Integrand size = 105, antiderivative size = 31 \begin {dmath*} \int \frac {162 x^5+9 x^6+e^{2 e^x} \left (135 x^4-108 e^x x^5\right )}{27 e^{6 e^x}+216 x^3+108 x^4+18 x^5+x^6+e^{4 e^x} \left (162 x+27 x^2\right )+e^{2 e^x} \left (324 x^2+108 x^3+9 x^4\right )} \, dx=\frac {x^3}{\left (\frac {x}{3}+\left (\frac {e^{2 e^x}}{x^2}+\frac {2}{x}\right ) x\right )^2} \end {dmath*}
Time = 0.59 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.74 \begin {dmath*} \int \frac {162 x^5+9 x^6+e^{2 e^x} \left (135 x^4-108 e^x x^5\right )}{27 e^{6 e^x}+216 x^3+108 x^4+18 x^5+x^6+e^{4 e^x} \left (162 x+27 x^2\right )+e^{2 e^x} \left (324 x^2+108 x^3+9 x^4\right )} \, dx=\frac {9 x^5}{\left (3 e^{2 e^x}+6 x+x^2\right )^2} \end {dmath*}
Integrate[(162*x^5 + 9*x^6 + E^(2*E^x)*(135*x^4 - 108*E^x*x^5))/(27*E^(6*E ^x) + 216*x^3 + 108*x^4 + 18*x^5 + x^6 + E^(4*E^x)*(162*x + 27*x^2) + E^(2 *E^x)*(324*x^2 + 108*x^3 + 9*x^4)),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.
\(\displaystyle \int \frac {9 x^6+162 x^5+e^{2 e^x} \left (135 x^4-108 e^x x^5\right )}{x^6+18 x^5+108 x^4+216 x^3+e^{4 e^x} \left (27 x^2+162 x\right )+e^{2 e^x} \left (9 x^4+108 x^3+324 x^2\right )+27 e^{6 e^x}} \, dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \frac {9 x^4 \left (-12 e^{x+2 e^x} x+(x+18) x+15 e^{2 e^x}\right )}{\left (x (x+6)+3 e^{2 e^x}\right )^3}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle 9 \int \frac {x^4 \left (-12 e^{x+2 e^x} x+(x+18) x+15 e^{2 e^x}\right )}{\left (x (x+6)+3 e^{2 e^x}\right )^3}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle 9 \int \left (\frac {x^4 \left (x^2+18 x+15 e^{2 e^x}\right )}{\left (x^2+6 x+3 e^{2 e^x}\right )^3}-\frac {12 e^{x+2 e^x} x^5}{\left (x^2+6 x+3 e^{2 e^x}\right )^3}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 9 \left (-4 \int \frac {x^6}{\left (x^2+6 x+3 e^{2 e^x}\right )^3}dx-12 \int \frac {x^5}{\left (x^2+6 x+3 e^{2 e^x}\right )^3}dx-12 \int \frac {e^{x+2 e^x} x^5}{\left (x^2+6 x+3 e^{2 e^x}\right )^3}dx+5 \int \frac {x^4}{\left (x^2+6 x+3 e^{2 e^x}\right )^2}dx\right )\) |
Int[(162*x^5 + 9*x^6 + E^(2*E^x)*(135*x^4 - 108*E^x*x^5))/(27*E^(6*E^x) + 216*x^3 + 108*x^4 + 18*x^5 + x^6 + E^(4*E^x)*(162*x + 27*x^2) + E^(2*E^x)* (324*x^2 + 108*x^3 + 9*x^4)),x]
3.14.35.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 0.60 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.71
method | result | size |
risch | \(\frac {9 x^{5}}{\left (3 \,{\mathrm e}^{2 \,{\mathrm e}^{x}}+x^{2}+6 x \right )^{2}}\) | \(22\) |
parallelrisch | \(\frac {9 x^{5}}{x^{4}+6 \,{\mathrm e}^{2 \,{\mathrm e}^{x}} x^{2}+9 \,{\mathrm e}^{4 \,{\mathrm e}^{x}}+12 x^{3}+36 \,{\mathrm e}^{2 \,{\mathrm e}^{x}} x +36 x^{2}}\) | \(47\) |
int(((-108*x^5*exp(x)+135*x^4)*exp(exp(x))^2+9*x^6+162*x^5)/(27*exp(exp(x) )^6+(27*x^2+162*x)*exp(exp(x))^4+(9*x^4+108*x^3+324*x^2)*exp(exp(x))^2+x^6 +18*x^5+108*x^4+216*x^3),x,method=_RETURNVERBOSE)
Time = 0.25 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.35 \begin {dmath*} \int \frac {162 x^5+9 x^6+e^{2 e^x} \left (135 x^4-108 e^x x^5\right )}{27 e^{6 e^x}+216 x^3+108 x^4+18 x^5+x^6+e^{4 e^x} \left (162 x+27 x^2\right )+e^{2 e^x} \left (324 x^2+108 x^3+9 x^4\right )} \, dx=\frac {9 \, x^{5}}{x^{4} + 12 \, x^{3} + 36 \, x^{2} + 6 \, {\left (x^{2} + 6 \, x\right )} e^{\left (2 \, e^{x}\right )} + 9 \, e^{\left (4 \, e^{x}\right )}} \end {dmath*}
integrate(((-108*x^5*exp(x)+135*x^4)*exp(exp(x))^2+9*x^6+162*x^5)/(27*exp( exp(x))^6+(27*x^2+162*x)*exp(exp(x))^4+(9*x^4+108*x^3+324*x^2)*exp(exp(x)) ^2+x^6+18*x^5+108*x^4+216*x^3),x, algorithm=\
Time = 0.11 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.35 \begin {dmath*} \int \frac {162 x^5+9 x^6+e^{2 e^x} \left (135 x^4-108 e^x x^5\right )}{27 e^{6 e^x}+216 x^3+108 x^4+18 x^5+x^6+e^{4 e^x} \left (162 x+27 x^2\right )+e^{2 e^x} \left (324 x^2+108 x^3+9 x^4\right )} \, dx=\frac {x^{5}}{\frac {x^{4}}{9} + \frac {4 x^{3}}{3} + 4 x^{2} + \left (\frac {2 x^{2}}{3} + 4 x\right ) e^{2 e^{x}} + e^{4 e^{x}}} \end {dmath*}
integrate(((-108*x**5*exp(x)+135*x**4)*exp(exp(x))**2+9*x**6+162*x**5)/(27 *exp(exp(x))**6+(27*x**2+162*x)*exp(exp(x))**4+(9*x**4+108*x**3+324*x**2)* exp(exp(x))**2+x**6+18*x**5+108*x**4+216*x**3),x)
Time = 0.27 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.35 \begin {dmath*} \int \frac {162 x^5+9 x^6+e^{2 e^x} \left (135 x^4-108 e^x x^5\right )}{27 e^{6 e^x}+216 x^3+108 x^4+18 x^5+x^6+e^{4 e^x} \left (162 x+27 x^2\right )+e^{2 e^x} \left (324 x^2+108 x^3+9 x^4\right )} \, dx=\frac {9 \, x^{5}}{x^{4} + 12 \, x^{3} + 36 \, x^{2} + 6 \, {\left (x^{2} + 6 \, x\right )} e^{\left (2 \, e^{x}\right )} + 9 \, e^{\left (4 \, e^{x}\right )}} \end {dmath*}
integrate(((-108*x^5*exp(x)+135*x^4)*exp(exp(x))^2+9*x^6+162*x^5)/(27*exp( exp(x))^6+(27*x^2+162*x)*exp(exp(x))^4+(9*x^4+108*x^3+324*x^2)*exp(exp(x)) ^2+x^6+18*x^5+108*x^4+216*x^3),x, algorithm=\
Time = 0.31 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.48 \begin {dmath*} \int \frac {162 x^5+9 x^6+e^{2 e^x} \left (135 x^4-108 e^x x^5\right )}{27 e^{6 e^x}+216 x^3+108 x^4+18 x^5+x^6+e^{4 e^x} \left (162 x+27 x^2\right )+e^{2 e^x} \left (324 x^2+108 x^3+9 x^4\right )} \, dx=\frac {9 \, x^{5}}{x^{4} + 12 \, x^{3} + 6 \, x^{2} e^{\left (2 \, e^{x}\right )} + 36 \, x^{2} + 36 \, x e^{\left (2 \, e^{x}\right )} + 9 \, e^{\left (4 \, e^{x}\right )}} \end {dmath*}
integrate(((-108*x^5*exp(x)+135*x^4)*exp(exp(x))^2+9*x^6+162*x^5)/(27*exp( exp(x))^6+(27*x^2+162*x)*exp(exp(x))^4+(9*x^4+108*x^3+324*x^2)*exp(exp(x)) ^2+x^6+18*x^5+108*x^4+216*x^3),x, algorithm=\
Timed out. \begin {dmath*} \int \frac {162 x^5+9 x^6+e^{2 e^x} \left (135 x^4-108 e^x x^5\right )}{27 e^{6 e^x}+216 x^3+108 x^4+18 x^5+x^6+e^{4 e^x} \left (162 x+27 x^2\right )+e^{2 e^x} \left (324 x^2+108 x^3+9 x^4\right )} \, dx=\int \frac {162\,x^5-{\mathrm {e}}^{2\,{\mathrm {e}}^x}\,\left (108\,x^5\,{\mathrm {e}}^x-135\,x^4\right )+9\,x^6}{27\,{\mathrm {e}}^{6\,{\mathrm {e}}^x}+{\mathrm {e}}^{2\,{\mathrm {e}}^x}\,\left (9\,x^4+108\,x^3+324\,x^2\right )+{\mathrm {e}}^{4\,{\mathrm {e}}^x}\,\left (27\,x^2+162\,x\right )+216\,x^3+108\,x^4+18\,x^5+x^6} \,d x \end {dmath*}
int((162*x^5 - exp(2*exp(x))*(108*x^5*exp(x) - 135*x^4) + 9*x^6)/(27*exp(6 *exp(x)) + exp(2*exp(x))*(324*x^2 + 108*x^3 + 9*x^4) + exp(4*exp(x))*(162* x + 27*x^2) + 216*x^3 + 108*x^4 + 18*x^5 + x^6),x)