\(\int \frac {\frac {1}{27} e^{3 x} x^2+\frac {1}{3} e^{2 x} x^3+e^x x^4+x^5+e^{x+\frac {100}{\frac {e^{2 x}}{9}+\frac {2 e^x x}{3}+x^2}} (\frac {1}{27} e^{3 x} (1-x)+200 x+x^3-x^4+\frac {1}{9} e^{2 x} (3 x-3 x^2)+\frac {1}{3} e^x (200 x+3 x^2-3 x^3))}{\frac {1}{27} e^{3 x} x^2+\frac {1}{3} e^{2 x} x^3+e^x x^4+x^5} \, dx\) [2370]

Optimal result

Integrand size = 168, antiderivative size = 24 \[ \int \frac {\frac {1}{27} e^{3 x} x^2+\frac {1}{3} e^{2 x} x^3+e^x x^4+x^5+e^{x+\frac {100}{\frac {e^{2 x}}{9}+\frac {2 e^x x}{3}+x^2}} \left (\frac {1}{27} e^{3 x} (1-x)+200 x+x^3-x^4+\frac {1}{9} e^{2 x} \left (3 x-3 x^2\right )+\frac {1}{3} e^x \left (200 x+3 x^2-3 x^3\right )\right )}{\frac {1}{27} e^{3 x} x^2+\frac {1}{3} e^{2 x} x^3+e^x x^4+x^5} \, dx=-\frac {e^{x+\frac {100}{\left (\frac {e^x}{3}+x\right )^2}}}{x}+x \]

[Out]

x-exp(x)/x*exp(25/(1/2*exp(-ln(3)+x)+1/2*x)^2)

Rubi [F]

\[ \int \frac {\frac {1}{27} e^{3 x} x^2+\frac {1}{3} e^{2 x} x^3+e^x x^4+x^5+e^{x+\frac {100}{\frac {e^{2 x}}{9}+\frac {2 e^x x}{3}+x^2}} \left (\frac {1}{27} e^{3 x} (1-x)+200 x+x^3-x^4+\frac {1}{9} e^{2 x} \left (3 x-3 x^2\right )+\frac {1}{3} e^x \left (200 x+3 x^2-3 x^3\right )\right )}{\frac {1}{27} e^{3 x} x^2+\frac {1}{3} e^{2 x} x^3+e^x x^4+x^5} \, dx=\int \frac {\frac {1}{27} e^{3 x} x^2+\frac {1}{3} e^{2 x} x^3+e^x x^4+x^5+e^{x+\frac {100}{\frac {e^{2 x}}{9}+\frac {2 e^x x}{3}+x^2}} \left (\frac {1}{27} e^{3 x} (1-x)+200 x+x^3-x^4+\frac {1}{9} e^{2 x} \left (3 x-3 x^2\right )+\frac {1}{3} e^x \left (200 x+3 x^2-3 x^3\right )\right )}{\frac {1}{27} e^{3 x} x^2+\frac {1}{3} e^{2 x} x^3+e^x x^4+x^5} \, dx \]

[In]

Int[((E^(3*x)*x^2)/27 + (E^(2*x)*x^3)/3 + E^x*x^4 + x^5 + E^(x + 100/(E^(2*x)/9 + (2*E^x*x)/3 + x^2))*((E^(3*x
)*(1 - x))/27 + 200*x + x^3 - x^4 + (E^(2*x)*(3*x - 3*x^2))/9 + (E^x*(200*x + 3*x^2 - 3*x^3))/3))/((E^(3*x)*x^
2)/27 + (E^(2*x)*x^3)/3 + E^x*x^4 + x^5),x]

[Out]

(-27*x^2)/(2*(E^x + 3*x)^2) + (E^(x + 900/(E^x + 3*x)^2)*(5400*x + 1800*E^x*x - E^(3*x)*x - 9*E^(2*x)*x^2 - 27
*E^x*x^3 - 27*x^4))/(x^2*(E^x + 3*x)^3*(1 - (1800*(3 + E^x))/(E^x + 3*x)^3)) + Defer[Int][E^(3*x)/(E^x + 3*x)^
3, x] + 9*Defer[Int][(E^(2*x)*x)/(E^x + 3*x)^3, x] - 81*Defer[Int][x^2/(E^x + 3*x)^3, x] + 27*Defer[Int][x^3/(
E^x + 3*x)^3, x] + 27*Defer[Int][x/(E^x + 3*x)^2, x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {27 \left (\frac {1}{27} e^{3 x} x^2+\frac {1}{3} e^{2 x} x^3+e^x x^4+x^5-\frac {1}{27} e^{x+\frac {900}{\left (e^x+3 x\right )^2}} \left (e^{3 x} (-1+x)+9 e^{2 x} (-1+x) x+9 e^x x \left (-200-3 x+3 x^2\right )+27 x \left (-200-x^2+x^3\right )\right )\right )}{x^2 \left (e^x+3 x\right )^3} \, dx \\ & = 27 \int \frac {\frac {1}{27} e^{3 x} x^2+\frac {1}{3} e^{2 x} x^3+e^x x^4+x^5-\frac {1}{27} e^{x+\frac {900}{\left (e^x+3 x\right )^2}} \left (e^{3 x} (-1+x)+9 e^{2 x} (-1+x) x+9 e^x x \left (-200-3 x+3 x^2\right )+27 x \left (-200-x^2+x^3\right )\right )}{x^2 \left (e^x+3 x\right )^3} \, dx \\ & = 27 \int \left (\frac {e^{3 x}}{27 \left (e^x+3 x\right )^3}+\frac {e^{2 x} x}{3 \left (e^x+3 x\right )^3}+\frac {e^x x^2}{\left (e^x+3 x\right )^3}+\frac {x^3}{\left (e^x+3 x\right )^3}-\frac {e^{x+\frac {900}{\left (e^x+3 x\right )^2}} \left (-e^{3 x}-5400 x-1800 e^x x-9 e^{2 x} x+e^{3 x} x-27 e^x x^2+9 e^{2 x} x^2-27 x^3+27 e^x x^3+27 x^4\right )}{27 x^2 \left (e^x+3 x\right )^3}\right ) \, dx \\ & = 9 \int \frac {e^{2 x} x}{\left (e^x+3 x\right )^3} \, dx+27 \int \frac {e^x x^2}{\left (e^x+3 x\right )^3} \, dx+27 \int \frac {x^3}{\left (e^x+3 x\right )^3} \, dx+\int \frac {e^{3 x}}{\left (e^x+3 x\right )^3} \, dx-\int \frac {e^{x+\frac {900}{\left (e^x+3 x\right )^2}} \left (-e^{3 x}-5400 x-1800 e^x x-9 e^{2 x} x+e^{3 x} x-27 e^x x^2+9 e^{2 x} x^2-27 x^3+27 e^x x^3+27 x^4\right )}{x^2 \left (e^x+3 x\right )^3} \, dx \\ & = -\frac {27 x^2}{2 \left (e^x+3 x\right )^2}+\frac {e^{x+\frac {900}{\left (e^x+3 x\right )^2}} \left (5400 x+1800 e^x x-e^{3 x} x-9 e^{2 x} x^2-27 e^x x^3-27 x^4\right )}{x^2 \left (e^x+3 x\right )^3 \left (1-\frac {1800 \left (3+e^x\right )}{\left (e^x+3 x\right )^3}\right )}+9 \int \frac {e^{2 x} x}{\left (e^x+3 x\right )^3} \, dx+27 \int \frac {x^3}{\left (e^x+3 x\right )^3} \, dx+27 \int \frac {x}{\left (e^x+3 x\right )^2} \, dx-81 \int \frac {x^2}{\left (e^x+3 x\right )^3} \, dx+\int \frac {e^{3 x}}{\left (e^x+3 x\right )^3} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.24 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.25 \[ \int \frac {\frac {1}{27} e^{3 x} x^2+\frac {1}{3} e^{2 x} x^3+e^x x^4+x^5+e^{x+\frac {100}{\frac {e^{2 x}}{9}+\frac {2 e^x x}{3}+x^2}} \left (\frac {1}{27} e^{3 x} (1-x)+200 x+x^3-x^4+\frac {1}{9} e^{2 x} \left (3 x-3 x^2\right )+\frac {1}{3} e^x \left (200 x+3 x^2-3 x^3\right )\right )}{\frac {1}{27} e^{3 x} x^2+\frac {1}{3} e^{2 x} x^3+e^x x^4+x^5} \, dx=27 \left (-\frac {e^{x+\frac {900}{\left (e^x+3 x\right )^2}}}{27 x}+\frac {x}{27}\right ) \]

[In]

Integrate[((E^(3*x)*x^2)/27 + (E^(2*x)*x^3)/3 + E^x*x^4 + x^5 + E^(x + 100/(E^(2*x)/9 + (2*E^x*x)/3 + x^2))*((
E^(3*x)*(1 - x))/27 + 200*x + x^3 - x^4 + (E^(2*x)*(3*x - 3*x^2))/9 + (E^x*(200*x + 3*x^2 - 3*x^3))/3))/((E^(3
*x)*x^2)/27 + (E^(2*x)*x^3)/3 + E^x*x^4 + x^5),x]

[Out]

27*(-1/27*E^(x + 900/(E^x + 3*x)^2)/x + x/27)

Maple [A] (verified)

Time = 3.11 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.83

method	result	size
parallelrisch	\(\frac {15 x^{2}-15 \,{\mathrm e}^{x} {\mathrm e}^{\frac {100}{\frac {{\mathrm e}^{2 x}}{9}+2 x \,{\mathrm e}^{-\ln \left (3\right )+x}+x^{2}}}}{15 x}\)	\(44\)
risch	\(x -\frac {{\mathrm e}^{\frac {6 \,{\mathrm e}^{x} x^{2}+9 x^{3}+x \,{\mathrm e}^{2 x}+900}{6 \,{\mathrm e}^{x} x +9 x^{2}+{\mathrm e}^{2 x}}}}{x}\)	\(47\)

[In]

int((((1-x)*exp(-ln(3)+x)^3+(-3*x^2+3*x)*exp(-ln(3)+x)^2+(-3*x^3+3*x^2+200*x)*exp(-ln(3)+x)-x^4+x^3+200*x)*exp
(x)*exp(100/(exp(-ln(3)+x)^2+2*x*exp(-ln(3)+x)+x^2))+x^2*exp(-ln(3)+x)^3+3*x^3*exp(-ln(3)+x)^2+3*x^4*exp(-ln(3
)+x)+x^5)/(x^2*exp(-ln(3)+x)^3+3*x^3*exp(-ln(3)+x)^2+3*x^4*exp(-ln(3)+x)+x^5),x,method=_RETURNVERBOSE)

[Out]

1/15*(15*x^2-15*exp(x)*exp(100/(exp(-ln(3)+x)^2+2*x*exp(-ln(3)+x)+x^2)))/x

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 65 vs. \(2 (23) = 46\).

Time = 0.26 (sec) , antiderivative size = 65, normalized size of antiderivative = 2.71 \[ \int \frac {\frac {1}{27} e^{3 x} x^2+\frac {1}{3} e^{2 x} x^3+e^x x^4+x^5+e^{x+\frac {100}{\frac {e^{2 x}}{9}+\frac {2 e^x x}{3}+x^2}} \left (\frac {1}{27} e^{3 x} (1-x)+200 x+x^3-x^4+\frac {1}{9} e^{2 x} \left (3 x-3 x^2\right )+\frac {1}{3} e^x \left (200 x+3 x^2-3 x^3\right )\right )}{\frac {1}{27} e^{3 x} x^2+\frac {1}{3} e^{2 x} x^3+e^x x^4+x^5} \, dx=\frac {x^{2} - e^{\left (\frac {x^{3} + 2 \, x^{2} e^{\left (x - \log \left (3\right )\right )} + x e^{\left (2 \, x - 2 \, \log \left (3\right )\right )} + 100}{x^{2} + 2 \, x e^{\left (x - \log \left (3\right )\right )} + e^{\left (2 \, x - 2 \, \log \left (3\right )\right )}}\right )}}{x} \]

[In]

integrate((((1-x)*exp(-log(3)+x)^3+(-3*x^2+3*x)*exp(-log(3)+x)^2+(-3*x^3+3*x^2+200*x)*exp(-log(3)+x)-x^4+x^3+2
00*x)*exp(x)*exp(100/(exp(-log(3)+x)^2+2*x*exp(-log(3)+x)+x^2))+x^2*exp(-log(3)+x)^3+3*x^3*exp(-log(3)+x)^2+3*
x^4*exp(-log(3)+x)+x^5)/(x^2*exp(-log(3)+x)^3+3*x^3*exp(-log(3)+x)^2+3*x^4*exp(-log(3)+x)+x^5),x, algorithm="f
ricas")

[Out]

(x^2 - e^((x^3 + 2*x^2*e^(x - log(3)) + x*e^(2*x - 2*log(3)) + 100)/(x^2 + 2*x*e^(x - log(3)) + e^(2*x - 2*log
(3)))))/x

Sympy [A] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.12 \[ \int \frac {\frac {1}{27} e^{3 x} x^2+\frac {1}{3} e^{2 x} x^3+e^x x^4+x^5+e^{x+\frac {100}{\frac {e^{2 x}}{9}+\frac {2 e^x x}{3}+x^2}} \left (\frac {1}{27} e^{3 x} (1-x)+200 x+x^3-x^4+\frac {1}{9} e^{2 x} \left (3 x-3 x^2\right )+\frac {1}{3} e^x \left (200 x+3 x^2-3 x^3\right )\right )}{\frac {1}{27} e^{3 x} x^2+\frac {1}{3} e^{2 x} x^3+e^x x^4+x^5} \, dx=x - \frac {e^{x} e^{\frac {100}{x^{2} + \frac {2 x e^{x}}{3} + \frac {e^{2 x}}{9}}}}{x} \]

[In]

integrate((((1-x)*exp(-ln(3)+x)**3+(-3*x**2+3*x)*exp(-ln(3)+x)**2+(-3*x**3+3*x**2+200*x)*exp(-ln(3)+x)-x**4+x*
*3+200*x)*exp(x)*exp(100/(exp(-ln(3)+x)**2+2*x*exp(-ln(3)+x)+x**2))+x**2*exp(-ln(3)+x)**3+3*x**3*exp(-ln(3)+x)
**2+3*x**4*exp(-ln(3)+x)+x**5)/(x**2*exp(-ln(3)+x)**3+3*x**3*exp(-ln(3)+x)**2+3*x**4*exp(-ln(3)+x)+x**5),x)

[Out]

x - exp(x)*exp(100/(x**2 + 2*x*exp(x)/3 + exp(2*x)/9))/x

Maxima [F]

\[ \int \frac {\frac {1}{27} e^{3 x} x^2+\frac {1}{3} e^{2 x} x^3+e^x x^4+x^5+e^{x+\frac {100}{\frac {e^{2 x}}{9}+\frac {2 e^x x}{3}+x^2}} \left (\frac {1}{27} e^{3 x} (1-x)+200 x+x^3-x^4+\frac {1}{9} e^{2 x} \left (3 x-3 x^2\right )+\frac {1}{3} e^x \left (200 x+3 x^2-3 x^3\right )\right )}{\frac {1}{27} e^{3 x} x^2+\frac {1}{3} e^{2 x} x^3+e^x x^4+x^5} \, dx=\int { \frac {x^{5} + 3 \, x^{4} e^{\left (x - \log \left (3\right )\right )} + 3 \, x^{3} e^{\left (2 \, x - 2 \, \log \left (3\right )\right )} + x^{2} e^{\left (3 \, x - 3 \, \log \left (3\right )\right )} - {\left (x^{4} - x^{3} + {\left (x - 1\right )} e^{\left (3 \, x - 3 \, \log \left (3\right )\right )} + 3 \, {\left (x^{2} - x\right )} e^{\left (2 \, x - 2 \, \log \left (3\right )\right )} + {\left (3 \, x^{3} - 3 \, x^{2} - 200 \, x\right )} e^{\left (x - \log \left (3\right )\right )} - 200 \, x\right )} e^{\left (x + \frac {100}{x^{2} + 2 \, x e^{\left (x - \log \left (3\right )\right )} + e^{\left (2 \, x - 2 \, \log \left (3\right )\right )}}\right )}}{x^{5} + 3 \, x^{4} e^{\left (x - \log \left (3\right )\right )} + 3 \, x^{3} e^{\left (2 \, x - 2 \, \log \left (3\right )\right )} + x^{2} e^{\left (3 \, x - 3 \, \log \left (3\right )\right )}} \,d x } \]

[In]

integrate((((1-x)*exp(-log(3)+x)^3+(-3*x^2+3*x)*exp(-log(3)+x)^2+(-3*x^3+3*x^2+200*x)*exp(-log(3)+x)-x^4+x^3+2
00*x)*exp(x)*exp(100/(exp(-log(3)+x)^2+2*x*exp(-log(3)+x)+x^2))+x^2*exp(-log(3)+x)^3+3*x^3*exp(-log(3)+x)^2+3*
x^4*exp(-log(3)+x)+x^5)/(x^2*exp(-log(3)+x)^3+3*x^3*exp(-log(3)+x)^2+3*x^4*exp(-log(3)+x)+x^5),x, algorithm="m
axima")

[Out]

x - integrate(((x - 1)*e^(4*x) + 9*(x^2 - x)*e^(3*x) + 9*(3*x^3 - 3*x^2 - 200*x)*e^(2*x) + 27*(x^4 - x^3 - 200
*x)*e^x)*e^(900/(9*x^2 + 6*x*e^x + e^(2*x)))/(27*x^5 + 27*x^4*e^x + 9*x^3*e^(2*x) + x^2*e^(3*x)), x)

Giac [F]

\[ \int \frac {\frac {1}{27} e^{3 x} x^2+\frac {1}{3} e^{2 x} x^3+e^x x^4+x^5+e^{x+\frac {100}{\frac {e^{2 x}}{9}+\frac {2 e^x x}{3}+x^2}} \left (\frac {1}{27} e^{3 x} (1-x)+200 x+x^3-x^4+\frac {1}{9} e^{2 x} \left (3 x-3 x^2\right )+\frac {1}{3} e^x \left (200 x+3 x^2-3 x^3\right )\right )}{\frac {1}{27} e^{3 x} x^2+\frac {1}{3} e^{2 x} x^3+e^x x^4+x^5} \, dx=\int { \frac {x^{5} + 3 \, x^{4} e^{\left (x - \log \left (3\right )\right )} + 3 \, x^{3} e^{\left (2 \, x - 2 \, \log \left (3\right )\right )} + x^{2} e^{\left (3 \, x - 3 \, \log \left (3\right )\right )} - {\left (x^{4} - x^{3} + {\left (x - 1\right )} e^{\left (3 \, x - 3 \, \log \left (3\right )\right )} + 3 \, {\left (x^{2} - x\right )} e^{\left (2 \, x - 2 \, \log \left (3\right )\right )} + {\left (3 \, x^{3} - 3 \, x^{2} - 200 \, x\right )} e^{\left (x - \log \left (3\right )\right )} - 200 \, x\right )} e^{\left (x + \frac {100}{x^{2} + 2 \, x e^{\left (x - \log \left (3\right )\right )} + e^{\left (2 \, x - 2 \, \log \left (3\right )\right )}}\right )}}{x^{5} + 3 \, x^{4} e^{\left (x - \log \left (3\right )\right )} + 3 \, x^{3} e^{\left (2 \, x - 2 \, \log \left (3\right )\right )} + x^{2} e^{\left (3 \, x - 3 \, \log \left (3\right )\right )}} \,d x } \]

[In]

integrate((((1-x)*exp(-log(3)+x)^3+(-3*x^2+3*x)*exp(-log(3)+x)^2+(-3*x^3+3*x^2+200*x)*exp(-log(3)+x)-x^4+x^3+2
00*x)*exp(x)*exp(100/(exp(-log(3)+x)^2+2*x*exp(-log(3)+x)+x^2))+x^2*exp(-log(3)+x)^3+3*x^3*exp(-log(3)+x)^2+3*
x^4*exp(-log(3)+x)+x^5)/(x^2*exp(-log(3)+x)^3+3*x^3*exp(-log(3)+x)^2+3*x^4*exp(-log(3)+x)+x^5),x, algorithm="g
iac")

[Out]

undef

Mupad [B] (verification not implemented)

Time = 10.73 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.21 \[ \int \frac {\frac {1}{27} e^{3 x} x^2+\frac {1}{3} e^{2 x} x^3+e^x x^4+x^5+e^{x+\frac {100}{\frac {e^{2 x}}{9}+\frac {2 e^x x}{3}+x^2}} \left (\frac {1}{27} e^{3 x} (1-x)+200 x+x^3-x^4+\frac {1}{9} e^{2 x} \left (3 x-3 x^2\right )+\frac {1}{3} e^x \left (200 x+3 x^2-3 x^3\right )\right )}{\frac {1}{27} e^{3 x} x^2+\frac {1}{3} e^{2 x} x^3+e^x x^4+x^5} \, dx=x-\frac {{\mathrm {e}}^{\frac {100}{\frac {{\mathrm {e}}^{2\,x}}{9}+\frac {2\,x\,{\mathrm {e}}^x}{3}+x^2}}\,{\mathrm {e}}^x}{x} \]

[In]

int((3*x^4*exp(x - log(3)) + 3*x^3*exp(2*x - 2*log(3)) + x^2*exp(3*x - 3*log(3)) + x^5 + exp(100/(exp(2*x - 2*
log(3)) + x^2 + 2*x*exp(x - log(3))))*exp(x)*(200*x + exp(x - log(3))*(200*x + 3*x^2 - 3*x^3) + exp(2*x - 2*lo
g(3))*(3*x - 3*x^2) - exp(3*x - 3*log(3))*(x - 1) + x^3 - x^4))/(3*x^4*exp(x - log(3)) + 3*x^3*exp(2*x - 2*log
(3)) + x^2*exp(3*x - 3*log(3)) + x^5),x)

[Out]

x - (exp(100/(exp(2*x)/9 + (2*x*exp(x))/3 + x^2))*exp(x))/x