Integrand size = 102, antiderivative size = 22 \[ \int \frac {e^{\frac {-x^2+6 x^3+6 x^4+x^5}{-9+6 x+6 x^2+x^3}} \left (18 x-168 x^2-144 x^3+100 x^4+96 x^5+24 x^6+2 x^7\right )}{81-108 x-72 x^2+54 x^3+48 x^4+12 x^5+x^6} \, dx=e^{x \left (x+\frac {8}{-3-\frac {9}{x}+(3+x)^2}\right )} \] Output:
exp(x*(x+8/((3+x)^2-9/x-3)))
Time = 1.38 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.18 \[ \int \frac {e^{\frac {-x^2+6 x^3+6 x^4+x^5}{-9+6 x+6 x^2+x^3}} \left (18 x-168 x^2-144 x^3+100 x^4+96 x^5+24 x^6+2 x^7\right )}{81-108 x-72 x^2+54 x^3+48 x^4+12 x^5+x^6} \, dx=e^{x^2+\frac {8 x^2}{-9+6 x+6 x^2+x^3}} \] Input:
Integrate[(E^((-x^2 + 6*x^3 + 6*x^4 + x^5)/(-9 + 6*x + 6*x^2 + x^3))*(18*x - 168*x^2 - 144*x^3 + 100*x^4 + 96*x^5 + 24*x^6 + 2*x^7))/(81 - 108*x - 7 2*x^2 + 54*x^3 + 48*x^4 + 12*x^5 + x^6),x]
Output:
E^(x^2 + (8*x^2)/(-9 + 6*x + 6*x^2 + x^3))
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.
\(\displaystyle \int \frac {e^{\frac {x^5+6 x^4+6 x^3-x^2}{x^3+6 x^2+6 x-9}} \left (2 x^7+24 x^6+96 x^5+100 x^4-144 x^3-168 x^2+18 x\right )}{x^6+12 x^5+48 x^4+54 x^3-72 x^2-108 x+81} \, dx\) |
\(\Big \downarrow \) 2463 |
\(\displaystyle \int \left (-\frac {2 e^{\frac {x^5+6 x^4+6 x^3-x^2}{x^3+6 x^2+6 x-9}} \left (2 x^7+24 x^6+96 x^5+100 x^4-144 x^3-168 x^2+18 x\right )}{9 (x+3)}+\frac {e^{\frac {x^5+6 x^4+6 x^3-x^2}{x^3+6 x^2+6 x-9}} (2 x-1) \left (2 x^7+24 x^6+96 x^5+100 x^4-144 x^3-168 x^2+18 x\right )}{9 \left (x^2+3 x-3\right )}+\frac {e^{\frac {x^5+6 x^4+6 x^3-x^2}{x^3+6 x^2+6 x-9}} \left (2 x^7+24 x^6+96 x^5+100 x^4-144 x^3-168 x^2+18 x\right )}{9 (x+3)^2}+\frac {e^{\frac {x^5+6 x^4+6 x^3-x^2}{x^3+6 x^2+6 x-9}} (1-x) \left (2 x^7+24 x^6+96 x^5+100 x^4-144 x^3-168 x^2+18 x\right )}{3 \left (x^2+3 x-3\right )^2}\right )dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \frac {2 x \left (x^6+12 x^5+48 x^4+50 x^3-72 x^2-84 x+9\right ) \exp \left (\frac {x^2 \left (x^3+6 x^2+6 x-1\right )}{x^3+6 x^2+6 x-9}\right )}{(x+3)^2 \left (-x^2-3 x+3\right )^2}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle 2 \int \frac {\exp \left (\frac {x^2 \left (-x^3-6 x^2-6 x+1\right )}{-x^3-6 x^2-6 x+9}\right ) x \left (x^6+12 x^5+48 x^4+50 x^3-72 x^2-84 x+9\right )}{(x+3)^2 \left (-x^2-3 x+3\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle 2 \int \left (\exp \left (\frac {x^2 \left (-x^3-6 x^2-6 x+1\right )}{-x^3-6 x^2-6 x+9}\right ) x-\frac {16 \exp \left (\frac {x^2 \left (-x^3-6 x^2-6 x+1\right )}{-x^3-6 x^2-6 x+9}\right )}{x^2+3 x-3}+\frac {12 \exp \left (\frac {x^2 \left (-x^3-6 x^2-6 x+1\right )}{-x^3-6 x^2-6 x+9}\right )}{(x+3)^2}+\frac {12 \exp \left (\frac {x^2 \left (-x^3-6 x^2-6 x+1\right )}{-x^3-6 x^2-6 x+9}\right ) (6 x-5)}{\left (x^2+3 x-3\right )^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 2 \left (-\frac {48}{7} \left (3-\sqrt {21}\right ) \int \frac {\exp \left (\frac {x^2 \left (-x^3-6 x^2-6 x+1\right )}{-x^3-6 x^2-6 x+9}\right )}{\left (-2 x+\sqrt {21}-3\right )^2}dx-\frac {80}{7} \int \frac {\exp \left (\frac {x^2 \left (-x^3-6 x^2-6 x+1\right )}{-x^3-6 x^2-6 x+9}\right )}{\left (-2 x+\sqrt {21}-3\right )^2}dx+\int \exp \left (\frac {x^2 \left (-x^3-6 x^2-6 x+1\right )}{-x^3-6 x^2-6 x+9}\right ) xdx+12 \int \frac {\exp \left (\frac {x^2 \left (-x^3-6 x^2-6 x+1\right )}{-x^3-6 x^2-6 x+9}\right )}{(x+3)^2}dx-\frac {48}{7} \left (3+\sqrt {21}\right ) \int \frac {\exp \left (\frac {x^2 \left (-x^3-6 x^2-6 x+1\right )}{-x^3-6 x^2-6 x+9}\right )}{\left (2 x+\sqrt {21}+3\right )^2}dx-\frac {80}{7} \int \frac {\exp \left (\frac {x^2 \left (-x^3-6 x^2-6 x+1\right )}{-x^3-6 x^2-6 x+9}\right )}{\left (2 x+\sqrt {21}+3\right )^2}dx\right )\) |
Input:
Int[(E^((-x^2 + 6*x^3 + 6*x^4 + x^5)/(-9 + 6*x + 6*x^2 + x^3))*(18*x - 168 *x^2 - 144*x^3 + 100*x^4 + 96*x^5 + 24*x^6 + 2*x^7))/(81 - 108*x - 72*x^2 + 54*x^3 + 48*x^4 + 12*x^5 + x^6),x]
Output:
$Aborted
Time = 0.39 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.55
method | result | size |
gosper | \({\mathrm e}^{\frac {x^{2} \left (x^{3}+6 x^{2}+6 x -1\right )}{x^{3}+6 x^{2}+6 x -9}}\) | \(34\) |
risch | \({\mathrm e}^{\frac {x^{2} \left (x^{3}+6 x^{2}+6 x -1\right )}{\left (3+x \right ) \left (x^{2}+3 x -3\right )}}\) | \(34\) |
parallelrisch | \({\mathrm e}^{\frac {x^{2} \left (x^{3}+6 x^{2}+6 x -1\right )}{x^{3}+6 x^{2}+6 x -9}}\) | \(34\) |
orering | \(\frac {\left (3+x \right )^{2} \left (x^{2}+3 x -3\right )^{2} \left (2 x^{7}+24 x^{6}+96 x^{5}+100 x^{4}-144 x^{3}-168 x^{2}+18 x \right ) {\mathrm e}^{\frac {x^{5}+6 x^{4}+6 x^{3}-x^{2}}{x^{3}+6 x^{2}+6 x -9}}}{2 x \left (x^{6}+12 x^{5}+48 x^{4}+50 x^{3}-72 x^{2}-84 x +9\right ) \left (x^{6}+12 x^{5}+48 x^{4}+54 x^{3}-72 x^{2}-108 x +81\right )}\) | \(151\) |
norman | \(\frac {x^{3} {\mathrm e}^{\frac {x^{5}+6 x^{4}+6 x^{3}-x^{2}}{x^{3}+6 x^{2}+6 x -9}}+6 x \,{\mathrm e}^{\frac {x^{5}+6 x^{4}+6 x^{3}-x^{2}}{x^{3}+6 x^{2}+6 x -9}}+6 x^{2} {\mathrm e}^{\frac {x^{5}+6 x^{4}+6 x^{3}-x^{2}}{x^{3}+6 x^{2}+6 x -9}}-9 \,{\mathrm e}^{\frac {x^{5}+6 x^{4}+6 x^{3}-x^{2}}{x^{3}+6 x^{2}+6 x -9}}}{x^{3}+6 x^{2}+6 x -9}\) | \(176\) |
Input:
int((2*x^7+24*x^6+96*x^5+100*x^4-144*x^3-168*x^2+18*x)*exp((x^5+6*x^4+6*x^ 3-x^2)/(x^3+6*x^2+6*x-9))/(x^6+12*x^5+48*x^4+54*x^3-72*x^2-108*x+81),x,met hod=_RETURNVERBOSE)
Output:
exp(x^2*(x^3+6*x^2+6*x-1)/(x^3+6*x^2+6*x-9))
Time = 0.06 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.64 \[ \int \frac {e^{\frac {-x^2+6 x^3+6 x^4+x^5}{-9+6 x+6 x^2+x^3}} \left (18 x-168 x^2-144 x^3+100 x^4+96 x^5+24 x^6+2 x^7\right )}{81-108 x-72 x^2+54 x^3+48 x^4+12 x^5+x^6} \, dx=e^{\left (\frac {x^{5} + 6 \, x^{4} + 6 \, x^{3} - x^{2}}{x^{3} + 6 \, x^{2} + 6 \, x - 9}\right )} \] Input:
integrate((2*x^7+24*x^6+96*x^5+100*x^4-144*x^3-168*x^2+18*x)*exp((x^5+6*x^ 4+6*x^3-x^2)/(x^3+6*x^2+6*x-9))/(x^6+12*x^5+48*x^4+54*x^3-72*x^2-108*x+81) ,x, algorithm="fricas")
Output:
e^((x^5 + 6*x^4 + 6*x^3 - x^2)/(x^3 + 6*x^2 + 6*x - 9))
Leaf count of result is larger than twice the leaf count of optimal. 31 vs. \(2 (15) = 30\).
Time = 0.11 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.41 \[ \int \frac {e^{\frac {-x^2+6 x^3+6 x^4+x^5}{-9+6 x+6 x^2+x^3}} \left (18 x-168 x^2-144 x^3+100 x^4+96 x^5+24 x^6+2 x^7\right )}{81-108 x-72 x^2+54 x^3+48 x^4+12 x^5+x^6} \, dx=e^{\frac {x^{5} + 6 x^{4} + 6 x^{3} - x^{2}}{x^{3} + 6 x^{2} + 6 x - 9}} \] Input:
integrate((2*x**7+24*x**6+96*x**5+100*x**4-144*x**3-168*x**2+18*x)*exp((x* *5+6*x**4+6*x**3-x**2)/(x**3+6*x**2+6*x-9))/(x**6+12*x**5+48*x**4+54*x**3- 72*x**2-108*x+81),x)
Output:
exp((x**5 + 6*x**4 + 6*x**3 - x**2)/(x**3 + 6*x**2 + 6*x - 9))
Time = 0.30 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.68 \[ \int \frac {e^{\frac {-x^2+6 x^3+6 x^4+x^5}{-9+6 x+6 x^2+x^3}} \left (18 x-168 x^2-144 x^3+100 x^4+96 x^5+24 x^6+2 x^7\right )}{81-108 x-72 x^2+54 x^3+48 x^4+12 x^5+x^6} \, dx=e^{\left (x^{2} + \frac {32 \, x}{x^{2} + 3 \, x - 3} - \frac {24}{x^{2} + 3 \, x - 3} - \frac {24}{x + 3}\right )} \] Input:
integrate((2*x^7+24*x^6+96*x^5+100*x^4-144*x^3-168*x^2+18*x)*exp((x^5+6*x^ 4+6*x^3-x^2)/(x^3+6*x^2+6*x-9))/(x^6+12*x^5+48*x^4+54*x^3-72*x^2-108*x+81) ,x, algorithm="maxima")
Output:
e^(x^2 + 32*x/(x^2 + 3*x - 3) - 24/(x^2 + 3*x - 3) - 24/(x + 3))
Leaf count of result is larger than twice the leaf count of optimal. 81 vs. \(2 (21) = 42\).
Time = 0.12 (sec) , antiderivative size = 81, normalized size of antiderivative = 3.68 \[ \int \frac {e^{\frac {-x^2+6 x^3+6 x^4+x^5}{-9+6 x+6 x^2+x^3}} \left (18 x-168 x^2-144 x^3+100 x^4+96 x^5+24 x^6+2 x^7\right )}{81-108 x-72 x^2+54 x^3+48 x^4+12 x^5+x^6} \, dx=e^{\left (\frac {x^{5}}{x^{3} + 6 \, x^{2} + 6 \, x - 9} + \frac {6 \, x^{4}}{x^{3} + 6 \, x^{2} + 6 \, x - 9} + \frac {6 \, x^{3}}{x^{3} + 6 \, x^{2} + 6 \, x - 9} - \frac {x^{2}}{x^{3} + 6 \, x^{2} + 6 \, x - 9}\right )} \] Input:
integrate((2*x^7+24*x^6+96*x^5+100*x^4-144*x^3-168*x^2+18*x)*exp((x^5+6*x^ 4+6*x^3-x^2)/(x^3+6*x^2+6*x-9))/(x^6+12*x^5+48*x^4+54*x^3-72*x^2-108*x+81) ,x, algorithm="giac")
Output:
e^(x^5/(x^3 + 6*x^2 + 6*x - 9) + 6*x^4/(x^3 + 6*x^2 + 6*x - 9) + 6*x^3/(x^ 3 + 6*x^2 + 6*x - 9) - x^2/(x^3 + 6*x^2 + 6*x - 9))
Time = 0.40 (sec) , antiderivative size = 84, normalized size of antiderivative = 3.82 \[ \int \frac {e^{\frac {-x^2+6 x^3+6 x^4+x^5}{-9+6 x+6 x^2+x^3}} \left (18 x-168 x^2-144 x^3+100 x^4+96 x^5+24 x^6+2 x^7\right )}{81-108 x-72 x^2+54 x^3+48 x^4+12 x^5+x^6} \, dx={\mathrm {e}}^{-\frac {x^2}{x^3+6\,x^2+6\,x-9}}\,{\mathrm {e}}^{\frac {x^5}{x^3+6\,x^2+6\,x-9}}\,{\mathrm {e}}^{\frac {6\,x^3}{x^3+6\,x^2+6\,x-9}}\,{\mathrm {e}}^{\frac {6\,x^4}{x^3+6\,x^2+6\,x-9}} \] Input:
int((exp((6*x^3 - x^2 + 6*x^4 + x^5)/(6*x + 6*x^2 + x^3 - 9))*(18*x - 168* x^2 - 144*x^3 + 100*x^4 + 96*x^5 + 24*x^6 + 2*x^7))/(54*x^3 - 72*x^2 - 108 *x + 48*x^4 + 12*x^5 + x^6 + 81),x)
Output:
exp(-x^2/(6*x + 6*x^2 + x^3 - 9))*exp(x^5/(6*x + 6*x^2 + x^3 - 9))*exp((6* x^3)/(6*x + 6*x^2 + x^3 - 9))*exp((6*x^4)/(6*x + 6*x^2 + x^3 - 9))
Time = 0.39 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.68 \[ \int \frac {e^{\frac {-x^2+6 x^3+6 x^4+x^5}{-9+6 x+6 x^2+x^3}} \left (18 x-168 x^2-144 x^3+100 x^4+96 x^5+24 x^6+2 x^7\right )}{81-108 x-72 x^2+54 x^3+48 x^4+12 x^5+x^6} \, dx=e^{\frac {x^{5}+6 x^{4}+6 x^{3}-x^{2}}{x^{3}+6 x^{2}+6 x -9}} \] Input:
int((2*x^7+24*x^6+96*x^5+100*x^4-144*x^3-168*x^2+18*x)*exp((x^5+6*x^4+6*x^ 3-x^2)/(x^3+6*x^2+6*x-9))/(x^6+12*x^5+48*x^4+54*x^3-72*x^2-108*x+81),x)
Output:
e**((x**5 + 6*x**4 + 6*x**3 - x**2)/(x**3 + 6*x**2 + 6*x - 9))