Integrand size = 74, antiderivative size = 31 \[ \int \frac {4 x^3+2 x^4+e^{\frac {-x^3+e^{x^2} \left (25-10 x+x^2\right )}{x^2}} \left (-x^3+e^{x^2} \left (-50+10 x+50 x^2-20 x^3+2 x^4\right )\right )}{x^3} \, dx=e^{e^{x^2} \left (2+\frac {-5-x}{x}\right )^2-x}+(2+x)^2 \] Output:
(2+x)^2+exp(exp(x^2)*((-x-5)/x+2)^2-x)
Time = 1.20 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.87 \[ \int \frac {4 x^3+2 x^4+e^{\frac {-x^3+e^{x^2} \left (25-10 x+x^2\right )}{x^2}} \left (-x^3+e^{x^2} \left (-50+10 x+50 x^2-20 x^3+2 x^4\right )\right )}{x^3} \, dx=e^{\frac {e^{x^2} (-5+x)^2}{x^2}-x}+4 x+x^2 \] Input:
Integrate[(4*x^3 + 2*x^4 + E^((-x^3 + E^x^2*(25 - 10*x + x^2))/x^2)*(-x^3 + E^x^2*(-50 + 10*x + 50*x^2 - 20*x^3 + 2*x^4)))/x^3,x]
Output:
E^((E^x^2*(-5 + x)^2)/x^2 - x) + 4*x + 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.
\(\displaystyle \int \frac {2 x^4+4 x^3+e^{\frac {e^{x^2} \left (x^2-10 x+25\right )-x^3}{x^2}} \left (e^{x^2} \left (2 x^4-20 x^3+50 x^2+10 x-50\right )-x^3\right )}{x^3} \, dx\) |
\(\Big \downarrow \) 2010 |
\(\displaystyle \int \left (\frac {e^{\frac {e^{x^2} (x-5)^2}{x^2}-x} \left (-x^3+50 e^{x^2} x^2+10 e^{x^2} x-50 e^{x^2}+2 e^{x^2} x^4-20 e^{x^2} x^3\right )}{x^3}+2 (x+2)\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\int e^{\frac {e^{x^2} (x-5)^2}{x^2}-x}dx-20 \int e^{\frac {e^{x^2} (x-5)^2}{x^2}+x^2-x}dx+10 \int \frac {e^{\frac {e^{x^2} (x-5)^2}{x^2}+x^2-x}}{x^2}dx+50 \int \frac {e^{\frac {e^{x^2} (x-5)^2}{x^2}+x^2-x}}{x}dx+2 \int e^{\frac {e^{x^2} (x-5)^2}{x^2}+x^2-x} xdx-50 \int \frac {e^{\frac {e^{x^2} (x-5)^2}{x^2}+x^2-x}}{x^3}dx+(x+2)^2\) |
Input:
Int[(4*x^3 + 2*x^4 + E^((-x^3 + E^x^2*(25 - 10*x + x^2))/x^2)*(-x^3 + E^x^ 2*(-50 + 10*x + 50*x^2 - 20*x^3 + 2*x^4)))/x^3,x]
Output:
$Aborted
Time = 0.30 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.03
method | result | size |
parallelrisch | \(x^{2}+4 x +{\mathrm e}^{\frac {\left (x^{2}-10 x +25\right ) {\mathrm e}^{x^{2}}-x^{3}}{x^{2}}}\) | \(32\) |
risch | \(x^{2}+4 x +{\mathrm e}^{-\frac {-x^{2} {\mathrm e}^{x^{2}}+x^{3}+10 \,{\mathrm e}^{x^{2}} x -25 \,{\mathrm e}^{x^{2}}}{x^{2}}}\) | \(40\) |
norman | \(\frac {x^{4}+x^{2} {\mathrm e}^{\frac {\left (x^{2}-10 x +25\right ) {\mathrm e}^{x^{2}}-x^{3}}{x^{2}}}+4 x^{3}}{x^{2}}\) | \(42\) |
Input:
int((((2*x^4-20*x^3+50*x^2+10*x-50)*exp(x^2)-x^3)*exp(((x^2-10*x+25)*exp(x ^2)-x^3)/x^2)+2*x^4+4*x^3)/x^3,x,method=_RETURNVERBOSE)
Output:
x^2+4*x+exp(((x^2-10*x+25)*exp(x^2)-x^3)/x^2)
Time = 0.08 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00 \[ \int \frac {4 x^3+2 x^4+e^{\frac {-x^3+e^{x^2} \left (25-10 x+x^2\right )}{x^2}} \left (-x^3+e^{x^2} \left (-50+10 x+50 x^2-20 x^3+2 x^4\right )\right )}{x^3} \, dx=x^{2} + 4 \, x + e^{\left (-\frac {x^{3} - {\left (x^{2} - 10 \, x + 25\right )} e^{\left (x^{2}\right )}}{x^{2}}\right )} \] Input:
integrate((((2*x^4-20*x^3+50*x^2+10*x-50)*exp(x^2)-x^3)*exp(((x^2-10*x+25) *exp(x^2)-x^3)/x^2)+2*x^4+4*x^3)/x^3,x, algorithm="fricas")
Output:
x^2 + 4*x + e^(-(x^3 - (x^2 - 10*x + 25)*e^(x^2))/x^2)
Time = 0.15 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.87 \[ \int \frac {4 x^3+2 x^4+e^{\frac {-x^3+e^{x^2} \left (25-10 x+x^2\right )}{x^2}} \left (-x^3+e^{x^2} \left (-50+10 x+50 x^2-20 x^3+2 x^4\right )\right )}{x^3} \, dx=x^{2} + 4 x + e^{\frac {- x^{3} + \left (x^{2} - 10 x + 25\right ) e^{x^{2}}}{x^{2}}} \] Input:
integrate((((2*x**4-20*x**3+50*x**2+10*x-50)*exp(x**2)-x**3)*exp(((x**2-10 *x+25)*exp(x**2)-x**3)/x**2)+2*x**4+4*x**3)/x**3,x)
Output:
x**2 + 4*x + exp((-x**3 + (x**2 - 10*x + 25)*exp(x**2))/x**2)
Time = 0.15 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.10 \[ \int \frac {4 x^3+2 x^4+e^{\frac {-x^3+e^{x^2} \left (25-10 x+x^2\right )}{x^2}} \left (-x^3+e^{x^2} \left (-50+10 x+50 x^2-20 x^3+2 x^4\right )\right )}{x^3} \, dx=x^{2} + 4 \, x + e^{\left (-x - \frac {10 \, e^{\left (x^{2}\right )}}{x} + \frac {25 \, e^{\left (x^{2}\right )}}{x^{2}} + e^{\left (x^{2}\right )}\right )} \] Input:
integrate((((2*x^4-20*x^3+50*x^2+10*x-50)*exp(x^2)-x^3)*exp(((x^2-10*x+25) *exp(x^2)-x^3)/x^2)+2*x^4+4*x^3)/x^3,x, algorithm="maxima")
Output:
x^2 + 4*x + e^(-x - 10*e^(x^2)/x + 25*e^(x^2)/x^2 + e^(x^2))
\[ \int \frac {4 x^3+2 x^4+e^{\frac {-x^3+e^{x^2} \left (25-10 x+x^2\right )}{x^2}} \left (-x^3+e^{x^2} \left (-50+10 x+50 x^2-20 x^3+2 x^4\right )\right )}{x^3} \, dx=\int { \frac {2 \, x^{4} + 4 \, x^{3} - {\left (x^{3} - 2 \, {\left (x^{4} - 10 \, x^{3} + 25 \, x^{2} + 5 \, x - 25\right )} e^{\left (x^{2}\right )}\right )} e^{\left (-\frac {x^{3} - {\left (x^{2} - 10 \, x + 25\right )} e^{\left (x^{2}\right )}}{x^{2}}\right )}}{x^{3}} \,d x } \] Input:
integrate((((2*x^4-20*x^3+50*x^2+10*x-50)*exp(x^2)-x^3)*exp(((x^2-10*x+25) *exp(x^2)-x^3)/x^2)+2*x^4+4*x^3)/x^3,x, algorithm="giac")
Output:
integrate((2*x^4 + 4*x^3 - (x^3 - 2*(x^4 - 10*x^3 + 25*x^2 + 5*x - 25)*e^( x^2))*e^(-(x^3 - (x^2 - 10*x + 25)*e^(x^2))/x^2))/x^3, x)
Time = 4.13 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.19 \[ \int \frac {4 x^3+2 x^4+e^{\frac {-x^3+e^{x^2} \left (25-10 x+x^2\right )}{x^2}} \left (-x^3+e^{x^2} \left (-50+10 x+50 x^2-20 x^3+2 x^4\right )\right )}{x^3} \, dx=4\,x+x^2+{\mathrm {e}}^{-x}\,{\mathrm {e}}^{{\mathrm {e}}^{x^2}}\,{\mathrm {e}}^{-\frac {10\,{\mathrm {e}}^{x^2}}{x}}\,{\mathrm {e}}^{\frac {25\,{\mathrm {e}}^{x^2}}{x^2}} \] Input:
int((exp((exp(x^2)*(x^2 - 10*x + 25) - x^3)/x^2)*(exp(x^2)*(10*x + 50*x^2 - 20*x^3 + 2*x^4 - 50) - x^3) + 4*x^3 + 2*x^4)/x^3,x)
Output:
4*x + x^2 + exp(-x)*exp(exp(x^2))*exp(-(10*exp(x^2))/x)*exp((25*exp(x^2))/ x^2)
Time = 0.22 (sec) , antiderivative size = 85, normalized size of antiderivative = 2.74 \[ \int \frac {4 x^3+2 x^4+e^{\frac {-x^3+e^{x^2} \left (25-10 x+x^2\right )}{x^2}} \left (-x^3+e^{x^2} \left (-50+10 x+50 x^2-20 x^3+2 x^4\right )\right )}{x^3} \, dx=\frac {e^{\frac {e^{x^{2}} x^{2}+25 e^{x^{2}}}{x^{2}}}+e^{\frac {10 e^{x^{2}}+x^{2}}{x}} x^{2}+4 e^{\frac {10 e^{x^{2}}+x^{2}}{x}} x}{e^{\frac {10 e^{x^{2}}+x^{2}}{x}}} \] Input:
int((((2*x^4-20*x^3+50*x^2+10*x-50)*exp(x^2)-x^3)*exp(((x^2-10*x+25)*exp(x ^2)-x^3)/x^2)+2*x^4+4*x^3)/x^3,x)
Output:
(e**((e**(x**2)*x**2 + 25*e**(x**2))/x**2) + e**((10*e**(x**2) + x**2)/x)* x**2 + 4*e**((10*e**(x**2) + x**2)/x)*x)/e**((10*e**(x**2) + x**2)/x)