Integrand size = 91, antiderivative size = 27 \[ \int \frac {e^{-\frac {5 x^2}{5 e^{2 x}-6 e^x x}} \left (100 e^{3 x}-240 e^{2 x} x+120 x^3-120 x^4+e^x \left (-56 x^2+200 x^3\right )\right )}{25 e^{3 x}-60 e^{2 x} x+36 e^x x^2} \, dx=4 e^{\frac {e^{-x} x^2}{-e^x+\frac {6 x}{5}}} x \] Output:
4*exp(x^2/(6/5*x-exp(x))/exp(x))*x
Time = 2.71 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.96 \[ \int \frac {e^{-\frac {5 x^2}{5 e^{2 x}-6 e^x x}} \left (100 e^{3 x}-240 e^{2 x} x+120 x^3-120 x^4+e^x \left (-56 x^2+200 x^3\right )\right )}{25 e^{3 x}-60 e^{2 x} x+36 e^x x^2} \, dx=4 e^{-\frac {5 e^{-x} x^2}{5 e^x-6 x}} x \] Input:
Integrate[(100*E^(3*x) - 240*E^(2*x)*x + 120*x^3 - 120*x^4 + E^x*(-56*x^2 + 200*x^3))/(E^((5*x^2)/(5*E^(2*x) - 6*E^x*x))*(25*E^(3*x) - 60*E^(2*x)*x + 36*E^x*x^2)),x]
Output:
(4*x)/E^((5*x^2)/(E^x*(5*E^x - 6*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.
\(\displaystyle \int \frac {e^{-\frac {5 x^2}{5 e^{2 x}-6 e^x x}} \left (-120 x^4+120 x^3+e^x \left (200 x^3-56 x^2\right )-240 e^{2 x} x+100 e^{3 x}\right )}{36 e^x x^2-60 e^{2 x} x+25 e^{3 x}} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {e^{-\frac {5 e^{-x} x^2}{5 e^x-6 x}-x} \left (-120 x^4+120 x^3+e^x \left (200 x^3-56 x^2\right )-240 e^{2 x} x+100 e^{3 x}\right )}{\left (5 e^x-6 x\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {40 e^{-\frac {5 e^{-x} x^2}{5 e^x-6 x}-x} (x-1) x^2}{5 e^x-6 x}+4 e^{-\frac {5 e^{-x} x^2}{5 e^x-6 x}}+\frac {120 e^{-\frac {5 e^{-x} x^2}{5 e^x-6 x}-x} (x-1) x^3}{\left (5 e^x-6 x\right )^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 4 \int e^{-\frac {5 e^{-x} x^2}{5 e^x-6 x}}dx-40 \int \frac {e^{-\frac {5 e^{-x} x^2}{5 e^x-6 x}-x} x^2}{5 e^x-6 x}dx+120 \int \frac {e^{-\frac {5 e^{-x} x^2}{5 e^x-6 x}-x} x^4}{\left (5 e^x-6 x\right )^2}dx-120 \int \frac {e^{-\frac {5 e^{-x} x^2}{5 e^x-6 x}-x} x^3}{\left (5 e^x-6 x\right )^2}dx+40 \int \frac {e^{-\frac {5 e^{-x} x^2}{5 e^x-6 x}-x} x^3}{5 e^x-6 x}dx\) |
Input:
Int[(100*E^(3*x) - 240*E^(2*x)*x + 120*x^3 - 120*x^4 + E^x*(-56*x^2 + 200* x^3))/(E^((5*x^2)/(5*E^(2*x) - 6*E^x*x))*(25*E^(3*x) - 60*E^(2*x)*x + 36*E ^x*x^2)),x]
Output:
$Aborted
Time = 0.44 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.89
method | result | size |
risch | \(4 x \,{\mathrm e}^{\frac {5 x^{2}}{-5 \,{\mathrm e}^{2 x}+6 \,{\mathrm e}^{x} x}}\) | \(24\) |
parallelrisch | \(4 \,{\mathrm e}^{-\frac {5 x^{2} {\mathrm e}^{-x}}{5 \,{\mathrm e}^{x}-6 x}} x\) | \(24\) |
Input:
int((100*exp(x)^3-240*x*exp(x)^2+(200*x^3-56*x^2)*exp(x)-120*x^4+120*x^3)* exp(-5*x^2/(5*exp(x)^2-6*exp(x)*x))/(25*exp(x)^3-60*x*exp(x)^2+36*exp(x)*x ^2),x,method=_RETURNVERBOSE)
Output:
4*x*exp(5*x^2/(-5*exp(2*x)+6*exp(x)*x))
Time = 0.10 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.85 \[ \int \frac {e^{-\frac {5 x^2}{5 e^{2 x}-6 e^x x}} \left (100 e^{3 x}-240 e^{2 x} x+120 x^3-120 x^4+e^x \left (-56 x^2+200 x^3\right )\right )}{25 e^{3 x}-60 e^{2 x} x+36 e^x x^2} \, dx=4 \, x e^{\left (\frac {5 \, x^{2}}{6 \, x e^{x} - 5 \, e^{\left (2 \, x\right )}}\right )} \] Input:
integrate((100*exp(x)^3-240*x*exp(x)^2+(200*x^3-56*x^2)*exp(x)-120*x^4+120 *x^3)*exp(-5*x^2/(5*exp(x)^2-6*exp(x)*x))/(25*exp(x)^3-60*x*exp(x)^2+36*ex p(x)*x^2),x, algorithm="fricas")
Output:
4*x*e^(5*x^2/(6*x*e^x - 5*e^(2*x)))
Time = 2.86 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.81 \[ \int \frac {e^{-\frac {5 x^2}{5 e^{2 x}-6 e^x x}} \left (100 e^{3 x}-240 e^{2 x} x+120 x^3-120 x^4+e^x \left (-56 x^2+200 x^3\right )\right )}{25 e^{3 x}-60 e^{2 x} x+36 e^x x^2} \, dx=4 x e^{- \frac {5 x^{2}}{- 6 x e^{x} + 5 e^{2 x}}} \] Input:
integrate((100*exp(x)**3-240*x*exp(x)**2+(200*x**3-56*x**2)*exp(x)-120*x** 4+120*x**3)*exp(-5*x**2/(5*exp(x)**2-6*exp(x)*x))/(25*exp(x)**3-60*x*exp(x )**2+36*exp(x)*x**2),x)
Output:
4*x*exp(-5*x**2/(-6*x*exp(x) + 5*exp(2*x)))
\[ \int \frac {e^{-\frac {5 x^2}{5 e^{2 x}-6 e^x x}} \left (100 e^{3 x}-240 e^{2 x} x+120 x^3-120 x^4+e^x \left (-56 x^2+200 x^3\right )\right )}{25 e^{3 x}-60 e^{2 x} x+36 e^x x^2} \, dx=\int { -\frac {4 \, {\left (30 \, x^{4} - 30 \, x^{3} + 60 \, x e^{\left (2 \, x\right )} - 2 \, {\left (25 \, x^{3} - 7 \, x^{2}\right )} e^{x} - 25 \, e^{\left (3 \, x\right )}\right )} e^{\left (\frac {5 \, x^{2}}{6 \, x e^{x} - 5 \, e^{\left (2 \, x\right )}}\right )}}{36 \, x^{2} e^{x} - 60 \, x e^{\left (2 \, x\right )} + 25 \, e^{\left (3 \, x\right )}} \,d x } \] Input:
integrate((100*exp(x)^3-240*x*exp(x)^2+(200*x^3-56*x^2)*exp(x)-120*x^4+120 *x^3)*exp(-5*x^2/(5*exp(x)^2-6*exp(x)*x))/(25*exp(x)^3-60*x*exp(x)^2+36*ex p(x)*x^2),x, algorithm="maxima")
Output:
-4*integrate((30*x^4 - 30*x^3 + 60*x*e^(2*x) - 2*(25*x^3 - 7*x^2)*e^x - 25 *e^(3*x))*e^(5*x^2/(6*x*e^x - 5*e^(2*x)))/(36*x^2*e^x - 60*x*e^(2*x) + 25* e^(3*x)), x)
Time = 0.14 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.59 \[ \int \frac {e^{-\frac {5 x^2}{5 e^{2 x}-6 e^x x}} \left (100 e^{3 x}-240 e^{2 x} x+120 x^3-120 x^4+e^x \left (-56 x^2+200 x^3\right )\right )}{25 e^{3 x}-60 e^{2 x} x+36 e^x x^2} \, dx=4 \, x e^{\left (-x + \frac {6 \, x^{2} e^{x} + 5 \, x^{2} - 5 \, x e^{\left (2 \, x\right )}}{6 \, x e^{x} - 5 \, e^{\left (2 \, x\right )}}\right )} \] Input:
integrate((100*exp(x)^3-240*x*exp(x)^2+(200*x^3-56*x^2)*exp(x)-120*x^4+120 *x^3)*exp(-5*x^2/(5*exp(x)^2-6*exp(x)*x))/(25*exp(x)^3-60*x*exp(x)^2+36*ex p(x)*x^2),x, algorithm="giac")
Output:
4*x*e^(-x + (6*x^2*e^x + 5*x^2 - 5*x*e^(2*x))/(6*x*e^x - 5*e^(2*x)))
Time = 3.48 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.85 \[ \int \frac {e^{-\frac {5 x^2}{5 e^{2 x}-6 e^x x}} \left (100 e^{3 x}-240 e^{2 x} x+120 x^3-120 x^4+e^x \left (-56 x^2+200 x^3\right )\right )}{25 e^{3 x}-60 e^{2 x} x+36 e^x x^2} \, dx=4\,x\,{\mathrm {e}}^{-\frac {5\,x^2}{5\,{\mathrm {e}}^{2\,x}-6\,x\,{\mathrm {e}}^x}} \] Input:
int(-(exp(-(5*x^2)/(5*exp(2*x) - 6*x*exp(x)))*(exp(x)*(56*x^2 - 200*x^3) - 100*exp(3*x) + 240*x*exp(2*x) - 120*x^3 + 120*x^4))/(25*exp(3*x) - 60*x*e xp(2*x) + 36*x^2*exp(x)),x)
Output:
4*x*exp(-(5*x^2)/(5*exp(2*x) - 6*x*exp(x)))
Time = 0.20 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.04 \[ \int \frac {e^{-\frac {5 x^2}{5 e^{2 x}-6 e^x x}} \left (100 e^{3 x}-240 e^{2 x} x+120 x^3-120 x^4+e^x \left (-56 x^2+200 x^3\right )\right )}{25 e^{3 x}-60 e^{2 x} x+36 e^x x^2} \, dx=\frac {4 x}{e^{\frac {5 x^{2}}{5 e^{2 x}-6 e^{x} x}}} \] Input:
int((100*exp(x)^3-240*x*exp(x)^2+(200*x^3-56*x^2)*exp(x)-120*x^4+120*x^3)* exp(-5*x^2/(5*exp(x)^2-6*exp(x)*x))/(25*exp(x)^3-60*x*exp(x)^2+36*exp(x)*x ^2),x)
Output:
(4*x)/e**((5*x**2)/(5*e**(2*x) - 6*e**x*x))