxima")
Output:
e^(81*x^6 - 27*x^5 + 297*x^4 - 795*x^3 + 2455*x^2 - 7293*x - 65712/(x + 3) + 3/x + 21903)
Leaf count of result is larger than twice the leaf count of optimal. 111 vs. \(2 (27) = 54\).
Time = 0.13 (sec) , antiderivative size = 111, normalized size of antiderivative = 3.70 \[ \int \frac {e^{\frac {9+24 x^2+72 x^3+70 x^4+96 x^5+216 x^6+216 x^7+81 x^8}{3 x+x^2}} \left (-27-18 x+72 x^2+432 x^3+702 x^4+1292 x^5+3528 x^6+4752 x^7+2781 x^8+486 x^9\right )}{9 x^2+6 x^3+x^4} \, dx=e^{\left (\frac {81 \, x^{8}}{x^{2} + 3 \, x} + \frac {216 \, x^{7}}{x^{2} + 3 \, x} + \frac {216 \, x^{6}}{x^{2} + 3 \, x} + \frac {96 \, x^{5}}{x^{2} + 3 \, x} + \frac {70 \, x^{4}}{x^{2} + 3 \, x} + \frac {72 \, x^{3}}{x^{2} + 3 \, x} + \frac {24 \, x^{2}}{x^{2} + 3 \, x} + \frac {9}{x^{2} + 3 \, x}\right )} \] Input:
integrate((486*x^9+2781*x^8+4752*x^7+3528*x^6+1292*x^5+702*x^4+432*x^3+72* x^2-18*x-27)*exp((81*x^8+216*x^7+216*x^6+96*x^5+70*x^4+72*x^3+24*x^2+9)/(x ^2+3*x))/(x^4+6*x^3+9*x^2),x, algorithm="giac")
Output:
e^(81*x^8/(x^2 + 3*x) + 216*x^7/(x^2 + 3*x) + 216*x^6/(x^2 + 3*x) + 96*x^5 /(x^2 + 3*x) + 70*x^4/(x^2 + 3*x) + 72*x^3/(x^2 + 3*x) + 24*x^2/(x^2 + 3*x ) + 9/(x^2 + 3*x))
Time = 3.89 (sec) , antiderivative size = 88, normalized size of antiderivative = 2.93 \[ \int \frac {e^{\frac {9+24 x^2+72 x^3+70 x^4+96 x^5+216 x^6+216 x^7+81 x^8}{3 x+x^2}} \left (-27-18 x+72 x^2+432 x^3+702 x^4+1292 x^5+3528 x^6+4752 x^7+2781 x^8+486 x^9\right )}{9 x^2+6 x^3+x^4} \, dx={\mathrm {e}}^{\frac {9}{x^2+3\,x}}\,{\mathrm {e}}^{\frac {24\,x}{x+3}}\,{\mathrm {e}}^{\frac {70\,x^3}{x+3}}\,{\mathrm {e}}^{\frac {72\,x^2}{x+3}}\,{\mathrm {e}}^{\frac {81\,x^7}{x+3}}\,{\mathrm {e}}^{\frac {96\,x^4}{x+3}}\,{\mathrm {e}}^{\frac {216\,x^5}{x+3}}\,{\mathrm {e}}^{\frac {216\,x^6}{x+3}} \] Input:
int((exp((24*x^2 + 72*x^3 + 70*x^4 + 96*x^5 + 216*x^6 + 216*x^7 + 81*x^8 + 9)/(3*x + x^2))*(72*x^2 - 18*x + 432*x^3 + 702*x^4 + 1292*x^5 + 3528*x^6 + 4752*x^7 + 2781*x^8 + 486*x^9 - 27))/(9*x^2 + 6*x^3 + x^4),x)
Output:
exp(9/(3*x + x^2))*exp((24*x)/(x + 3))*exp((70*x^3)/(x + 3))*exp((72*x^2)/ (x + 3))*exp((81*x^7)/(x + 3))*exp((96*x^4)/(x + 3))*exp((216*x^5)/(x + 3) )*exp((216*x^6)/(x + 3))
\[ \int \frac {e^{\frac {9+24 x^2+72 x^3+70 x^4+96 x^5+216 x^6+216 x^7+81 x^8}{3 x+x^2}} \left (-27-18 x+72 x^2+432 x^3+702 x^4+1292 x^5+3528 x^6+4752 x^7+2781 x^8+486 x^9\right )}{9 x^2+6 x^3+x^4} \, dx=\int \frac {\left (486 x^{9}+2781 x^{8}+4752 x^{7}+3528 x^{6}+1292 x^{5}+702 x^{4}+432 x^{3}+72 x^{2}-18 x -27\right ) {\mathrm e}^{\frac {81 x^{8}+216 x^{7}+216 x^{6}+96 x^{5}+70 x^{4}+72 x^{3}+24 x^{2}+9}{x^{2}+3 x}}}{x^{4}+6 x^{3}+9 x^{2}}d x \] Input:
int((486*x^9+2781*x^8+4752*x^7+3528*x^6+1292*x^5+702*x^4+432*x^3+72*x^2-18 *x-27)*exp((81*x^8+216*x^7+216*x^6+96*x^5+70*x^4+72*x^3+24*x^2+9)/(x^2+3*x ))/(x^4+6*x^3+9*x^2),x)
Output:
int((486*x^9+2781*x^8+4752*x^7+3528*x^6+1292*x^5+702*x^4+432*x^3+72*x^2-18 *x-27)*exp((81*x^8+216*x^7+216*x^6+96*x^5+70*x^4+72*x^3+24*x^2+9)/(x^2+3*x ))/(x^4+6*x^3+9*x^2),x)