Integrand size = 64, antiderivative size = 23 \[ \int \frac {-x^3+e^{e^{\frac {20+20 x+125 x^2+5 x^3}{x^2}}+\frac {20+20 x+125 x^2+5 x^3}{x^2}} \left (-40-20 x+5 x^3\right )}{x^3} \, dx=-3+e^{e^{5 \left (24+x+\frac {(2+x)^2}{x^2}\right )}}-x \] Output:
exp(exp(5*x+120+5*(2+x)^2/x^2))-3-x
Time = 0.88 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {-x^3+e^{e^{\frac {20+20 x+125 x^2+5 x^3}{x^2}}+\frac {20+20 x+125 x^2+5 x^3}{x^2}} \left (-40-20 x+5 x^3\right )}{x^3} \, dx=e^{e^{125+\frac {20}{x^2}+\frac {20}{x}+5 x}}-x \] Input:
Integrate[(-x^3 + E^(E^((20 + 20*x + 125*x^2 + 5*x^3)/x^2) + (20 + 20*x + 125*x^2 + 5*x^3)/x^2)*(-40 - 20*x + 5*x^3))/x^3,x]
Output:
E^E^(125 + 20/x^2 + 20/x + 5*x) - 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.
\(\displaystyle \int \frac {\left (5 x^3-20 x-40\right ) \exp \left (\frac {5 x^3+125 x^2+20 x+20}{x^2}+e^{\frac {5 x^3+125 x^2+20 x+20}{x^2}}\right )-x^3}{x^3} \, dx\) |
\(\Big \downarrow \) 2010 |
\(\displaystyle \int \left (\frac {5 e^{e^{\frac {20}{x^2}+5 x+\frac {20}{x}+125}+\frac {20}{x^2}+5 x+\frac {20}{x}+125} \left (x^3-4 x-8\right )}{x^3}-1\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 5 \int e^{5 x+e^{5 x+125+\frac {20}{x}+\frac {20}{x^2}}+125+\frac {20}{x}+\frac {20}{x^2}}dx-20 \int \frac {e^{5 x+e^{5 x+125+\frac {20}{x}+\frac {20}{x^2}}+125+\frac {20}{x}+\frac {20}{x^2}}}{x^2}dx-40 \int \frac {e^{5 x+e^{5 x+125+\frac {20}{x}+\frac {20}{x^2}}+125+\frac {20}{x}+\frac {20}{x^2}}}{x^3}dx-x\) |
Input:
Int[(-x^3 + E^(E^((20 + 20*x + 125*x^2 + 5*x^3)/x^2) + (20 + 20*x + 125*x^ 2 + 5*x^3)/x^2)*(-40 - 20*x + 5*x^3))/x^3,x]
Output:
$Aborted
Time = 0.30 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09
method | result | size |
risch | \(-x +{\mathrm e}^{{\mathrm e}^{\frac {5 x^{3}+125 x^{2}+20 x +20}{x^{2}}}}\) | \(25\) |
parallelrisch | \(-x +{\mathrm e}^{{\mathrm e}^{\frac {5 x^{3}+125 x^{2}+20 x +20}{x^{2}}}}\) | \(25\) |
parts | \(-x +{\mathrm e}^{{\mathrm e}^{\frac {5 x^{3}+125 x^{2}+20 x +20}{x^{2}}}}\) | \(26\) |
norman | \(\frac {x^{2} {\mathrm e}^{{\mathrm e}^{\frac {5 x^{3}+125 x^{2}+20 x +20}{x^{2}}}}-x^{3}}{x^{2}}\) | \(36\) |
Input:
int(((5*x^3-20*x-40)*exp((5*x^3+125*x^2+20*x+20)/x^2)*exp(exp((5*x^3+125*x ^2+20*x+20)/x^2))-x^3)/x^3,x,method=_RETURNVERBOSE)
Output:
-x+exp(exp(5*(x^3+25*x^2+4*x+4)/x^2))
Leaf count of result is larger than twice the leaf count of optimal. 88 vs. \(2 (22) = 44\).
Time = 0.09 (sec) , antiderivative size = 88, normalized size of antiderivative = 3.83 \[ \int \frac {-x^3+e^{e^{\frac {20+20 x+125 x^2+5 x^3}{x^2}}+\frac {20+20 x+125 x^2+5 x^3}{x^2}} \left (-40-20 x+5 x^3\right )}{x^3} \, dx=-{\left (x e^{\left (\frac {5 \, {\left (x^{3} + 25 \, x^{2} + 4 \, x + 4\right )}}{x^{2}}\right )} - e^{\left (\frac {5 \, x^{3} + x^{2} e^{\left (\frac {5 \, {\left (x^{3} + 25 \, x^{2} + 4 \, x + 4\right )}}{x^{2}}\right )} + 125 \, x^{2} + 20 \, x + 20}{x^{2}}\right )}\right )} e^{\left (-\frac {5 \, {\left (x^{3} + 25 \, x^{2} + 4 \, x + 4\right )}}{x^{2}}\right )} \] Input:
integrate(((5*x^3-20*x-40)*exp((5*x^3+125*x^2+20*x+20)/x^2)*exp(exp((5*x^3 +125*x^2+20*x+20)/x^2))-x^3)/x^3,x, algorithm="fricas")
Output:
-(x*e^(5*(x^3 + 25*x^2 + 4*x + 4)/x^2) - e^((5*x^3 + x^2*e^(5*(x^3 + 25*x^ 2 + 4*x + 4)/x^2) + 125*x^2 + 20*x + 20)/x^2))*e^(-5*(x^3 + 25*x^2 + 4*x + 4)/x^2)
Time = 0.12 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96 \[ \int \frac {-x^3+e^{e^{\frac {20+20 x+125 x^2+5 x^3}{x^2}}+\frac {20+20 x+125 x^2+5 x^3}{x^2}} \left (-40-20 x+5 x^3\right )}{x^3} \, dx=- x + e^{e^{\frac {5 x^{3} + 125 x^{2} + 20 x + 20}{x^{2}}}} \] Input:
integrate(((5*x**3-20*x-40)*exp((5*x**3+125*x**2+20*x+20)/x**2)*exp(exp((5 *x**3+125*x**2+20*x+20)/x**2))-x**3)/x**3,x)
Output:
-x + exp(exp((5*x**3 + 125*x**2 + 20*x + 20)/x**2))
Time = 0.16 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91 \[ \int \frac {-x^3+e^{e^{\frac {20+20 x+125 x^2+5 x^3}{x^2}}+\frac {20+20 x+125 x^2+5 x^3}{x^2}} \left (-40-20 x+5 x^3\right )}{x^3} \, dx=-x + e^{\left (e^{\left (5 \, x + \frac {20}{x} + \frac {20}{x^{2}} + 125\right )}\right )} \] Input:
integrate(((5*x^3-20*x-40)*exp((5*x^3+125*x^2+20*x+20)/x^2)*exp(exp((5*x^3 +125*x^2+20*x+20)/x^2))-x^3)/x^3,x, algorithm="maxima")
Output:
-x + e^(e^(5*x + 20/x + 20/x^2 + 125))
\[ \int \frac {-x^3+e^{e^{\frac {20+20 x+125 x^2+5 x^3}{x^2}}+\frac {20+20 x+125 x^2+5 x^3}{x^2}} \left (-40-20 x+5 x^3\right )}{x^3} \, dx=\int { -\frac {x^{3} - 5 \, {\left (x^{3} - 4 \, x - 8\right )} e^{\left (\frac {5 \, {\left (x^{3} + 25 \, x^{2} + 4 \, x + 4\right )}}{x^{2}} + e^{\left (\frac {5 \, {\left (x^{3} + 25 \, x^{2} + 4 \, x + 4\right )}}{x^{2}}\right )}\right )}}{x^{3}} \,d x } \] Input:
integrate(((5*x^3-20*x-40)*exp((5*x^3+125*x^2+20*x+20)/x^2)*exp(exp((5*x^3 +125*x^2+20*x+20)/x^2))-x^3)/x^3,x, algorithm="giac")
Output:
integrate(-(x^3 - 5*(x^3 - 4*x - 8)*e^(5*(x^3 + 25*x^2 + 4*x + 4)/x^2 + e^ (5*(x^3 + 25*x^2 + 4*x + 4)/x^2)))/x^3, x)
Time = 3.06 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.04 \[ \int \frac {-x^3+e^{e^{\frac {20+20 x+125 x^2+5 x^3}{x^2}}+\frac {20+20 x+125 x^2+5 x^3}{x^2}} \left (-40-20 x+5 x^3\right )}{x^3} \, dx={\mathrm {e}}^{{\mathrm {e}}^{5\,x}\,{\mathrm {e}}^{125}\,{\mathrm {e}}^{20/x}\,{\mathrm {e}}^{\frac {20}{x^2}}}-x \] Input:
int(-(x^3 + exp((20*x + 125*x^2 + 5*x^3 + 20)/x^2)*exp(exp((20*x + 125*x^2 + 5*x^3 + 20)/x^2))*(20*x - 5*x^3 + 40))/x^3,x)
Output:
exp(exp(5*x)*exp(125)*exp(20/x)*exp(20/x^2)) - x
\[ \int \frac {-x^3+e^{e^{\frac {20+20 x+125 x^2+5 x^3}{x^2}}+\frac {20+20 x+125 x^2+5 x^3}{x^2}} \left (-40-20 x+5 x^3\right )}{x^3} \, dx=\int \frac {\left (5 x^{3}-20 x -40\right ) {\mathrm e}^{\frac {5 x^{3}+125 x^{2}+20 x +20}{x^{2}}} {\mathrm e}^{{\mathrm e}^{\frac {5 x^{3}+125 x^{2}+20 x +20}{x^{2}}}}-x^{3}}{x^{3}}d x \] Input:
int(((5*x^3-20*x-40)*exp((5*x^3+125*x^2+20*x+20)/x^2)*exp(exp((5*x^3+125*x ^2+20*x+20)/x^2))-x^3)/x^3,x)
Output:
int(((5*x^3-20*x-40)*exp((5*x^3+125*x^2+20*x+20)/x^2)*exp(exp((5*x^3+125*x ^2+20*x+20)/x^2))-x^3)/x^3,x)