3.83.83 \(\int \frac {e^{e^{\frac {-137-34 x-x^2}{5+x}}} (e (25+10 x+x^2)+e^{1+\frac {-137-34 x-x^2}{5+x}} (-33 x-10 x^2-x^3))}{300+120 x+12 x^2} \, dx\)

Optimal. Leaf size=23 \[ \frac {1}{12} e^{1+e^{-29-x+\frac {8}{5+x}}} x \]

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Rubi [B]  time = 0.51, antiderivative size = 70, normalized size of antiderivative = 3.04, number of steps used = 3, number of rules used = 3, integrand size = 79, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {27, 12, 2288} \begin {gather*} \frac {e^{e^{-\frac {x^2+34 x+137}{x+5}}+1} \left (x^3+10 x^2+33 x\right )}{12 (x+5)^2 \left (\frac {2 (x+17)}{x+5}-\frac {x^2+34 x+137}{(x+5)^2}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^E^((-137 - 34*x - x^2)/(5 + x))*(E*(25 + 10*x + x^2) + E^(1 + (-137 - 34*x - x^2)/(5 + x))*(-33*x - 10*
x^2 - x^3)))/(300 + 120*x + 12*x^2),x]

[Out]

(E^(1 + E^(-((137 + 34*x + x^2)/(5 + x))))*(33*x + 10*x^2 + x^3))/(12*(5 + x)^2*((2*(17 + x))/(5 + x) - (137 +
34*x + x^2)/(5 + x)^2))

Rule 12

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

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 2288

Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = (v*y)/(Log[F]*D[u, x])}, Simp[F^u*z, x] /; EqQ[D[z,
x], w*y]] /; FreeQ[F, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{e^{\frac {-137-34 x-x^2}{5+x}}} \left (e \left (25+10 x+x^2\right )+e^{1+\frac {-137-34 x-x^2}{5+x}} \left (-33 x-10 x^2-x^3\right )\right )}{12 (5+x)^2} \, dx\\ &=\frac {1}{12} \int \frac {e^{e^{\frac {-137-34 x-x^2}{5+x}}} \left (e \left (25+10 x+x^2\right )+e^{1+\frac {-137-34 x-x^2}{5+x}} \left (-33 x-10 x^2-x^3\right )\right )}{(5+x)^2} \, dx\\ &=\frac {e^{1+e^{-\frac {137+34 x+x^2}{5+x}}} \left (33 x+10 x^2+x^3\right )}{12 (5+x)^2 \left (\frac {2 (17+x)}{5+x}-\frac {137+34 x+x^2}{(5+x)^2}\right )}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.11, size = 26, normalized size = 1.13 \begin {gather*} \frac {1}{12} e^{1+e^{-\frac {137+34 x+x^2}{5+x}}} x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^E^((-137 - 34*x - x^2)/(5 + x))*(E*(25 + 10*x + x^2) + E^(1 + (-137 - 34*x - x^2)/(5 + x))*(-33*x
- 10*x^2 - x^3)))/(300 + 120*x + 12*x^2),x]

[Out]

(E^(1 + E^(-((137 + 34*x + x^2)/(5 + x))))*x)/12

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fricas [A]  time = 0.75, size = 24, normalized size = 1.04 \begin {gather*} \frac {1}{12} \, x e^{\left (e^{\left (-\frac {x^{2} + 33 \, x + 132}{x + 5} - 1\right )} + 1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^3-10*x^2-33*x)*exp(1)*exp((-x^2-34*x-137)/(5+x))+(x^2+10*x+25)*exp(1))*exp(exp((-x^2-34*x-137)/
(5+x)))/(12*x^2+120*x+300),x, algorithm="fricas")

[Out]

1/12*x*e^(e^(-(x^2 + 33*x + 132)/(x + 5) - 1) + 1)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left ({\left (x^{2} + 10 \, x + 25\right )} e - {\left (x^{3} + 10 \, x^{2} + 33 \, x\right )} e^{\left (-\frac {x^{2} + 34 \, x + 137}{x + 5} + 1\right )}\right )} e^{\left (e^{\left (-\frac {x^{2} + 34 \, x + 137}{x + 5}\right )}\right )}}{12 \, {\left (x^{2} + 10 \, x + 25\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^3-10*x^2-33*x)*exp(1)*exp((-x^2-34*x-137)/(5+x))+(x^2+10*x+25)*exp(1))*exp(exp((-x^2-34*x-137)/
(5+x)))/(12*x^2+120*x+300),x, algorithm="giac")

[Out]

integrate(1/12*((x^2 + 10*x + 25)*e - (x^3 + 10*x^2 + 33*x)*e^(-(x^2 + 34*x + 137)/(x + 5) + 1))*e^(e^(-(x^2 +
34*x + 137)/(x + 5)))/(x^2 + 10*x + 25), x)

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maple [A]  time = 0.36, size = 23, normalized size = 1.00

method	result	size
risch	\(\frac {x \,{\mathrm e}^{1+{\mathrm e}^{-\frac {x^{2}+34 x +137}{5+x}}}}{12}\)	\(23\)
norman	\(\frac {\frac {5 x \,{\mathrm e} \,{\mathrm e}^{{\mathrm e}^{\frac {-x^{2}-34 x -137}{5+x}}}}{12}+\frac {x^{2} {\mathrm e} \,{\mathrm e}^{{\mathrm e}^{\frac {-x^{2}-34 x -137}{5+x}}}}{12}}{5+x}\)	\(56\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-x^3-10*x^2-33*x)*exp(1)*exp((-x^2-34*x-137)/(5+x))+(x^2+10*x+25)*exp(1))*exp(exp((-x^2-34*x-137)/(5+x))
)/(12*x^2+120*x+300),x,method=_RETURNVERBOSE)

[Out]

1/12*x*exp(1+exp(-(x^2+34*x+137)/(5+x)))

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maxima [A]  time = 0.44, size = 19, normalized size = 0.83 \begin {gather*} \frac {1}{12} \, x e^{\left (e^{\left (-x + \frac {8}{x + 5} - 29\right )} + 1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^3-10*x^2-33*x)*exp(1)*exp((-x^2-34*x-137)/(5+x))+(x^2+10*x+25)*exp(1))*exp(exp((-x^2-34*x-137)/
(5+x)))/(12*x^2+120*x+300),x, algorithm="maxima")

[Out]

1/12*x*e^(e^(-x + 8/(x + 5) - 29) + 1)

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mupad [B]  time = 5.42, size = 35, normalized size = 1.52 \begin {gather*} \frac {x\,{\mathrm {e}}^{{\mathrm {e}}^{-\frac {34\,x}{x+5}}\,{\mathrm {e}}^{-\frac {x^2}{x+5}}\,{\mathrm {e}}^{-\frac {137}{x+5}}}\,\mathrm {e}}{12} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(exp(-(34*x + x^2 + 137)/(x + 5)))*(exp(1)*(10*x + x^2 + 25) - exp(-(34*x + x^2 + 137)/(x + 5))*exp(1)
*(33*x + 10*x^2 + x^3)))/(120*x + 12*x^2 + 300),x)

[Out]

(x*exp(exp(-(34*x)/(x + 5))*exp(-x^2/(x + 5))*exp(-137/(x + 5)))*exp(1))/12

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x**3-10*x**2-33*x)*exp(1)*exp((-x**2-34*x-137)/(5+x))+(x**2+10*x+25)*exp(1))*exp(exp((-x**2-34*x-
137)/(5+x)))/(12*x**2+120*x+300),x)

[Out]

Timed out

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