Integrand size = 83, antiderivative size = 28 \[ \int \frac {e^{\frac {6+6 x-15 x^2-5 x^3}{3 x^2+x^3}} \left (-180-216 x-96 x^2-21 x^3-6 x^4-x^5\right )}{7875 x^3+8400 x^4+3290 x^5+560 x^6+35 x^7} \, dx=\frac {e^{-5+\frac {2}{x^2}+\frac {4}{x (3+x)}}}{35 (5+x)} \] Output:
1/35*exp(2/x^2+4/(3+x)/x-5)/(5+x)
Time = 1.05 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.21 \[ \int \frac {e^{\frac {6+6 x-15 x^2-5 x^3}{3 x^2+x^3}} \left (-180-216 x-96 x^2-21 x^3-6 x^4-x^5\right )}{7875 x^3+8400 x^4+3290 x^5+560 x^6+35 x^7} \, dx=\frac {e^{-5+\frac {2}{x^2}+\frac {4}{3 x}-\frac {4}{3 (3+x)}}}{35 (5+x)} \] Input:
Integrate[(E^((6 + 6*x - 15*x^2 - 5*x^3)/(3*x^2 + x^3))*(-180 - 216*x - 96 *x^2 - 21*x^3 - 6*x^4 - x^5))/(7875*x^3 + 8400*x^4 + 3290*x^5 + 560*x^6 + 35*x^7),x]
Output:
E^(-5 + 2/x^2 + 4/(3*x) - 4/(3*(3 + x)))/(35*(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.
\(\displaystyle \int \frac {e^{\frac {-5 x^3-15 x^2+6 x+6}{x^3+3 x^2}} \left (-x^5-6 x^4-21 x^3-96 x^2-216 x-180\right )}{35 x^7+560 x^6+3290 x^5+8400 x^4+7875 x^3} \, dx\) |
\(\Big \downarrow \) 2026 |
\(\displaystyle \int \frac {e^{\frac {-5 x^3-15 x^2+6 x+6}{x^3+3 x^2}} \left (-x^5-6 x^4-21 x^3-96 x^2-216 x-180\right )}{x^3 \left (35 x^4+560 x^3+3290 x^2+8400 x+7875\right )}dx\) |
\(\Big \downarrow \) 2463 |
\(\displaystyle \int \left (-\frac {e^{\frac {-5 x^3-15 x^2+6 x+6}{x^3+3 x^2}} \left (-x^5-6 x^4-21 x^3-96 x^2-216 x-180\right )}{140 x^3 (x+3)}+\frac {e^{\frac {-5 x^3-15 x^2+6 x+6}{x^3+3 x^2}} \left (-x^5-6 x^4-21 x^3-96 x^2-216 x-180\right )}{140 x^3 (x+5)}+\frac {e^{\frac {-5 x^3-15 x^2+6 x+6}{x^3+3 x^2}} \left (-x^5-6 x^4-21 x^3-96 x^2-216 x-180\right )}{140 x^3 (x+3)^2}+\frac {e^{\frac {-5 x^3-15 x^2+6 x+6}{x^3+3 x^2}} \left (-x^5-6 x^4-21 x^3-96 x^2-216 x-180\right )}{140 x^3 (x+5)^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {4}{175} \int \frac {e^{\frac {-5 x^3-15 x^2+6 x+6}{x^2 (x+3)}}}{x^3}dx-\frac {8 \int \frac {e^{\frac {-5 x^3-15 x^2+6 x+6}{x^2 (x+3)}}}{x^2}dx}{2625}+\frac {8 \int \frac {e^{\frac {-5 x^3-15 x^2+6 x+6}{x^2 (x+3)}}}{x}dx}{13125}+\frac {2}{105} \int \frac {e^{\frac {-5 x^3-15 x^2+6 x+6}{x^2 (x+3)}}}{(x+3)^2}dx-\frac {1}{105} \int \frac {e^{\frac {-5 x^3-15 x^2+6 x+6}{x^2 (x+3)}}}{x+3}dx-\frac {1}{35} \int \frac {e^{\frac {-5 x^3-15 x^2+6 x+6}{x^2 (x+3)}}}{(x+5)^2}dx+\frac {39 \int \frac {e^{\frac {-5 x^3-15 x^2+6 x+6}{x^2 (x+3)}}}{x+5}dx}{4375}\) |
Input:
Int[(E^((6 + 6*x - 15*x^2 - 5*x^3)/(3*x^2 + x^3))*(-180 - 216*x - 96*x^2 - 21*x^3 - 6*x^4 - x^5))/(7875*x^3 + 8400*x^4 + 3290*x^5 + 560*x^6 + 35*x^7 ),x]
Output:
$Aborted
Time = 0.25 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.21
method | result | size |
gosper | \(\frac {{\mathrm e}^{-\frac {5 x^{3}+15 x^{2}-6 x -6}{x^{2} \left (3+x \right )}}}{175+35 x}\) | \(34\) |
risch | \(\frac {{\mathrm e}^{-\frac {5 x^{3}+15 x^{2}-6 x -6}{x^{2} \left (3+x \right )}}}{175+35 x}\) | \(34\) |
parallelrisch | \(\frac {{\mathrm e}^{-\frac {5 x^{3}+15 x^{2}-6 x -6}{x^{2} \left (3+x \right )}}}{175+35 x}\) | \(34\) |
norman | \(\frac {\frac {3 x^{2} {\mathrm e}^{\frac {-5 x^{3}-15 x^{2}+6 x +6}{x^{3}+3 x^{2}}}}{35}+\frac {x^{3} {\mathrm e}^{\frac {-5 x^{3}-15 x^{2}+6 x +6}{x^{3}+3 x^{2}}}}{35}}{x^{2} \left (x^{2}+8 x +15\right )}\) | \(82\) |
orering | \(-\frac {x^{3} \left (3+x \right )^{2} \left (5+x \right ) \left (-x^{5}-6 x^{4}-21 x^{3}-96 x^{2}-216 x -180\right ) {\mathrm e}^{\frac {-5 x^{3}-15 x^{2}+6 x +6}{x^{3}+3 x^{2}}}}{\left (x^{5}+6 x^{4}+21 x^{3}+96 x^{2}+216 x +180\right ) \left (35 x^{7}+560 x^{6}+3290 x^{5}+8400 x^{4}+7875 x^{3}\right )}\) | \(120\) |
Input:
int((-x^5-6*x^4-21*x^3-96*x^2-216*x-180)*exp((-5*x^3-15*x^2+6*x+6)/(x^3+3* x^2))/(35*x^7+560*x^6+3290*x^5+8400*x^4+7875*x^3),x,method=_RETURNVERBOSE)
Output:
1/35*exp(-(5*x^3+15*x^2-6*x-6)/x^2/(3+x))/(5+x)
Time = 0.11 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.29 \[ \int \frac {e^{\frac {6+6 x-15 x^2-5 x^3}{3 x^2+x^3}} \left (-180-216 x-96 x^2-21 x^3-6 x^4-x^5\right )}{7875 x^3+8400 x^4+3290 x^5+560 x^6+35 x^7} \, dx=\frac {e^{\left (-\frac {5 \, x^{3} + 15 \, x^{2} - 6 \, x - 6}{x^{3} + 3 \, x^{2}}\right )}}{35 \, {\left (x + 5\right )}} \] Input:
integrate((-x^5-6*x^4-21*x^3-96*x^2-216*x-180)*exp((-5*x^3-15*x^2+6*x+6)/( x^3+3*x^2))/(35*x^7+560*x^6+3290*x^5+8400*x^4+7875*x^3),x, algorithm="fric as")
Output:
1/35*e^(-(5*x^3 + 15*x^2 - 6*x - 6)/(x^3 + 3*x^2))/(x + 5)
Time = 0.10 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.04 \[ \int \frac {e^{\frac {6+6 x-15 x^2-5 x^3}{3 x^2+x^3}} \left (-180-216 x-96 x^2-21 x^3-6 x^4-x^5\right )}{7875 x^3+8400 x^4+3290 x^5+560 x^6+35 x^7} \, dx=\frac {e^{\frac {- 5 x^{3} - 15 x^{2} + 6 x + 6}{x^{3} + 3 x^{2}}}}{35 x + 175} \] Input:
integrate((-x**5-6*x**4-21*x**3-96*x**2-216*x-180)*exp((-5*x**3-15*x**2+6* x+6)/(x**3+3*x**2))/(35*x**7+560*x**6+3290*x**5+8400*x**4+7875*x**3),x)
Output:
exp((-5*x**3 - 15*x**2 + 6*x + 6)/(x**3 + 3*x**2))/(35*x + 175)
Time = 0.18 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.14 \[ \int \frac {e^{\frac {6+6 x-15 x^2-5 x^3}{3 x^2+x^3}} \left (-180-216 x-96 x^2-21 x^3-6 x^4-x^5\right )}{7875 x^3+8400 x^4+3290 x^5+560 x^6+35 x^7} \, dx=\frac {e^{\left (-\frac {4}{3 \, {\left (x + 3\right )}} + \frac {4}{3 \, x} + \frac {2}{x^{2}}\right )}}{35 \, {\left (x e^{5} + 5 \, e^{5}\right )}} \] Input:
integrate((-x^5-6*x^4-21*x^3-96*x^2-216*x-180)*exp((-5*x^3-15*x^2+6*x+6)/( x^3+3*x^2))/(35*x^7+560*x^6+3290*x^5+8400*x^4+7875*x^3),x, algorithm="maxi ma")
Output:
1/35*e^(-4/3/(x + 3) + 4/3/x + 2/x^2)/(x*e^5 + 5*e^5)
Time = 0.18 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.29 \[ \int \frac {e^{\frac {6+6 x-15 x^2-5 x^3}{3 x^2+x^3}} \left (-180-216 x-96 x^2-21 x^3-6 x^4-x^5\right )}{7875 x^3+8400 x^4+3290 x^5+560 x^6+35 x^7} \, dx=\frac {e^{\left (-\frac {5 \, x^{3} + 15 \, x^{2} - 6 \, x - 6}{x^{3} + 3 \, x^{2}}\right )}}{35 \, {\left (x + 5\right )}} \] Input:
integrate((-x^5-6*x^4-21*x^3-96*x^2-216*x-180)*exp((-5*x^3-15*x^2+6*x+6)/( x^3+3*x^2))/(35*x^7+560*x^6+3290*x^5+8400*x^4+7875*x^3),x, algorithm="giac ")
Output:
1/35*e^(-(5*x^3 + 15*x^2 - 6*x - 6)/(x^3 + 3*x^2))/(x + 5)
Time = 4.36 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.82 \[ \int \frac {e^{\frac {6+6 x-15 x^2-5 x^3}{3 x^2+x^3}} \left (-180-216 x-96 x^2-21 x^3-6 x^4-x^5\right )}{7875 x^3+8400 x^4+3290 x^5+560 x^6+35 x^7} \, dx=\frac {{\mathrm {e}}^{\frac {6}{x^2+3\,x}}\,{\mathrm {e}}^{-\frac {5\,x}{x+3}}\,{\mathrm {e}}^{\frac {6}{x^3+3\,x^2}}\,{\mathrm {e}}^{-\frac {15}{x+3}}}{35\,\left (x+5\right )} \] Input:
int(-(exp((6*x - 15*x^2 - 5*x^3 + 6)/(3*x^2 + x^3))*(216*x + 96*x^2 + 21*x ^3 + 6*x^4 + x^5 + 180))/(7875*x^3 + 8400*x^4 + 3290*x^5 + 560*x^6 + 35*x^ 7),x)
Output:
(exp(6/(3*x + x^2))*exp(-(5*x)/(x + 3))*exp(6/(3*x^2 + x^3))*exp(-15/(x + 3)))/(35*(x + 5))
Time = 0.17 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.04 \[ \int \frac {e^{\frac {6+6 x-15 x^2-5 x^3}{3 x^2+x^3}} \left (-180-216 x-96 x^2-21 x^3-6 x^4-x^5\right )}{7875 x^3+8400 x^4+3290 x^5+560 x^6+35 x^7} \, dx=\frac {e^{\frac {6 x +6}{x^{3}+3 x^{2}}}}{35 e^{5} \left (x +5\right )} \] Input:
int((-x^5-6*x^4-21*x^3-96*x^2-216*x-180)*exp((-5*x^3-15*x^2+6*x+6)/(x^3+3* x^2))/(35*x^7+560*x^6+3290*x^5+8400*x^4+7875*x^3),x)
Output:
e**((6*x + 6)/(x**3 + 3*x**2))/(35*e**5*(x + 5))