Integrand size = 127, antiderivative size = 27 \[ \int \frac {-9 e^{15} x+9 e^{30} x+e^{30+\frac {4}{x}+\frac {x}{9}} \left (36+9 x-x^2\right )}{9 x+9 e^{30+\frac {8}{x}+\frac {2 x}{9}} x+e^{15} \left (-18 x+18 x^2\right )+e^{30} \left (9 x-18 x^2+9 x^3\right )+e^{\frac {4}{x}+\frac {x}{9}} \left (-18 e^{15} x+e^{30} \left (18 x-18 x^2\right )\right )} \, dx=\frac {x}{1-\frac {1}{e^{15}}+e^{\frac {4}{x}+\frac {x}{9}}-x} \] Output:
x/(exp(1/9*x)*exp(4/x)-exp(-15)-x+1)
Time = 0.66 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.19 \[ \int \frac {-9 e^{15} x+9 e^{30} x+e^{30+\frac {4}{x}+\frac {x}{9}} \left (36+9 x-x^2\right )}{9 x+9 e^{30+\frac {8}{x}+\frac {2 x}{9}} x+e^{15} \left (-18 x+18 x^2\right )+e^{30} \left (9 x-18 x^2+9 x^3\right )+e^{\frac {4}{x}+\frac {x}{9}} \left (-18 e^{15} x+e^{30} \left (18 x-18 x^2\right )\right )} \, dx=\frac {e^{15} x}{-1+e^{15}+e^{15+\frac {4}{x}+\frac {x}{9}}-e^{15} x} \] Input:
Integrate[(-9*E^15*x + 9*E^30*x + E^(30 + 4/x + x/9)*(36 + 9*x - x^2))/(9* x + 9*E^(30 + 8/x + (2*x)/9)*x + E^15*(-18*x + 18*x^2) + E^30*(9*x - 18*x^ 2 + 9*x^3) + E^(4/x + x/9)*(-18*E^15*x + E^30*(18*x - 18*x^2))),x]
Output:
(E^15*x)/(-1 + E^15 + E^(15 + 4/x + x/9) - E^15*x)
Time = 0.68 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.33, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.039, Rules used = {6, 7239, 27, 25, 7238}
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.
\(\displaystyle \int \frac {e^{\frac {x}{9}+\frac {4}{x}+30} \left (-x^2+9 x+36\right )+9 e^{30} x-9 e^{15} x}{e^{15} \left (18 x^2-18 x\right )+e^{\frac {x}{9}+\frac {4}{x}} \left (e^{30} \left (18 x-18 x^2\right )-18 e^{15} x\right )+e^{30} \left (9 x^3-18 x^2+9 x\right )+9 e^{\frac {2 x}{9}+\frac {8}{x}+30} x+9 x} \, dx\) |
\(\Big \downarrow \) 6 |
\(\displaystyle \int \frac {e^{\frac {x}{9}+\frac {4}{x}+30} \left (-x^2+9 x+36\right )+\left (9 e^{30}-9 e^{15}\right ) x}{e^{15} \left (18 x^2-18 x\right )+e^{\frac {x}{9}+\frac {4}{x}} \left (e^{30} \left (18 x-18 x^2\right )-18 e^{15} x\right )+e^{30} \left (9 x^3-18 x^2+9 x\right )+9 e^{\frac {2 x}{9}+\frac {8}{x}+30} x+9 x}dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \frac {e^{\frac {x}{9}+\frac {4}{x}+30} \left (-x^2+9 x+36\right )-9 e^{15} \left (1-e^{15}\right ) x}{9 \left (e^{15} (x-1)-e^{\frac {x}{9}+\frac {4}{x}+15}+1\right )^2 x}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{9} \int -\frac {9 e^{15} \left (1-e^{15}\right ) x-e^{\frac {x}{9}+30+\frac {4}{x}} \left (-x^2+9 x+36\right )}{\left (-e^{15} (1-x)-e^{\frac {x}{9}+15+\frac {4}{x}}+1\right )^2 x}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {1}{9} \int \frac {9 e^{15} \left (1-e^{15}\right ) x-e^{\frac {x}{9}+30+\frac {4}{x}} \left (-x^2+9 x+36\right )}{\left (-e^{15} (1-x)-e^{\frac {x}{9}+15+\frac {4}{x}}+1\right )^2 x}dx\) |
\(\Big \downarrow \) 7238 |
\(\displaystyle -\frac {e^{15} x}{-e^{15} (1-x)-e^{\frac {x}{9}+\frac {4}{x}+15}+1}\) |
Input:
Int[(-9*E^15*x + 9*E^30*x + E^(30 + 4/x + x/9)*(36 + 9*x - x^2))/(9*x + 9* E^(30 + 8/x + (2*x)/9)*x + E^15*(-18*x + 18*x^2) + E^30*(9*x - 18*x^2 + 9* x^3) + E^(4/x + x/9)*(-18*E^15*x + E^30*(18*x - 18*x^2))),x]
Output:
-((E^15*x)/(1 - E^(15 + 4/x + x/9) - E^15*(1 - x)))
Int[(u_.)*((v_.) + (a_.)*(Fx_) + (b_.)*(Fx_))^(p_.), x_Symbol] :> Int[u*(v + (a + b)*Fx)^p, x] /; FreeQ[{a, b}, x] && !FreeQ[Fx, x]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(u_)*(y_)^(m_.)*(z_)^(n_.), x_Symbol] :> With[{q = DerivativeDivides[y* z, u*z^(n - m), x]}, Simp[q*y^(m + 1)*(z^(m + 1)/(m + 1)), x] /; !FalseQ[q ]] /; FreeQ[{m, n}, x] && NeQ[m, -1]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 1.28 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.19
method | result | size |
parallelrisch | \(-\frac {x \,{\mathrm e}^{15}}{-{\mathrm e}^{15} {\mathrm e}^{\frac {4}{x}} {\mathrm e}^{\frac {x}{9}}+x \,{\mathrm e}^{15}-{\mathrm e}^{15}+1}\) | \(32\) |
risch | \(-\frac {x \,{\mathrm e}^{15}}{-{\mathrm e}^{\frac {x^{2}+135 x +36}{9 x}}+x \,{\mathrm e}^{15}-{\mathrm e}^{15}+1}\) | \(34\) |
Input:
int(((-x^2+9*x+36)*exp(15)^2*exp(4/x)*exp(1/9*x)+9*x*exp(15)^2-9*x*exp(15) )/(9*x*exp(15)^2*exp(4/x)^2*exp(1/9*x)^2+((-18*x^2+18*x)*exp(15)^2-18*x*ex p(15))*exp(4/x)*exp(1/9*x)+(9*x^3-18*x^2+9*x)*exp(15)^2+(18*x^2-18*x)*exp( 15)+9*x),x,method=_RETURNVERBOSE)
Output:
-x*exp(15)/(-exp(15)*exp(4/x)*exp(1/9*x)+x*exp(15)-exp(15)+1)
Time = 0.10 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.19 \[ \int \frac {-9 e^{15} x+9 e^{30} x+e^{30+\frac {4}{x}+\frac {x}{9}} \left (36+9 x-x^2\right )}{9 x+9 e^{30+\frac {8}{x}+\frac {2 x}{9}} x+e^{15} \left (-18 x+18 x^2\right )+e^{30} \left (9 x-18 x^2+9 x^3\right )+e^{\frac {4}{x}+\frac {x}{9}} \left (-18 e^{15} x+e^{30} \left (18 x-18 x^2\right )\right )} \, dx=-\frac {x e^{30}}{{\left (x - 1\right )} e^{30} + e^{15} - e^{\left (\frac {x^{2} + 270 \, x + 36}{9 \, x}\right )}} \] Input:
integrate(((-x^2+9*x+36)*exp(15)^2*exp(4/x)*exp(1/9*x)+9*x*exp(15)^2-9*x*e xp(15))/(9*x*exp(15)^2*exp(4/x)^2*exp(1/9*x)^2+((-18*x^2+18*x)*exp(15)^2-1 8*x*exp(15))*exp(4/x)*exp(1/9*x)+(9*x^3-18*x^2+9*x)*exp(15)^2+(18*x^2-18*x )*exp(15)+9*x),x, algorithm="fricas")
Output:
-x*e^30/((x - 1)*e^30 + e^15 - e^(1/9*(x^2 + 270*x + 36)/x))
Time = 0.13 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \frac {-9 e^{15} x+9 e^{30} x+e^{30+\frac {4}{x}+\frac {x}{9}} \left (36+9 x-x^2\right )}{9 x+9 e^{30+\frac {8}{x}+\frac {2 x}{9}} x+e^{15} \left (-18 x+18 x^2\right )+e^{30} \left (9 x-18 x^2+9 x^3\right )+e^{\frac {4}{x}+\frac {x}{9}} \left (-18 e^{15} x+e^{30} \left (18 x-18 x^2\right )\right )} \, dx=\frac {x e^{15}}{- x e^{15} + e^{15} e^{\frac {4}{x}} e^{\frac {x}{9}} - 1 + e^{15}} \] Input:
integrate(((-x**2+9*x+36)*exp(15)**2*exp(4/x)*exp(1/9*x)+9*x*exp(15)**2-9* x*exp(15))/(9*x*exp(15)**2*exp(4/x)**2*exp(1/9*x)**2+((-18*x**2+18*x)*exp( 15)**2-18*x*exp(15))*exp(4/x)*exp(1/9*x)+(9*x**3-18*x**2+9*x)*exp(15)**2+( 18*x**2-18*x)*exp(15)+9*x),x)
Output:
x*exp(15)/(-x*exp(15) + exp(15)*exp(4/x)*exp(x/9) - 1 + exp(15))
Time = 0.07 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.11 \[ \int \frac {-9 e^{15} x+9 e^{30} x+e^{30+\frac {4}{x}+\frac {x}{9}} \left (36+9 x-x^2\right )}{9 x+9 e^{30+\frac {8}{x}+\frac {2 x}{9}} x+e^{15} \left (-18 x+18 x^2\right )+e^{30} \left (9 x-18 x^2+9 x^3\right )+e^{\frac {4}{x}+\frac {x}{9}} \left (-18 e^{15} x+e^{30} \left (18 x-18 x^2\right )\right )} \, dx=-\frac {x e^{15}}{x e^{15} - e^{15} - e^{\left (\frac {1}{9} \, x + \frac {4}{x} + 15\right )} + 1} \] Input:
integrate(((-x^2+9*x+36)*exp(15)^2*exp(4/x)*exp(1/9*x)+9*x*exp(15)^2-9*x*e xp(15))/(9*x*exp(15)^2*exp(4/x)^2*exp(1/9*x)^2+((-18*x^2+18*x)*exp(15)^2-1 8*x*exp(15))*exp(4/x)*exp(1/9*x)+(9*x^3-18*x^2+9*x)*exp(15)^2+(18*x^2-18*x )*exp(15)+9*x),x, algorithm="maxima")
Output:
-x*e^15/(x*e^15 - e^15 - e^(1/9*x + 4/x + 15) + 1)
Time = 0.19 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.22 \[ \int \frac {-9 e^{15} x+9 e^{30} x+e^{30+\frac {4}{x}+\frac {x}{9}} \left (36+9 x-x^2\right )}{9 x+9 e^{30+\frac {8}{x}+\frac {2 x}{9}} x+e^{15} \left (-18 x+18 x^2\right )+e^{30} \left (9 x-18 x^2+9 x^3\right )+e^{\frac {4}{x}+\frac {x}{9}} \left (-18 e^{15} x+e^{30} \left (18 x-18 x^2\right )\right )} \, dx=-\frac {x e^{15}}{x e^{15} - e^{15} - e^{\left (\frac {x^{2} + 135 \, x + 36}{9 \, x}\right )} + 1} \] Input:
integrate(((-x^2+9*x+36)*exp(15)^2*exp(4/x)*exp(1/9*x)+9*x*exp(15)^2-9*x*e xp(15))/(9*x*exp(15)^2*exp(4/x)^2*exp(1/9*x)^2+((-18*x^2+18*x)*exp(15)^2-1 8*x*exp(15))*exp(4/x)*exp(1/9*x)+(9*x^3-18*x^2+9*x)*exp(15)^2+(18*x^2-18*x )*exp(15)+9*x),x, algorithm="giac")
Output:
-x*e^15/(x*e^15 - e^15 - e^(1/9*(x^2 + 135*x + 36)/x) + 1)
Time = 2.84 (sec) , antiderivative size = 140, normalized size of antiderivative = 5.19 \[ \int \frac {-9 e^{15} x+9 e^{30} x+e^{30+\frac {4}{x}+\frac {x}{9}} \left (36+9 x-x^2\right )}{9 x+9 e^{30+\frac {8}{x}+\frac {2 x}{9}} x+e^{15} \left (-18 x+18 x^2\right )+e^{30} \left (9 x-18 x^2+9 x^3\right )+e^{\frac {4}{x}+\frac {x}{9}} \left (-18 e^{15} x+e^{30} \left (18 x-18 x^2\right )\right )} \, dx=\frac {36\,x^3\,{\mathrm {e}}^{15}-x^5\,{\mathrm {e}}^{15}-36\,x^3\,{\mathrm {e}}^{30}+36\,x^4\,{\mathrm {e}}^{30}+10\,x^5\,{\mathrm {e}}^{30}-x^6\,{\mathrm {e}}^{30}}{\left ({\mathrm {e}}^{4/x}-{\mathrm {e}}^{-\frac {x}{9}-15}\,\left (x\,{\mathrm {e}}^{15}-{\mathrm {e}}^{15}+1\right )\right )\,\left (36\,x^2\,{\mathrm {e}}^{x/9}\,{\mathrm {e}}^{15}-x^4\,{\mathrm {e}}^{x/9}\,{\mathrm {e}}^{15}-36\,x^2\,{\mathrm {e}}^{x/9}\,{\mathrm {e}}^{30}+36\,x^3\,{\mathrm {e}}^{x/9}\,{\mathrm {e}}^{30}+10\,x^4\,{\mathrm {e}}^{x/9}\,{\mathrm {e}}^{30}-x^5\,{\mathrm {e}}^{x/9}\,{\mathrm {e}}^{30}\right )} \] Input:
int((9*x*exp(30) - 9*x*exp(15) + exp(x/9)*exp(30)*exp(4/x)*(9*x - x^2 + 36 ))/(9*x - exp(15)*(18*x - 18*x^2) + exp(30)*(9*x - 18*x^2 + 9*x^3) + exp(x /9)*exp(4/x)*(exp(30)*(18*x - 18*x^2) - 18*x*exp(15)) + 9*x*exp((2*x)/9)*e xp(30)*exp(8/x)),x)
Output:
(36*x^3*exp(15) - x^5*exp(15) - 36*x^3*exp(30) + 36*x^4*exp(30) + 10*x^5*e xp(30) - x^6*exp(30))/((exp(4/x) - exp(- x/9 - 15)*(x*exp(15) - exp(15) + 1))*(36*x^2*exp(x/9)*exp(15) - x^4*exp(x/9)*exp(15) - 36*x^2*exp(x/9)*exp( 30) + 36*x^3*exp(x/9)*exp(30) + 10*x^4*exp(x/9)*exp(30) - x^5*exp(x/9)*exp (30)))
Time = 2.64 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.89 \[ \int \frac {-9 e^{15} x+9 e^{30} x+e^{30+\frac {4}{x}+\frac {x}{9}} \left (36+9 x-x^2\right )}{9 x+9 e^{30+\frac {8}{x}+\frac {2 x}{9}} x+e^{15} \left (-18 x+18 x^2\right )+e^{30} \left (9 x-18 x^2+9 x^3\right )+e^{\frac {4}{x}+\frac {x}{9}} \left (-18 e^{15} x+e^{30} \left (18 x-18 x^2\right )\right )} \, dx=\frac {e^{\frac {x^{2}+36}{9 x}} e^{15}+e^{15}-1}{e^{\frac {x^{2}+36}{9 x}} e^{15}-e^{15} x +e^{15}-1} \] Input:
int(((-x^2+9*x+36)*exp(15)^2*exp(4/x)*exp(1/9*x)+9*x*exp(15)^2-9*x*exp(15) )/(9*x*exp(15)^2*exp(4/x)^2*exp(1/9*x)^2+((-18*x^2+18*x)*exp(15)^2-18*x*ex p(15))*exp(4/x)*exp(1/9*x)+(9*x^3-18*x^2+9*x)*exp(15)^2+(18*x^2-18*x)*exp( 15)+9*x),x)
Output:
(e**((x**2 + 36)/(9*x))*e**15 + e**15 - 1)/(e**((x**2 + 36)/(9*x))*e**15 - e**15*x + e**15 - 1)