Integrand size = 70, antiderivative size = 25 \[ \int \frac {e^{\frac {x^2}{6+14 x+6 x^2+4 x^3}} \left (6 x+7 x^2-2 x^4\right )}{18+84 x+134 x^2+108 x^3+74 x^4+24 x^5+8 x^6} \, dx=e^{\frac {x}{2 (1+2 x) \left (x+\frac {3+x}{x}\right )}} \] Output:
exp(1/2*x/((3+x)/x+x)/(1+2*x))
Time = 0.11 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92 \[ \int \frac {e^{\frac {x^2}{6+14 x+6 x^2+4 x^3}} \left (6 x+7 x^2-2 x^4\right )}{18+84 x+134 x^2+108 x^3+74 x^4+24 x^5+8 x^6} \, dx=e^{\frac {x^2}{6+14 x+6 x^2+4 x^3}} \] Input:
Integrate[(E^(x^2/(6 + 14*x + 6*x^2 + 4*x^3))*(6*x + 7*x^2 - 2*x^4))/(18 + 84*x + 134*x^2 + 108*x^3 + 74*x^4 + 24*x^5 + 8*x^6),x]
Output:
E^(x^2/(6 + 14*x + 6*x^2 + 4*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.
\(\displaystyle \int \frac {e^{\frac {x^2}{4 x^3+6 x^2+14 x+6}} \left (-2 x^4+7 x^2+6 x\right )}{8 x^6+24 x^5+74 x^4+108 x^3+134 x^2+84 x+18} \, dx\) |
\(\Big \downarrow \) 2028 |
\(\displaystyle \int \frac {e^{\frac {x^2}{4 x^3+6 x^2+14 x+6}} x \left (-2 x^3+7 x+6\right )}{8 x^6+24 x^5+74 x^4+108 x^3+134 x^2+84 x+18}dx\) |
\(\Big \downarrow \) 2463 |
\(\displaystyle \int \left (-\frac {2 e^{\frac {x^2}{4 x^3+6 x^2+14 x+6}} x \left (-2 x^3+7 x+6\right )}{121 \left (x^2+x+3\right )}+\frac {8 e^{\frac {x^2}{4 x^3+6 x^2+14 x+6}} x \left (-2 x^3+7 x+6\right )}{121 (2 x+1)^2}-\frac {e^{\frac {x^2}{4 x^3+6 x^2+14 x+6}} x \left (-2 x^3+7 x+6\right )}{22 \left (x^2+x+3\right )^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{11} \left (1-i \sqrt {11}\right ) \int \frac {e^{\frac {x^2}{4 x^3+6 x^2+14 x+6}}}{\left (-2 x+i \sqrt {11}-1\right )^2}dx-\frac {6}{11} \int \frac {e^{\frac {x^2}{4 x^3+6 x^2+14 x+6}}}{\left (-2 x+i \sqrt {11}-1\right )^2}dx+\frac {5 i \int \frac {e^{\frac {x^2}{4 x^3+6 x^2+14 x+6}}}{-2 x+i \sqrt {11}-1}dx}{11 \sqrt {11}}-\frac {1}{11} \int \frac {e^{\frac {x^2}{4 x^3+6 x^2+14 x+6}}}{(2 x+1)^2}dx-\frac {1}{121} \left (22-15 i \sqrt {11}\right ) \int \frac {e^{\frac {x^2}{4 x^3+6 x^2+14 x+6}}}{2 x-i \sqrt {11}+1}dx+\frac {2}{121} \left (11-5 i \sqrt {11}\right ) \int \frac {e^{\frac {x^2}{4 x^3+6 x^2+14 x+6}}}{2 x-i \sqrt {11}+1}dx+\frac {1}{11} \left (1+i \sqrt {11}\right ) \int \frac {e^{\frac {x^2}{4 x^3+6 x^2+14 x+6}}}{\left (2 x+i \sqrt {11}+1\right )^2}dx-\frac {6}{11} \int \frac {e^{\frac {x^2}{4 x^3+6 x^2+14 x+6}}}{\left (2 x+i \sqrt {11}+1\right )^2}dx-\frac {1}{121} \left (22+15 i \sqrt {11}\right ) \int \frac {e^{\frac {x^2}{4 x^3+6 x^2+14 x+6}}}{2 x+i \sqrt {11}+1}dx+\frac {2}{121} \left (11+5 i \sqrt {11}\right ) \int \frac {e^{\frac {x^2}{4 x^3+6 x^2+14 x+6}}}{2 x+i \sqrt {11}+1}dx+\frac {5 i \int \frac {e^{\frac {x^2}{4 x^3+6 x^2+14 x+6}}}{2 x+i \sqrt {11}+1}dx}{11 \sqrt {11}}\) |
Input:
Int[(E^(x^2/(6 + 14*x + 6*x^2 + 4*x^3))*(6*x + 7*x^2 - 2*x^4))/(18 + 84*x + 134*x^2 + 108*x^3 + 74*x^4 + 24*x^5 + 8*x^6),x]
Output:
$Aborted
Time = 0.28 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.88
method | result | size |
risch | \({\mathrm e}^{\frac {x^{2}}{2 \left (1+2 x \right ) \left (x^{2}+x +3\right )}}\) | \(22\) |
gosper | \({\mathrm e}^{\frac {x^{2}}{4 x^{3}+6 x^{2}+14 x +6}}\) | \(24\) |
parallelrisch | \({\mathrm e}^{\frac {x^{2}}{4 x^{3}+6 x^{2}+14 x +6}}\) | \(24\) |
orering | \(-\frac {2 \left (1+2 x \right )^{2} \left (x^{2}+x +3\right )^{2} \left (-2 x^{4}+7 x^{2}+6 x \right ) {\mathrm e}^{\frac {x^{2}}{4 x^{3}+6 x^{2}+14 x +6}}}{x \left (2 x^{3}-7 x -6\right ) \left (8 x^{6}+24 x^{5}+74 x^{4}+108 x^{3}+134 x^{2}+84 x +18\right )}\) | \(101\) |
norman | \(\frac {7 x \,{\mathrm e}^{\frac {x^{2}}{4 x^{3}+6 x^{2}+14 x +6}}+3 x^{2} {\mathrm e}^{\frac {x^{2}}{4 x^{3}+6 x^{2}+14 x +6}}+2 x^{3} {\mathrm e}^{\frac {x^{2}}{4 x^{3}+6 x^{2}+14 x +6}}+3 \,{\mathrm e}^{\frac {x^{2}}{4 x^{3}+6 x^{2}+14 x +6}}}{2 x^{3}+3 x^{2}+7 x +3}\) | \(123\) |
Input:
int((-2*x^4+7*x^2+6*x)*exp(x^2/(4*x^3+6*x^2+14*x+6))/(8*x^6+24*x^5+74*x^4+ 108*x^3+134*x^2+84*x+18),x,method=_RETURNVERBOSE)
Output:
exp(1/2*x^2/(1+2*x)/(x^2+x+3))
Time = 0.10 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92 \[ \int \frac {e^{\frac {x^2}{6+14 x+6 x^2+4 x^3}} \left (6 x+7 x^2-2 x^4\right )}{18+84 x+134 x^2+108 x^3+74 x^4+24 x^5+8 x^6} \, dx=e^{\left (\frac {x^{2}}{2 \, {\left (2 \, x^{3} + 3 \, x^{2} + 7 \, x + 3\right )}}\right )} \] Input:
integrate((-2*x^4+7*x^2+6*x)*exp(x^2/(4*x^3+6*x^2+14*x+6))/(8*x^6+24*x^5+7 4*x^4+108*x^3+134*x^2+84*x+18),x, algorithm="fricas")
Output:
e^(1/2*x^2/(2*x^3 + 3*x^2 + 7*x + 3))
Time = 0.09 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.76 \[ \int \frac {e^{\frac {x^2}{6+14 x+6 x^2+4 x^3}} \left (6 x+7 x^2-2 x^4\right )}{18+84 x+134 x^2+108 x^3+74 x^4+24 x^5+8 x^6} \, dx=e^{\frac {x^{2}}{4 x^{3} + 6 x^{2} + 14 x + 6}} \] Input:
integrate((-2*x**4+7*x**2+6*x)*exp(x**2/(4*x**3+6*x**2+14*x+6))/(8*x**6+24 *x**5+74*x**4+108*x**3+134*x**2+84*x+18),x)
Output:
exp(x**2/(4*x**3 + 6*x**2 + 14*x + 6))
Time = 0.14 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.28 \[ \int \frac {e^{\frac {x^2}{6+14 x+6 x^2+4 x^3}} \left (6 x+7 x^2-2 x^4\right )}{18+84 x+134 x^2+108 x^3+74 x^4+24 x^5+8 x^6} \, dx=e^{\left (\frac {5 \, x}{22 \, {\left (x^{2} + x + 3\right )}} - \frac {3}{22 \, {\left (x^{2} + x + 3\right )}} + \frac {1}{22 \, {\left (2 \, x + 1\right )}}\right )} \] Input:
integrate((-2*x^4+7*x^2+6*x)*exp(x^2/(4*x^3+6*x^2+14*x+6))/(8*x^6+24*x^5+7 4*x^4+108*x^3+134*x^2+84*x+18),x, algorithm="maxima")
Output:
e^(5/22*x/(x^2 + x + 3) - 3/22/(x^2 + x + 3) + 1/22/(2*x + 1))
Time = 0.11 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92 \[ \int \frac {e^{\frac {x^2}{6+14 x+6 x^2+4 x^3}} \left (6 x+7 x^2-2 x^4\right )}{18+84 x+134 x^2+108 x^3+74 x^4+24 x^5+8 x^6} \, dx=e^{\left (\frac {x^{2}}{2 \, {\left (2 \, x^{3} + 3 \, x^{2} + 7 \, x + 3\right )}}\right )} \] Input:
integrate((-2*x^4+7*x^2+6*x)*exp(x^2/(4*x^3+6*x^2+14*x+6))/(8*x^6+24*x^5+7 4*x^4+108*x^3+134*x^2+84*x+18),x, algorithm="giac")
Output:
e^(1/2*x^2/(2*x^3 + 3*x^2 + 7*x + 3))
Time = 7.30 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92 \[ \int \frac {e^{\frac {x^2}{6+14 x+6 x^2+4 x^3}} \left (6 x+7 x^2-2 x^4\right )}{18+84 x+134 x^2+108 x^3+74 x^4+24 x^5+8 x^6} \, dx={\mathrm {e}}^{\frac {1}{22\,\left (2\,x+1\right )}+\frac {\frac {5\,x}{22}-\frac {3}{22}}{x^2+x+3}} \] Input:
int((exp(x^2/(14*x + 6*x^2 + 4*x^3 + 6))*(6*x + 7*x^2 - 2*x^4))/(84*x + 13 4*x^2 + 108*x^3 + 74*x^4 + 24*x^5 + 8*x^6 + 18),x)
Output:
exp(1/(22*(2*x + 1)) + ((5*x)/22 - 3/22)/(x + x^2 + 3))
Time = 0.19 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92 \[ \int \frac {e^{\frac {x^2}{6+14 x+6 x^2+4 x^3}} \left (6 x+7 x^2-2 x^4\right )}{18+84 x+134 x^2+108 x^3+74 x^4+24 x^5+8 x^6} \, dx=e^{\frac {x^{2}}{4 x^{3}+6 x^{2}+14 x +6}} \] Input:
int((-2*x^4+7*x^2+6*x)*exp(x^2/(4*x^3+6*x^2+14*x+6))/(8*x^6+24*x^5+74*x^4+ 108*x^3+134*x^2+84*x+18),x)
Output:
e**(x**2/(4*x**3 + 6*x**2 + 14*x + 6))