Integrand size = 109, antiderivative size = 36 \[ \int \frac {e^{e^{\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}} x^2+\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}} \left (1568 x+3276 x^2-1462 x^3-54 x^4+42 x^5+4 x^6\right )}{784-280 x-31 x^2+10 x^3+x^4} \, dx=e^{e^{x \left (5+\frac {x}{-x+\frac {5 (5-x)}{-\frac {3}{x}+x}}\right )} x^2} \] Output:
exp(x^2*exp((x/(5*(5-x)/(x-3/x)-x)+5)*x))
Time = 0.12 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.83 \[ \int \frac {e^{e^{\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}} x^2+\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}} \left (1568 x+3276 x^2-1462 x^3-54 x^4+42 x^5+4 x^6\right )}{784-280 x-31 x^2+10 x^3+x^4} \, dx=e^{e^{5+4 x-\frac {10 (-14+5 x)}{-28+5 x+x^2}} x^2} \] Input:
Integrate[(E^(E^((-137*x + 25*x^2 + 4*x^3)/(-28 + 5*x + x^2))*x^2 + (-137* x + 25*x^2 + 4*x^3)/(-28 + 5*x + x^2))*(1568*x + 3276*x^2 - 1462*x^3 - 54* x^4 + 42*x^5 + 4*x^6))/(784 - 280*x - 31*x^2 + 10*x^3 + x^4),x]
Output:
E^(E^(5 + 4*x - (10*(-14 + 5*x))/(-28 + 5*x + x^2))*x^2)
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.
| \(\displaystyle \int \frac {\left (4 x^6+42 x^5-54 x^4-1462 x^3+3276 x^2+1568 x\right ) \exp \left (e^{\frac {4 x^3+25 x^2-137 x}{x^2+5 x-28}} x^2+\frac {4 x^3+25 x^2-137 x}{x^2+5 x-28}\right )}{x^4+10 x^3-31 x^2-280 x+784} \, dx\) |
| \(\Big \downarrow \) 2463 |
| \(\displaystyle \int \left (\frac {4 \left (4 x^6+42 x^5-54 x^4-1462 x^3+3276 x^2+1568 x\right ) \exp \left (e^{\frac {4 x^3+25 x^2-137 x}{x^2+5 x-28}} x^2+\frac {4 x^3+25 x^2-137 x}{x^2+5 x-28}\right )}{137 \sqrt {137} \left (-2 x+\sqrt {137}-5\right )}+\frac {4 \left (4 x^6+42 x^5-54 x^4-1462 x^3+3276 x^2+1568 x\right ) \exp \left (e^{\frac {4 x^3+25 x^2-137 x}{x^2+5 x-28}} x^2+\frac {4 x^3+25 x^2-137 x}{x^2+5 x-28}\right )}{137 \sqrt {137} \left (2 x+\sqrt {137}+5\right )}+\frac {4 \left (4 x^6+42 x^5-54 x^4-1462 x^3+3276 x^2+1568 x\right ) \exp \left (e^{\frac {4 x^3+25 x^2-137 x}{x^2+5 x-28}} x^2+\frac {4 x^3+25 x^2-137 x}{x^2+5 x-28}\right )}{137 \left (-2 x+\sqrt {137}-5\right )^2}+\frac {4 \left (4 x^6+42 x^5-54 x^4-1462 x^3+3276 x^2+1568 x\right ) \exp \left (e^{\frac {4 x^3+25 x^2-137 x}{x^2+5 x-28}} x^2+\frac {4 x^3+25 x^2-137 x}{x^2+5 x-28}\right )}{137 \left (2 x+\sqrt {137}+5\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {2 x \left (2 x^5+21 x^4-27 x^3-731 x^2+1638 x+784\right ) \exp \left (-\frac {x \left (4 x^2+e^{\frac {x \left (4 x^2+25 x-137\right )}{x^2+5 x-28}} \left (x^2+5 x-28\right ) x+25 x-137\right )}{-x^2-5 x+28}\right )}{\left (-x^2-5 x+28\right )^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 \int \frac {\exp \left (\frac {x \left (-4 x^2+e^{\frac {x \left (-4 x^2-25 x+137\right )}{-x^2-5 x+28}} \left (-x^2-5 x+28\right ) x-25 x+137\right )}{-x^2-5 x+28}\right ) x \left (2 x^5+21 x^4-27 x^3-731 x^2+1638 x+784\right )}{\left (-x^2-5 x+28\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 2 \int \left (2 \exp \left (\frac {x \left (-4 x^2+e^{\frac {x \left (-4 x^2-25 x+137\right )}{-x^2-5 x+28}} \left (-x^2-5 x+28\right ) x-25 x+137\right )}{-x^2-5 x+28}\right ) x^2+\exp \left (\frac {x \left (-4 x^2+e^{\frac {x \left (-4 x^2-25 x+137\right )}{-x^2-5 x+28}} \left (-x^2-5 x+28\right ) x-25 x+137\right )}{-x^2-5 x+28}\right ) x+25 \exp \left (\frac {x \left (-4 x^2+e^{\frac {x \left (-4 x^2-25 x+137\right )}{-x^2-5 x+28}} \left (-x^2-5 x+28\right ) x-25 x+137\right )}{-x^2-5 x+28}\right )-\frac {15 \exp \left (\frac {x \left (-4 x^2+e^{\frac {x \left (-4 x^2-25 x+137\right )}{-x^2-5 x+28}} \left (-x^2-5 x+28\right ) x-25 x+137\right )}{-x^2-5 x+28}\right ) (26 x-205)}{x^2+5 x-28}-\frac {5 \exp \left (\frac {x \left (-4 x^2+e^{\frac {x \left (-4 x^2-25 x+137\right )}{-x^2-5 x+28}} \left (-x^2-5 x+28\right ) x-25 x+137\right )}{-x^2-5 x+28}\right ) (3859 x-13300)}{\left (x^2+5 x-28\right )^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 2 \left (25 \int e^{\frac {x \left (-4 x^2+e^{\frac {x \left (-4 x^2-25 x+137\right )}{-x^2-5 x+28}} \left (-x^2-5 x+28\right ) x-25 x+137\right )}{-x^2-5 x+28}}dx+\frac {38590}{137} \left (5-\sqrt {137}\right ) \int \frac {e^{\frac {x \left (-4 x^2+e^{\frac {x \left (-4 x^2-25 x+137\right )}{-x^2-5 x+28}} \left (-x^2-5 x+28\right ) x-25 x+137\right )}{-x^2-5 x+28}}}{\left (-2 x+\sqrt {137}-5\right )^2}dx+\frac {266000}{137} \int \frac {e^{\frac {x \left (-4 x^2+e^{\frac {x \left (-4 x^2-25 x+137\right )}{-x^2-5 x+28}} \left (-x^2-5 x+28\right ) x-25 x+137\right )}{-x^2-5 x+28}}}{\left (-2 x+\sqrt {137}-5\right )^2}dx+\frac {3350 \int \frac {e^{\frac {x \left (-4 x^2+e^{\frac {x \left (-4 x^2-25 x+137\right )}{-x^2-5 x+28}} \left (-x^2-5 x+28\right ) x-25 x+137\right )}{-x^2-5 x+28}}}{-2 x+\sqrt {137}-5}dx}{\sqrt {137}}+\int e^{\frac {x \left (-4 x^2+e^{\frac {x \left (-4 x^2-25 x+137\right )}{-x^2-5 x+28}} \left (-x^2-5 x+28\right ) x-25 x+137\right )}{-x^2-5 x+28}} xdx+2 \int e^{\frac {x \left (-4 x^2+e^{\frac {x \left (-4 x^2-25 x+137\right )}{-x^2-5 x+28}} \left (-x^2-5 x+28\right ) x-25 x+137\right )}{-x^2-5 x+28}} x^2dx-\frac {30}{137} \left (1781-270 \sqrt {137}\right ) \int \frac {e^{\frac {x \left (-4 x^2+e^{\frac {x \left (-4 x^2-25 x+137\right )}{-x^2-5 x+28}} \left (-x^2-5 x+28\right ) x-25 x+137\right )}{-x^2-5 x+28}}}{2 x-\sqrt {137}+5}dx+\frac {38590}{137} \left (5+\sqrt {137}\right ) \int \frac {e^{\frac {x \left (-4 x^2+e^{\frac {x \left (-4 x^2-25 x+137\right )}{-x^2-5 x+28}} \left (-x^2-5 x+28\right ) x-25 x+137\right )}{-x^2-5 x+28}}}{\left (2 x+\sqrt {137}+5\right )^2}dx+\frac {266000}{137} \int \frac {e^{\frac {x \left (-4 x^2+e^{\frac {x \left (-4 x^2-25 x+137\right )}{-x^2-5 x+28}} \left (-x^2-5 x+28\right ) x-25 x+137\right )}{-x^2-5 x+28}}}{\left (2 x+\sqrt {137}+5\right )^2}dx-\frac {30}{137} \left (1781+270 \sqrt {137}\right ) \int \frac {e^{\frac {x \left (-4 x^2+e^{\frac {x \left (-4 x^2-25 x+137\right )}{-x^2-5 x+28}} \left (-x^2-5 x+28\right ) x-25 x+137\right )}{-x^2-5 x+28}}}{2 x+\sqrt {137}+5}dx+\frac {3350 \int \frac {e^{\frac {x \left (-4 x^2+e^{\frac {x \left (-4 x^2-25 x+137\right )}{-x^2-5 x+28}} \left (-x^2-5 x+28\right ) x-25 x+137\right )}{-x^2-5 x+28}}}{2 x+\sqrt {137}+5}dx}{\sqrt {137}}\right )\) |
Input:
Int[(E^(E^((-137*x + 25*x^2 + 4*x^3)/(-28 + 5*x + x^2))*x^2 + (-137*x + 25 *x^2 + 4*x^3)/(-28 + 5*x + x^2))*(1568*x + 3276*x^2 - 1462*x^3 - 54*x^4 + 42*x^5 + 4*x^6))/(784 - 280*x - 31*x^2 + 10*x^3 + x^4),x]
Output:
$Aborted
Time = 18.79 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.81
| method | result | size |
| risch | \({\mathrm e}^{x^{2} {\mathrm e}^{\frac {x \left (4 x^{2}+25 x -137\right )}{x^{2}+5 x -28}}}\) | \(29\) |
| norman | \(\frac {x^{2} {\mathrm e}^{x^{2} {\mathrm e}^{\frac {4 x^{3}+25 x^{2}-137 x}{x^{2}+5 x -28}}}+5 x \,{\mathrm e}^{x^{2} {\mathrm e}^{\frac {4 x^{3}+25 x^{2}-137 x}{x^{2}+5 x -28}}}-28 \,{\mathrm e}^{x^{2} {\mathrm e}^{\frac {4 x^{3}+25 x^{2}-137 x}{x^{2}+5 x -28}}}}{x^{2}+5 x -28}\) | \(115\) |
Input:
int((4*x^6+42*x^5-54*x^4-1462*x^3+3276*x^2+1568*x)*exp((4*x^3+25*x^2-137*x )/(x^2+5*x-28))*exp(x^2*exp((4*x^3+25*x^2-137*x)/(x^2+5*x-28)))/(x^4+10*x^ 3-31*x^2-280*x+784),x,method=_RETURNVERBOSE)
Output:
exp(x^2*exp(x*(4*x^2+25*x-137)/(x^2+5*x-28)))
Leaf count of result is larger than twice the leaf count of optimal. 94 vs. \(2 (31) = 62\).
Time = 0.08 (sec) , antiderivative size = 94, normalized size of antiderivative = 2.61 \[ \int \frac {e^{e^{\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}} x^2+\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}} \left (1568 x+3276 x^2-1462 x^3-54 x^4+42 x^5+4 x^6\right )}{784-280 x-31 x^2+10 x^3+x^4} \, dx=e^{\left (\frac {4 \, x^{3} + 25 \, x^{2} + {\left (x^{4} + 5 \, x^{3} - 28 \, x^{2}\right )} e^{\left (\frac {4 \, x^{3} + 25 \, x^{2} - 137 \, x}{x^{2} + 5 \, x - 28}\right )} - 137 \, x}{x^{2} + 5 \, x - 28} - \frac {4 \, x^{3} + 25 \, x^{2} - 137 \, x}{x^{2} + 5 \, x - 28}\right )} \] Input:
integrate((4*x^6+42*x^5-54*x^4-1462*x^3+3276*x^2+1568*x)*exp((4*x^3+25*x^2 -137*x)/(x^2+5*x-28))*exp(x^2*exp((4*x^3+25*x^2-137*x)/(x^2+5*x-28)))/(x^4 +10*x^3-31*x^2-280*x+784),x, algorithm="fricas")
Output:
e^((4*x^3 + 25*x^2 + (x^4 + 5*x^3 - 28*x^2)*e^((4*x^3 + 25*x^2 - 137*x)/(x ^2 + 5*x - 28)) - 137*x)/(x^2 + 5*x - 28) - (4*x^3 + 25*x^2 - 137*x)/(x^2 + 5*x - 28))
Time = 0.37 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.75 \[ \int \frac {e^{e^{\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}} x^2+\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}} \left (1568 x+3276 x^2-1462 x^3-54 x^4+42 x^5+4 x^6\right )}{784-280 x-31 x^2+10 x^3+x^4} \, dx=e^{x^{2} e^{\frac {4 x^{3} + 25 x^{2} - 137 x}{x^{2} + 5 x - 28}}} \] Input:
integrate((4*x**6+42*x**5-54*x**4-1462*x**3+3276*x**2+1568*x)*exp((4*x**3+ 25*x**2-137*x)/(x**2+5*x-28))*exp(x**2*exp((4*x**3+25*x**2-137*x)/(x**2+5* x-28)))/(x**4+10*x**3-31*x**2-280*x+784),x)
Output:
exp(x**2*exp((4*x**3 + 25*x**2 - 137*x)/(x**2 + 5*x - 28)))
Time = 0.48 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00 \[ \int \frac {e^{e^{\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}} x^2+\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}} \left (1568 x+3276 x^2-1462 x^3-54 x^4+42 x^5+4 x^6\right )}{784-280 x-31 x^2+10 x^3+x^4} \, dx=e^{\left (x^{2} e^{\left (4 \, x - \frac {50 \, x}{x^{2} + 5 \, x - 28} + \frac {140}{x^{2} + 5 \, x - 28} + 5\right )}\right )} \] Input:
integrate((4*x^6+42*x^5-54*x^4-1462*x^3+3276*x^2+1568*x)*exp((4*x^3+25*x^2 -137*x)/(x^2+5*x-28))*exp(x^2*exp((4*x^3+25*x^2-137*x)/(x^2+5*x-28)))/(x^4 +10*x^3-31*x^2-280*x+784),x, algorithm="maxima")
Output:
e^(x^2*e^(4*x - 50*x/(x^2 + 5*x - 28) + 140/(x^2 + 5*x - 28) + 5))
\[ \int \frac {e^{e^{\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}} x^2+\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}} \left (1568 x+3276 x^2-1462 x^3-54 x^4+42 x^5+4 x^6\right )}{784-280 x-31 x^2+10 x^3+x^4} \, dx=\int { \frac {2 \, {\left (2 \, x^{6} + 21 \, x^{5} - 27 \, x^{4} - 731 \, x^{3} + 1638 \, x^{2} + 784 \, x\right )} e^{\left (x^{2} e^{\left (\frac {4 \, x^{3} + 25 \, x^{2} - 137 \, x}{x^{2} + 5 \, x - 28}\right )} + \frac {4 \, x^{3} + 25 \, x^{2} - 137 \, x}{x^{2} + 5 \, x - 28}\right )}}{x^{4} + 10 \, x^{3} - 31 \, x^{2} - 280 \, x + 784} \,d x } \] Input:
integrate((4*x^6+42*x^5-54*x^4-1462*x^3+3276*x^2+1568*x)*exp((4*x^3+25*x^2 -137*x)/(x^2+5*x-28))*exp(x^2*exp((4*x^3+25*x^2-137*x)/(x^2+5*x-28)))/(x^4 +10*x^3-31*x^2-280*x+784),x, algorithm="giac")
Output:
integrate(2*(2*x^6 + 21*x^5 - 27*x^4 - 731*x^3 + 1638*x^2 + 784*x)*e^(x^2* e^((4*x^3 + 25*x^2 - 137*x)/(x^2 + 5*x - 28)) + (4*x^3 + 25*x^2 - 137*x)/( x^2 + 5*x - 28))/(x^4 + 10*x^3 - 31*x^2 - 280*x + 784), x)
Time = 0.21 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.86 \[ \int \frac {e^{e^{\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}} x^2+\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}} \left (1568 x+3276 x^2-1462 x^3-54 x^4+42 x^5+4 x^6\right )}{784-280 x-31 x^2+10 x^3+x^4} \, dx={\mathrm {e}}^{x^2\,{\mathrm {e}}^{\frac {4\,x^3+25\,x^2-137\,x}{x^2+5\,x-28}}} \] Input:
int((exp(x^2*exp((25*x^2 - 137*x + 4*x^3)/(5*x + x^2 - 28)))*exp((25*x^2 - 137*x + 4*x^3)/(5*x + x^2 - 28))*(1568*x + 3276*x^2 - 1462*x^3 - 54*x^4 + 42*x^5 + 4*x^6))/(10*x^3 - 31*x^2 - 280*x + x^4 + 784),x)
Output:
exp(x^2*exp((25*x^2 - 137*x + 4*x^3)/(5*x + x^2 - 28)))
\[ \int \frac {e^{e^{\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}} x^2+\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}} \left (1568 x+3276 x^2-1462 x^3-54 x^4+42 x^5+4 x^6\right )}{784-280 x-31 x^2+10 x^3+x^4} \, dx=\text {too large to display} \] Input:
int((4*x^6+42*x^5-54*x^4-1462*x^3+3276*x^2+1568*x)*exp((4*x^3+25*x^2-137*x )/(x^2+5*x-28))*exp(x^2*exp((4*x^3+25*x^2-137*x)/(x^2+5*x-28)))/(x^4+10*x^ 3-31*x^2-280*x+784),x)
Output:
2*e**5*(2*int((e**((4*e**((50*x)/(x**2 + 5*x - 28))*x**3 + 20*e**((50*x)/( x**2 + 5*x - 28))*x**2 - 112*e**((50*x)/(x**2 + 5*x - 28))*x + 140*e**((50 *x)/(x**2 + 5*x - 28)) + e**((4*x**3 + 20*x**2 - 112*x + 140)/(x**2 + 5*x - 28))*e**5*x**4 + 5*e**((4*x**3 + 20*x**2 - 112*x + 140)/(x**2 + 5*x - 28 ))*e**5*x**3)/(e**((50*x)/(x**2 + 5*x - 28))*x**2 + 5*e**((50*x)/(x**2 + 5 *x - 28))*x - 28*e**((50*x)/(x**2 + 5*x - 28))))*x**6)/(e**((50*e**((50*x) /(x**2 + 5*x - 28))*x + 28*e**((4*x**3 + 20*x**2 - 112*x + 140)/(x**2 + 5* x - 28))*e**5*x**2)/(e**((50*x)/(x**2 + 5*x - 28))*x**2 + 5*e**((50*x)/(x* *2 + 5*x - 28))*x - 28*e**((50*x)/(x**2 + 5*x - 28))))*x**4 + 10*e**((50*e **((50*x)/(x**2 + 5*x - 28))*x + 28*e**((4*x**3 + 20*x**2 - 112*x + 140)/( x**2 + 5*x - 28))*e**5*x**2)/(e**((50*x)/(x**2 + 5*x - 28))*x**2 + 5*e**(( 50*x)/(x**2 + 5*x - 28))*x - 28*e**((50*x)/(x**2 + 5*x - 28))))*x**3 - 31* e**((50*e**((50*x)/(x**2 + 5*x - 28))*x + 28*e**((4*x**3 + 20*x**2 - 112*x + 140)/(x**2 + 5*x - 28))*e**5*x**2)/(e**((50*x)/(x**2 + 5*x - 28))*x**2 + 5*e**((50*x)/(x**2 + 5*x - 28))*x - 28*e**((50*x)/(x**2 + 5*x - 28))))*x **2 - 280*e**((50*e**((50*x)/(x**2 + 5*x - 28))*x + 28*e**((4*x**3 + 20*x* *2 - 112*x + 140)/(x**2 + 5*x - 28))*e**5*x**2)/(e**((50*x)/(x**2 + 5*x - 28))*x**2 + 5*e**((50*x)/(x**2 + 5*x - 28))*x - 28*e**((50*x)/(x**2 + 5*x - 28))))*x + 784*e**((50*e**((50*x)/(x**2 + 5*x - 28))*x + 28*e**((4*x**3 + 20*x**2 - 112*x + 140)/(x**2 + 5*x - 28))*e**5*x**2)/(e**((50*x)/(x**...