Integrand size = 88, antiderivative size = 25 \[ \int \frac {e^{\frac {-80-129 x+9 x^2-81 x^3}{16+26 x-3 x^2+18 x^3}} \left (16-192 x+279 x^2+432 x^3+81 x^4\right )}{256+832 x+580 x^2+420 x^3+945 x^4-108 x^5+324 x^6} \, dx=e^{-5+\frac {x}{\left (1-\frac {5}{1-3 x}\right )^2+2 x}} \] Output:
exp(x/(2*x+(1-5/(1-3*x))^2)-5)
Time = 0.81 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.48 \[ \int \frac {e^{\frac {-80-129 x+9 x^2-81 x^3}{16+26 x-3 x^2+18 x^3}} \left (16-192 x+279 x^2+432 x^3+81 x^4\right )}{256+832 x+580 x^2+420 x^3+945 x^4-108 x^5+324 x^6} \, dx=e^{-\frac {9}{2}+\frac {-16-24 x-9 x^2}{2 \left (16+26 x-3 x^2+18 x^3\right )}} \] Input:
Integrate[(E^((-80 - 129*x + 9*x^2 - 81*x^3)/(16 + 26*x - 3*x^2 + 18*x^3)) *(16 - 192*x + 279*x^2 + 432*x^3 + 81*x^4))/(256 + 832*x + 580*x^2 + 420*x ^3 + 945*x^4 - 108*x^5 + 324*x^6),x]
Output:
E^(-9/2 + (-16 - 24*x - 9*x^2)/(2*(16 + 26*x - 3*x^2 + 18*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.
\(\displaystyle \int \frac {\left (81 x^4+432 x^3+279 x^2-192 x+16\right ) \exp \left (\frac {-81 x^3+9 x^2-129 x-80}{18 x^3-3 x^2+26 x+16}\right )}{324 x^6-108 x^5+945 x^4+420 x^3+580 x^2+832 x+256} \, dx\) |
\(\Big \downarrow \) 2463 |
\(\displaystyle \int \left (\frac {192 \left (81 x^4+432 x^3+279 x^2-192 x+16\right ) \exp \left (\frac {-81 x^3+9 x^2-129 x-80}{18 x^3-3 x^2+26 x+16}\right )}{122825 (2 x+1)}-\frac {36 (24 x-11) \left (81 x^4+432 x^3+279 x^2-192 x+16\right ) \exp \left (\frac {-81 x^3+9 x^2-129 x-80}{18 x^3-3 x^2+26 x+16}\right )}{122825 \left (9 x^2-6 x+16\right )}+\frac {16 \left (81 x^4+432 x^3+279 x^2-192 x+16\right ) \exp \left (\frac {-81 x^3+9 x^2-129 x-80}{18 x^3-3 x^2+26 x+16}\right )}{7225 (2 x+1)^2}-\frac {27 (4 x+1) \left (81 x^4+432 x^3+279 x^2-192 x+16\right ) \exp \left (\frac {-81 x^3+9 x^2-129 x-80}{18 x^3-3 x^2+26 x+16}\right )}{1445 \left (9 x^2-6 x+16\right )^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {378}{17} \left (1+i \sqrt {15}\right ) \int \frac {\exp \left (\frac {-81 x^3+9 x^2-129 x-80}{18 x^3-3 x^2+26 x+16}\right )}{\left (-18 x+6 i \sqrt {15}+6\right )^2}dx+\frac {864}{17} \int \frac {\exp \left (\frac {-81 x^3+9 x^2-129 x-80}{18 x^3-3 x^2+26 x+16}\right )}{\left (-18 x+6 i \sqrt {15}+6\right )^2}dx-\frac {27}{17} i \sqrt {\frac {3}{5}} \int \frac {\exp \left (\frac {-81 x^3+9 x^2-129 x-80}{18 x^3-3 x^2+26 x+16}\right )}{-18 x+6 i \sqrt {15}+6}dx+\frac {5}{17} \int \frac {\exp \left (\frac {-81 x^3+9 x^2-129 x-80}{18 x^3-3 x^2+26 x+16}\right )}{(2 x+1)^2}dx-\frac {27 \left (360+121 i \sqrt {15}\right ) \int \frac {\exp \left (\frac {-81 x^3+9 x^2-129 x-80}{18 x^3-3 x^2+26 x+16}\right )}{18 x-6 i \sqrt {15}-6}dx}{1445}+\frac {216 \left (45+13 i \sqrt {15}\right ) \int \frac {\exp \left (\frac {-81 x^3+9 x^2-129 x-80}{18 x^3-3 x^2+26 x+16}\right )}{18 x-6 i \sqrt {15}-6}dx}{1445}-\frac {378}{17} \left (1-i \sqrt {15}\right ) \int \frac {\exp \left (\frac {-81 x^3+9 x^2-129 x-80}{18 x^3-3 x^2+26 x+16}\right )}{\left (18 x+6 i \sqrt {15}-6\right )^2}dx+\frac {864}{17} \int \frac {\exp \left (\frac {-81 x^3+9 x^2-129 x-80}{18 x^3-3 x^2+26 x+16}\right )}{\left (18 x+6 i \sqrt {15}-6\right )^2}dx-\frac {27 \left (360-121 i \sqrt {15}\right ) \int \frac {\exp \left (\frac {-81 x^3+9 x^2-129 x-80}{18 x^3-3 x^2+26 x+16}\right )}{18 x+6 i \sqrt {15}-6}dx}{1445}+\frac {216 \left (45-13 i \sqrt {15}\right ) \int \frac {\exp \left (\frac {-81 x^3+9 x^2-129 x-80}{18 x^3-3 x^2+26 x+16}\right )}{18 x+6 i \sqrt {15}-6}dx}{1445}-\frac {27}{17} i \sqrt {\frac {3}{5}} \int \frac {\exp \left (\frac {-81 x^3+9 x^2-129 x-80}{18 x^3-3 x^2+26 x+16}\right )}{18 x+6 i \sqrt {15}-6}dx\) |
Input:
Int[(E^((-80 - 129*x + 9*x^2 - 81*x^3)/(16 + 26*x - 3*x^2 + 18*x^3))*(16 - 192*x + 279*x^2 + 432*x^3 + 81*x^4))/(256 + 832*x + 580*x^2 + 420*x^3 + 9 45*x^4 - 108*x^5 + 324*x^6),x]
Output:
$Aborted
Time = 0.54 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.44
method | result | size |
gosper | \({\mathrm e}^{-\frac {81 x^{3}-9 x^{2}+129 x +80}{18 x^{3}-3 x^{2}+26 x +16}}\) | \(36\) |
parallelrisch | \({\mathrm e}^{-\frac {81 x^{3}-9 x^{2}+129 x +80}{18 x^{3}-3 x^{2}+26 x +16}}\) | \(36\) |
risch | \({\mathrm e}^{-\frac {81 x^{3}-9 x^{2}+129 x +80}{\left (1+2 x \right ) \left (9 x^{2}-6 x +16\right )}}\) | \(38\) |
orering | \(\frac {\left (1+2 x \right )^{2} \left (9 x^{2}-6 x +16\right )^{2} \left (81 x^{4}+432 x^{3}+279 x^{2}-192 x +16\right ) {\mathrm e}^{\frac {-81 x^{3}+9 x^{2}-129 x -80}{18 x^{3}-3 x^{2}+26 x +16}}}{\left (-1+3 x \right ) \left (4+3 x \right ) \left (9 x^{2}+39 x -4\right ) \left (324 x^{6}-108 x^{5}+945 x^{4}+420 x^{3}+580 x^{2}+832 x +256\right )}\) | \(133\) |
norman | \(\frac {26 x \,{\mathrm e}^{\frac {-81 x^{3}+9 x^{2}-129 x -80}{18 x^{3}-3 x^{2}+26 x +16}}-3 x^{2} {\mathrm e}^{\frac {-81 x^{3}+9 x^{2}-129 x -80}{18 x^{3}-3 x^{2}+26 x +16}}+18 x^{3} {\mathrm e}^{\frac {-81 x^{3}+9 x^{2}-129 x -80}{18 x^{3}-3 x^{2}+26 x +16}}+16 \,{\mathrm e}^{\frac {-81 x^{3}+9 x^{2}-129 x -80}{18 x^{3}-3 x^{2}+26 x +16}}}{18 x^{3}-3 x^{2}+26 x +16}\) | \(171\) |
Input:
int((81*x^4+432*x^3+279*x^2-192*x+16)*exp((-81*x^3+9*x^2-129*x-80)/(18*x^3 -3*x^2+26*x+16))/(324*x^6-108*x^5+945*x^4+420*x^3+580*x^2+832*x+256),x,met hod=_RETURNVERBOSE)
Output:
exp(-(81*x^3-9*x^2+129*x+80)/(18*x^3-3*x^2+26*x+16))
Time = 0.07 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.40 \[ \int \frac {e^{\frac {-80-129 x+9 x^2-81 x^3}{16+26 x-3 x^2+18 x^3}} \left (16-192 x+279 x^2+432 x^3+81 x^4\right )}{256+832 x+580 x^2+420 x^3+945 x^4-108 x^5+324 x^6} \, dx=e^{\left (-\frac {81 \, x^{3} - 9 \, x^{2} + 129 \, x + 80}{18 \, x^{3} - 3 \, x^{2} + 26 \, x + 16}\right )} \] Input:
integrate((81*x^4+432*x^3+279*x^2-192*x+16)*exp((-81*x^3+9*x^2-129*x-80)/( 18*x^3-3*x^2+26*x+16))/(324*x^6-108*x^5+945*x^4+420*x^3+580*x^2+832*x+256) ,x, algorithm="fricas")
Output:
e^(-(81*x^3 - 9*x^2 + 129*x + 80)/(18*x^3 - 3*x^2 + 26*x + 16))
Time = 0.16 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.24 \[ \int \frac {e^{\frac {-80-129 x+9 x^2-81 x^3}{16+26 x-3 x^2+18 x^3}} \left (16-192 x+279 x^2+432 x^3+81 x^4\right )}{256+832 x+580 x^2+420 x^3+945 x^4-108 x^5+324 x^6} \, dx=e^{\frac {- 81 x^{3} + 9 x^{2} - 129 x - 80}{18 x^{3} - 3 x^{2} + 26 x + 16}} \] Input:
integrate((81*x**4+432*x**3+279*x**2-192*x+16)*exp((-81*x**3+9*x**2-129*x- 80)/(18*x**3-3*x**2+26*x+16))/(324*x**6-108*x**5+945*x**4+420*x**3+580*x** 2+832*x+256),x)
Output:
exp((-81*x**3 + 9*x**2 - 129*x - 80)/(18*x**3 - 3*x**2 + 26*x + 16))
Time = 0.24 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.64 \[ \int \frac {e^{\frac {-80-129 x+9 x^2-81 x^3}{16+26 x-3 x^2+18 x^3}} \left (16-192 x+279 x^2+432 x^3+81 x^4\right )}{256+832 x+580 x^2+420 x^3+945 x^4-108 x^5+324 x^6} \, dx=e^{\left (-\frac {27 \, x}{17 \, {\left (9 \, x^{2} - 6 \, x + 16\right )}} - \frac {96}{17 \, {\left (9 \, x^{2} - 6 \, x + 16\right )}} - \frac {5}{34 \, {\left (2 \, x + 1\right )}} - \frac {9}{2}\right )} \] Input:
integrate((81*x^4+432*x^3+279*x^2-192*x+16)*exp((-81*x^3+9*x^2-129*x-80)/( 18*x^3-3*x^2+26*x+16))/(324*x^6-108*x^5+945*x^4+420*x^3+580*x^2+832*x+256) ,x, algorithm="maxima")
Output:
e^(-27/17*x/(9*x^2 - 6*x + 16) - 96/17/(9*x^2 - 6*x + 16) - 5/34/(2*x + 1) - 9/2)
Leaf count of result is larger than twice the leaf count of optimal. 85 vs. \(2 (24) = 48\).
Time = 0.12 (sec) , antiderivative size = 85, normalized size of antiderivative = 3.40 \[ \int \frac {e^{\frac {-80-129 x+9 x^2-81 x^3}{16+26 x-3 x^2+18 x^3}} \left (16-192 x+279 x^2+432 x^3+81 x^4\right )}{256+832 x+580 x^2+420 x^3+945 x^4-108 x^5+324 x^6} \, dx=e^{\left (-\frac {81 \, x^{3}}{18 \, x^{3} - 3 \, x^{2} + 26 \, x + 16} + \frac {9 \, x^{2}}{18 \, x^{3} - 3 \, x^{2} + 26 \, x + 16} - \frac {129 \, x}{18 \, x^{3} - 3 \, x^{2} + 26 \, x + 16} - \frac {80}{18 \, x^{3} - 3 \, x^{2} + 26 \, x + 16}\right )} \] Input:
integrate((81*x^4+432*x^3+279*x^2-192*x+16)*exp((-81*x^3+9*x^2-129*x-80)/( 18*x^3-3*x^2+26*x+16))/(324*x^6-108*x^5+945*x^4+420*x^3+580*x^2+832*x+256) ,x, algorithm="giac")
Output:
e^(-81*x^3/(18*x^3 - 3*x^2 + 26*x + 16) + 9*x^2/(18*x^3 - 3*x^2 + 26*x + 1 6) - 129*x/(18*x^3 - 3*x^2 + 26*x + 16) - 80/(18*x^3 - 3*x^2 + 26*x + 16))
Time = 2.94 (sec) , antiderivative size = 88, normalized size of antiderivative = 3.52 \[ \int \frac {e^{\frac {-80-129 x+9 x^2-81 x^3}{16+26 x-3 x^2+18 x^3}} \left (16-192 x+279 x^2+432 x^3+81 x^4\right )}{256+832 x+580 x^2+420 x^3+945 x^4-108 x^5+324 x^6} \, dx={\mathrm {e}}^{-\frac {129\,x}{18\,x^3-3\,x^2+26\,x+16}}\,{\mathrm {e}}^{\frac {9\,x^2}{18\,x^3-3\,x^2+26\,x+16}}\,{\mathrm {e}}^{-\frac {81\,x^3}{18\,x^3-3\,x^2+26\,x+16}}\,{\mathrm {e}}^{-\frac {80}{18\,x^3-3\,x^2+26\,x+16}} \] Input:
int((exp(-(129*x - 9*x^2 + 81*x^3 + 80)/(26*x - 3*x^2 + 18*x^3 + 16))*(279 *x^2 - 192*x + 432*x^3 + 81*x^4 + 16))/(832*x + 580*x^2 + 420*x^3 + 945*x^ 4 - 108*x^5 + 324*x^6 + 256),x)
Output:
exp(-(129*x)/(26*x - 3*x^2 + 18*x^3 + 16))*exp((9*x^2)/(26*x - 3*x^2 + 18* x^3 + 16))*exp(-(81*x^3)/(26*x - 3*x^2 + 18*x^3 + 16))*exp(-80/(26*x - 3*x ^2 + 18*x^3 + 16))
Time = 0.20 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.64 \[ \int \frac {e^{\frac {-80-129 x+9 x^2-81 x^3}{16+26 x-3 x^2+18 x^3}} \left (16-192 x+279 x^2+432 x^3+81 x^4\right )}{256+832 x+580 x^2+420 x^3+945 x^4-108 x^5+324 x^6} \, dx=\frac {1}{e^{\frac {9 x^{3}+3 x^{2}+25 x +16}{18 x^{3}-3 x^{2}+26 x +16}} e^{4}} \] Input:
int((81*x^4+432*x^3+279*x^2-192*x+16)*exp((-81*x^3+9*x^2-129*x-80)/(18*x^3 -3*x^2+26*x+16))/(324*x^6-108*x^5+945*x^4+420*x^3+580*x^2+832*x+256),x)
Output:
1/(e**((9*x**3 + 3*x**2 + 25*x + 16)/(18*x**3 - 3*x**2 + 26*x + 16))*e**4)