3.60.53 \(\int \frac {-9 e^{15} x+9 e^{30} x+e^{30+\frac {4}{x}+\frac {x}{9}} (36+9 x-x^2)}{9 x+9 e^{30+\frac {8}{x}+\frac {2 x}{9}} x+e^{15} (-18 x+18 x^2)+e^{30} (9 x-18 x^2+9 x^3)+e^{\frac {4}{x}+\frac {x}{9}} (-18 e^{15} x+e^{30} (18 x-18 x^2))} \, dx\)

Optimal. Leaf size=27 \[ \frac {x}{1-\frac {1}{e^{15}}+e^{\frac {4}{x}+\frac {x}{9}}-x} \]

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Rubi [A]  time = 0.45, antiderivative size = 36, normalized size of antiderivative = 1.33, number of steps used = 4, number of rules used = 4, integrand size = 127, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.032, Rules used = {6, 6688, 12, 6687} \begin {gather*} -\frac {e^{15} x}{-e^{15} (1-x)-e^{\frac {x}{9}+\frac {4}{x}+15}+1} \end {gather*}

Antiderivative was successfully verified.

[In]

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]

[Out]

-((E^15*x)/(1 - E^(15 + 4/x + x/9) - E^15*(1 - x)))

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, x]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 6687

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]

Rule 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\left (-9 e^{15}+9 e^{30}\right ) 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\\ &=\int \frac {-9 e^{15} \left (1-e^{15}\right ) x+e^{30+\frac {4}{x}+\frac {x}{9}} \left (36+9 x-x^2\right )}{9 \left (1-e^{15+\frac {4}{x}+\frac {x}{9}}+e^{15} (-1+x)\right )^2 x} \, dx\\ &=\frac {1}{9} \int \frac {-9 e^{15} \left (1-e^{15}\right ) x+e^{30+\frac {4}{x}+\frac {x}{9}} \left (36+9 x-x^2\right )}{\left (1-e^{15+\frac {4}{x}+\frac {x}{9}}+e^{15} (-1+x)\right )^2 x} \, dx\\ &=-\frac {e^{15} x}{1-e^{15+\frac {4}{x}+\frac {x}{9}}-e^{15} (1-x)}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.63, size = 32, normalized size = 1.19 \begin {gather*} \frac {e^{15} x}{-1+e^{15}+e^{15+\frac {4}{x}+\frac {x}{9}}-e^{15} x} \end {gather*}

Antiderivative was successfully verified.

[In]

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^1
5*(-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]

[Out]

(E^15*x)/(-1 + E^15 + E^(15 + 4/x + x/9) - E^15*x)

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fricas [A]  time = 0.74, size = 32, normalized size = 1.19 \begin {gather*} -\frac {x e^{30}}{{\left (x - 1\right )} e^{30} + e^{15} - e^{\left (\frac {x^{2} + 270 \, x + 36}{9 \, x}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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*ex
p(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, algorithm="fricas")

[Out]

-x*e^30/((x - 1)*e^30 + e^15 - e^(1/9*(x^2 + 270*x + 36)/x))

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giac [A]  time = 0.46, size = 33, normalized size = 1.22 \begin {gather*} -\frac {x e^{15}}{x e^{15} - e^{15} - e^{\left (\frac {x^{2} + 135 \, x + 36}{9 \, x}\right )} + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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*ex
p(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, algorithm="giac")

[Out]

-x*e^15/(x*e^15 - e^15 - e^(1/9*(x^2 + 135*x + 36)/x) + 1)

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maple [A]  time = 0.20, size = 34, normalized size = 1.26

method	result	size
risch	\(-\frac {x \,{\mathrm e}^{15}}{-{\mathrm e}^{\frac {x^{2}+135 x +36}{9 x}}+x \,{\mathrm e}^{15}-{\mathrm e}^{15}+1}\)	\(34\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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*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)*ex
p(15)+9*x),x,method=_RETURNVERBOSE)

[Out]

-x*exp(15)/(-exp(1/9*(x^2+135*x+36)/x)+x*exp(15)-exp(15)+1)

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maxima [A]  time = 0.40, size = 30, normalized size = 1.11 \begin {gather*} -\frac {x e^{15}}{x e^{15} - e^{15} - e^{\left (\frac {1}{9} \, x + \frac {4}{x} + 15\right )} + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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*ex
p(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, algorithm="maxima")

[Out]

-x*e^15/(x*e^15 - e^15 - e^(1/9*x + 4/x + 15) + 1)

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mupad [B]  time = 4.78, size = 140, normalized size = 5.19 \begin {gather*} \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 )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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)
*exp(30)*exp(8/x)),x)

[Out]

(36*x^3*exp(15) - x^5*exp(15) - 36*x^3*exp(30) + 36*x^4*exp(30) + 10*x^5*exp(30) - x^6*exp(30))/((exp(4/x) - e
xp(- 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)*ex
p(30) + 36*x^3*exp(x/9)*exp(30) + 10*x^4*exp(x/9)*exp(30) - x^5*exp(x/9)*exp(30)))

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sympy [A]  time = 0.39, size = 27, normalized size = 1.00 \begin {gather*} \frac {x e^{15}}{- x e^{15} + e^{15} e^{\frac {4}{x}} e^{\frac {x}{9}} - 1 + e^{15}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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)

[Out]

x*exp(15)/(-x*exp(15) + exp(15)*exp(4/x)*exp(x/9) - 1 + exp(15))

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