\(\int \frac {-x^3+e^{e^{\frac {20+20 x+125 x^2+5 x^3}{x^2}}+\frac {20+20 x+125 x^2+5 x^3}{x^2}} (-40-20 x+5 x^3)}{x^3} \, dx\) [7765]

Optimal result

Integrand size = 64, antiderivative size = 23 \[ \int \frac {-x^3+e^{e^{\frac {20+20 x+125 x^2+5 x^3}{x^2}}+\frac {20+20 x+125 x^2+5 x^3}{x^2}} \left (-40-20 x+5 x^3\right )}{x^3} \, dx=-3+e^{e^{5 \left (24+x+\frac {(2+x)^2}{x^2}\right )}}-x \]

[Out]

exp(exp(5*x+120+5*(2+x)^2/x^2))-3-x

Rubi [F]

\[ \int \frac {-x^3+e^{e^{\frac {20+20 x+125 x^2+5 x^3}{x^2}}+\frac {20+20 x+125 x^2+5 x^3}{x^2}} \left (-40-20 x+5 x^3\right )}{x^3} \, dx=\int \frac {-x^3+\exp \left (e^{\frac {20+20 x+125 x^2+5 x^3}{x^2}}+\frac {20+20 x+125 x^2+5 x^3}{x^2}\right ) \left (-40-20 x+5 x^3\right )}{x^3} \, dx \]

[In]

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]

[Out]

-x + 5*Defer[Int][E^(125 + E^(125 + 20/x^2 + 20/x + 5*x) + 20/x^2 + 20/x + 5*x), x] - 40*Defer[Int][E^(125 + E
^(125 + 20/x^2 + 20/x + 5*x) + 20/x^2 + 20/x + 5*x)/x^3, x] - 20*Defer[Int][E^(125 + E^(125 + 20/x^2 + 20/x +
5*x) + 20/x^2 + 20/x + 5*x)/x^2, x]

Rubi steps \begin{align*} \text {integral}& = \int \left (-1+\frac {5 e^{125+e^{125+\frac {20}{x^2}+\frac {20}{x}+5 x}+\frac {20}{x^2}+\frac {20}{x}+5 x} \left (-8-4 x+x^3\right )}{x^3}\right ) \, dx \\ & = -x+5 \int \frac {e^{125+e^{125+\frac {20}{x^2}+\frac {20}{x}+5 x}+\frac {20}{x^2}+\frac {20}{x}+5 x} \left (-8-4 x+x^3\right )}{x^3} \, dx \\ & = -x+5 \int \left (e^{125+e^{125+\frac {20}{x^2}+\frac {20}{x}+5 x}+\frac {20}{x^2}+\frac {20}{x}+5 x}-\frac {8 e^{125+e^{125+\frac {20}{x^2}+\frac {20}{x}+5 x}+\frac {20}{x^2}+\frac {20}{x}+5 x}}{x^3}-\frac {4 e^{125+e^{125+\frac {20}{x^2}+\frac {20}{x}+5 x}+\frac {20}{x^2}+\frac {20}{x}+5 x}}{x^2}\right ) \, dx \\ & = -x+5 \int e^{125+e^{125+\frac {20}{x^2}+\frac {20}{x}+5 x}+\frac {20}{x^2}+\frac {20}{x}+5 x} \, dx-20 \int \frac {e^{125+e^{125+\frac {20}{x^2}+\frac {20}{x}+5 x}+\frac {20}{x^2}+\frac {20}{x}+5 x}}{x^2} \, dx-40 \int \frac {e^{125+e^{125+\frac {20}{x^2}+\frac {20}{x}+5 x}+\frac {20}{x^2}+\frac {20}{x}+5 x}}{x^3} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 1.46 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {-x^3+e^{e^{\frac {20+20 x+125 x^2+5 x^3}{x^2}}+\frac {20+20 x+125 x^2+5 x^3}{x^2}} \left (-40-20 x+5 x^3\right )}{x^3} \, dx=e^{e^{125+\frac {20}{x^2}+\frac {20}{x}+5 x}}-x \]

[In]

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]

[Out]

E^E^(125 + 20/x^2 + 20/x + 5*x) - x

Maple [A] (verified)

Time = 0.35 (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	\(\frac {x^{2} {\mathrm e}^{{\mathrm e}^{\frac {5 x^{3}+125 x^{2}+20 x +20}{x^{2}}}}-x^{3}}{x^{2}}\)	\(36\)

[In]

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)

[Out]

-x+exp(exp(5*(x^3+25*x^2+4*x+4)/x^2))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 88 vs. \(2 (22) = 44\).

Time = 0.25 (sec) , antiderivative size = 88, normalized size of antiderivative = 3.83 \[ \int \frac {-x^3+e^{e^{\frac {20+20 x+125 x^2+5 x^3}{x^2}}+\frac {20+20 x+125 x^2+5 x^3}{x^2}} \left (-40-20 x+5 x^3\right )}{x^3} \, dx=-{\left (x e^{\left (\frac {5 \, {\left (x^{3} + 25 \, x^{2} + 4 \, x + 4\right )}}{x^{2}}\right )} - e^{\left (\frac {5 \, x^{3} + x^{2} e^{\left (\frac {5 \, {\left (x^{3} + 25 \, x^{2} + 4 \, x + 4\right )}}{x^{2}}\right )} + 125 \, x^{2} + 20 \, x + 20}{x^{2}}\right )}\right )} e^{\left (-\frac {5 \, {\left (x^{3} + 25 \, x^{2} + 4 \, x + 4\right )}}{x^{2}}\right )} \]

[In]

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")

[Out]

-(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)

Sympy [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96 \[ \int \frac {-x^3+e^{e^{\frac {20+20 x+125 x^2+5 x^3}{x^2}}+\frac {20+20 x+125 x^2+5 x^3}{x^2}} \left (-40-20 x+5 x^3\right )}{x^3} \, dx=- x + e^{e^{\frac {5 x^{3} + 125 x^{2} + 20 x + 20}{x^{2}}}} \]

[In]

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)

[Out]

-x + exp(exp((5*x**3 + 125*x**2 + 20*x + 20)/x**2))

Maxima [A] (verification not implemented)

none

Time = 0.34 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91 \[ \int \frac {-x^3+e^{e^{\frac {20+20 x+125 x^2+5 x^3}{x^2}}+\frac {20+20 x+125 x^2+5 x^3}{x^2}} \left (-40-20 x+5 x^3\right )}{x^3} \, dx=-x + e^{\left (e^{\left (5 \, x + \frac {20}{x} + \frac {20}{x^{2}} + 125\right )}\right )} \]

[In]

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")

[Out]

-x + e^(e^(5*x + 20/x + 20/x^2 + 125))

Giac [F]

\[ \int \frac {-x^3+e^{e^{\frac {20+20 x+125 x^2+5 x^3}{x^2}}+\frac {20+20 x+125 x^2+5 x^3}{x^2}} \left (-40-20 x+5 x^3\right )}{x^3} \, dx=\int { -\frac {x^{3} - 5 \, {\left (x^{3} - 4 \, x - 8\right )} e^{\left (\frac {5 \, {\left (x^{3} + 25 \, x^{2} + 4 \, x + 4\right )}}{x^{2}} + e^{\left (\frac {5 \, {\left (x^{3} + 25 \, x^{2} + 4 \, x + 4\right )}}{x^{2}}\right )}\right )}}{x^{3}} \,d x } \]

[In]

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")

[Out]

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)

Mupad [B] (verification not implemented)

Time = 12.89 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.04 \[ \int \frac {-x^3+e^{e^{\frac {20+20 x+125 x^2+5 x^3}{x^2}}+\frac {20+20 x+125 x^2+5 x^3}{x^2}} \left (-40-20 x+5 x^3\right )}{x^3} \, dx={\mathrm {e}}^{{\mathrm {e}}^{5\,x}\,{\mathrm {e}}^{125}\,{\mathrm {e}}^{20/x}\,{\mathrm {e}}^{\frac {20}{x^2}}}-x \]

[In]

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)

[Out]

exp(exp(5*x)*exp(125)*exp(20/x)*exp(20/x^2)) - x