\(\int \frac {2 x^3+4 x^4+2 x^5+e^{\frac {e^{5+x}+e^{e^x} (-x-x^2)}{x+x^2}} (e^{5+x} (1+x-x^2)+e^{e^x+x} (x^2+2 x^3+x^4))}{x^2+2 x^3+x^4} \, dx\) [7653]

Optimal result

Integrand size = 98, antiderivative size = 32 \[ \int \frac {2 x^3+4 x^4+2 x^5+e^{\frac {e^{5+x}+e^{e^x} \left (-x-x^2\right )}{x+x^2}} \left (e^{5+x} \left (1+x-x^2\right )+e^{e^x+x} \left (x^2+2 x^3+x^4\right )\right )}{x^2+2 x^3+x^4} \, dx=x^2+\log \left (e^{-e^{-e^{e^x}+\frac {e^{5+x}}{x+x^2}}}\right ) \]

[Out]

ln(1/exp(exp(exp(5)*exp(x)/(x^2+x)-exp(exp(x)))))+x^2

Rubi [F]

\[ \int \frac {2 x^3+4 x^4+2 x^5+e^{\frac {e^{5+x}+e^{e^x} \left (-x-x^2\right )}{x+x^2}} \left (e^{5+x} \left (1+x-x^2\right )+e^{e^x+x} \left (x^2+2 x^3+x^4\right )\right )}{x^2+2 x^3+x^4} \, dx=\int \frac {2 x^3+4 x^4+2 x^5+e^{\frac {e^{5+x}+e^{e^x} \left (-x-x^2\right )}{x+x^2}} \left (e^{5+x} \left (1+x-x^2\right )+e^{e^x+x} \left (x^2+2 x^3+x^4\right )\right )}{x^2+2 x^3+x^4} \, dx \]

[In]

Int[(2*x^3 + 4*x^4 + 2*x^5 + E^((E^(5 + x) + E^E^x*(-x - x^2))/(x + x^2))*(E^(5 + x)*(1 + x - x^2) + E^(E^x +
x)*(x^2 + 2*x^3 + x^4)))/(x^2 + 2*x^3 + x^4),x]

[Out]

x^2 + Defer[Int][E^(-E^E^x + E^x + x + E^(5 + x)/(x + x^2)), x] - Defer[Int][E^(5 - E^E^x + x + E^(5 + x)/(x +
x^2))/(-1 - x), x] + Defer[Int][E^(5 - E^E^x + x + E^(5 + x)/(x + x^2))/x^2, x] - Defer[Int][E^(5 - E^E^x + x
+ E^(5 + x)/(x + x^2))/x, x] - Defer[Int][E^(5 - E^E^x + x + E^(5 + x)/(x + x^2))/(1 + x)^2, x]

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

Mathematica [A] (verified)

Time = 5.36 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.94 \[ \int \frac {2 x^3+4 x^4+2 x^5+e^{\frac {e^{5+x}+e^{e^x} \left (-x-x^2\right )}{x+x^2}} \left (e^{5+x} \left (1+x-x^2\right )+e^{e^x+x} \left (x^2+2 x^3+x^4\right )\right )}{x^2+2 x^3+x^4} \, dx=-e^{-e^{e^x}+\frac {e^{5+x}}{x (1+x)}}+x^2 \]

[In]

Integrate[(2*x^3 + 4*x^4 + 2*x^5 + E^((E^(5 + x) + E^E^x*(-x - x^2))/(x + x^2))*(E^(5 + x)*(1 + x - x^2) + E^(
E^x + x)*(x^2 + 2*x^3 + x^4)))/(x^2 + 2*x^3 + x^4),x]

[Out]

-E^(-E^E^x + E^(5 + x)/(x*(1 + x))) + x^2

Maple [A] (verified)

Time = 95.69 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.16

method	result	size
risch	\(x^{2}-{\mathrm e}^{-\frac {{\mathrm e}^{{\mathrm e}^{x}} x^{2}+x \,{\mathrm e}^{{\mathrm e}^{x}}-{\mathrm e}^{5+x}}{\left (1+x \right ) x}}\)	\(37\)
parallelrisch	\(x^{2}-{\mathrm e}^{\frac {\left (-x^{2}-x \right ) {\mathrm e}^{{\mathrm e}^{x}}+{\mathrm e}^{5} {\mathrm e}^{x}}{\left (1+x \right ) x}}-6\)	\(37\)

[In]

int((((x^4+2*x^3+x^2)*exp(x)*exp(exp(x))+(-x^2+x+1)*exp(5)*exp(x))*exp(((-x^2-x)*exp(exp(x))+exp(5)*exp(x))/(x
^2+x))+2*x^5+4*x^4+2*x^3)/(x^4+2*x^3+x^2),x,method=_RETURNVERBOSE)

[Out]

x^2-exp(-(exp(exp(x))*x^2+x*exp(exp(x))-exp(5+x))/(1+x)/x)

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.62 \[ \int \frac {2 x^3+4 x^4+2 x^5+e^{\frac {e^{5+x}+e^{e^x} \left (-x-x^2\right )}{x+x^2}} \left (e^{5+x} \left (1+x-x^2\right )+e^{e^x+x} \left (x^2+2 x^3+x^4\right )\right )}{x^2+2 x^3+x^4} \, dx=x^{2} - e^{\left (-\frac {{\left ({\left (x^{2} + x\right )} e^{\left ({\left (x e^{5} + e^{\left (x + 5\right )}\right )} e^{\left (-5\right )} + 5\right )} - e^{\left (2 \, x + 10\right )}\right )} e^{\left (-x - 5\right )}}{x^{2} + x}\right )} \]

[In]

integrate((((x^4+2*x^3+x^2)*exp(x)*exp(exp(x))+(-x^2+x+1)*exp(5)*exp(x))*exp(((-x^2-x)*exp(exp(x))+exp(5)*exp(
x))/(x^2+x))+2*x^5+4*x^4+2*x^3)/(x^4+2*x^3+x^2),x, algorithm="fricas")

[Out]

x^2 - e^(-((x^2 + x)*e^((x*e^5 + e^(x + 5))*e^(-5) + 5) - e^(2*x + 10))*e^(-x - 5)/(x^2 + x))

Sympy [A] (verification not implemented)

Time = 0.40 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.84 \[ \int \frac {2 x^3+4 x^4+2 x^5+e^{\frac {e^{5+x}+e^{e^x} \left (-x-x^2\right )}{x+x^2}} \left (e^{5+x} \left (1+x-x^2\right )+e^{e^x+x} \left (x^2+2 x^3+x^4\right )\right )}{x^2+2 x^3+x^4} \, dx=x^{2} - e^{\frac {\left (- x^{2} - x\right ) e^{e^{x}} + e^{5} e^{x}}{x^{2} + x}} \]

[In]

integrate((((x**4+2*x**3+x**2)*exp(x)*exp(exp(x))+(-x**2+x+1)*exp(5)*exp(x))*exp(((-x**2-x)*exp(exp(x))+exp(5)
*exp(x))/(x**2+x))+2*x**5+4*x**4+2*x**3)/(x**4+2*x**3+x**2),x)

[Out]

x**2 - exp(((-x**2 - x)*exp(exp(x)) + exp(5)*exp(x))/(x**2 + x))

Maxima [A] (verification not implemented)

none

Time = 0.36 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00 \[ \int \frac {2 x^3+4 x^4+2 x^5+e^{\frac {e^{5+x}+e^{e^x} \left (-x-x^2\right )}{x+x^2}} \left (e^{5+x} \left (1+x-x^2\right )+e^{e^x+x} \left (x^2+2 x^3+x^4\right )\right )}{x^2+2 x^3+x^4} \, dx=x^{2} - e^{\left (-\frac {e^{\left (x + 5\right )}}{x + 1} + \frac {e^{\left (x + 5\right )}}{x} - e^{\left (e^{x}\right )}\right )} \]

[In]

integrate((((x^4+2*x^3+x^2)*exp(x)*exp(exp(x))+(-x^2+x+1)*exp(5)*exp(x))*exp(((-x^2-x)*exp(exp(x))+exp(5)*exp(
x))/(x^2+x))+2*x^5+4*x^4+2*x^3)/(x^4+2*x^3+x^2),x, algorithm="maxima")

[Out]

x^2 - e^(-e^(x + 5)/(x + 1) + e^(x + 5)/x - e^(e^x))

Giac [F]

\[ \int \frac {2 x^3+4 x^4+2 x^5+e^{\frac {e^{5+x}+e^{e^x} \left (-x-x^2\right )}{x+x^2}} \left (e^{5+x} \left (1+x-x^2\right )+e^{e^x+x} \left (x^2+2 x^3+x^4\right )\right )}{x^2+2 x^3+x^4} \, dx=\int { \frac {2 \, x^{5} + 4 \, x^{4} + 2 \, x^{3} + {\left ({\left (x^{4} + 2 \, x^{3} + x^{2}\right )} e^{\left (x + e^{x}\right )} - {\left (x^{2} - x - 1\right )} e^{\left (x + 5\right )}\right )} e^{\left (-\frac {{\left (x^{2} + x\right )} e^{\left (e^{x}\right )} - e^{\left (x + 5\right )}}{x^{2} + x}\right )}}{x^{4} + 2 \, x^{3} + x^{2}} \,d x } \]

[In]

integrate((((x^4+2*x^3+x^2)*exp(x)*exp(exp(x))+(-x^2+x+1)*exp(5)*exp(x))*exp(((-x^2-x)*exp(exp(x))+exp(5)*exp(
x))/(x^2+x))+2*x^5+4*x^4+2*x^3)/(x^4+2*x^3+x^2),x, algorithm="giac")

[Out]

integrate((2*x^5 + 4*x^4 + 2*x^3 + ((x^4 + 2*x^3 + x^2)*e^(x + e^x) - (x^2 - x - 1)*e^(x + 5))*e^(-((x^2 + x)*
e^(e^x) - e^(x + 5))/(x^2 + x)))/(x^4 + 2*x^3 + x^2), x)

Mupad [B] (verification not implemented)

Time = 13.96 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.53 \[ \int \frac {2 x^3+4 x^4+2 x^5+e^{\frac {e^{5+x}+e^{e^x} \left (-x-x^2\right )}{x+x^2}} \left (e^{5+x} \left (1+x-x^2\right )+e^{e^x+x} \left (x^2+2 x^3+x^4\right )\right )}{x^2+2 x^3+x^4} \, dx=x^2-{\mathrm {e}}^{\frac {{\mathrm {e}}^5\,{\mathrm {e}}^x}{x^2+x}}\,{\mathrm {e}}^{-\frac {x\,{\mathrm {e}}^{{\mathrm {e}}^x}}{x^2+x}}\,{\mathrm {e}}^{-\frac {x^2\,{\mathrm {e}}^{{\mathrm {e}}^x}}{x^2+x}} \]

[In]

int((exp(-(exp(exp(x))*(x + x^2) - exp(5)*exp(x))/(x + x^2))*(exp(5)*exp(x)*(x - x^2 + 1) + exp(exp(x))*exp(x)
*(x^2 + 2*x^3 + x^4)) + 2*x^3 + 4*x^4 + 2*x^5)/(x^2 + 2*x^3 + x^4),x)

[Out]

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