3.36.11 \(\int \frac {e^{-2+e^x+\frac {x^3}{27+9 x}} (9 x^2+2 x^3+e^x (81+54 x+9 x^2))}{81+54 x+9 x^2} \, dx\)

Optimal. Leaf size=23 \[ e^{-2+e^x+\frac {x^2}{3 \left (3+\frac {9}{x}\right )}} \]

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Rubi [A]  time = 0.47, antiderivative size = 19, normalized size of antiderivative = 0.83, number of steps used = 3, number of rules used = 3, integrand size = 56, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.054, Rules used = {27, 12, 6706} \begin {gather*} e^{\frac {x^3}{9 (x+3)}+e^x-2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^(-2 + E^x + x^3/(27 + 9*x))*(9*x^2 + 2*x^3 + E^x*(81 + 54*x + 9*x^2)))/(81 + 54*x + 9*x^2),x]

[Out]

E^(-2 + E^x + x^3/(9*(3 + 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 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 6706

Int[(F_)^(v_)*(u_), x_Symbol] :> With[{q = DerivativeDivides[v, u, x]}, Simp[(q*F^v)/Log[F], x] /;  !FalseQ[q]
] /; FreeQ[F, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{-2+e^x+\frac {x^3}{27+9 x}} \left (9 x^2+2 x^3+e^x \left (81+54 x+9 x^2\right )\right )}{9 (3+x)^2} \, dx\\ &=\frac {1}{9} \int \frac {e^{-2+e^x+\frac {x^3}{27+9 x}} \left (9 x^2+2 x^3+e^x \left (81+54 x+9 x^2\right )\right )}{(3+x)^2} \, dx\\ &=e^{-2+e^x+\frac {x^3}{9 (3+x)}}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.32, size = 23, normalized size = 1.00 \begin {gather*} e^{e^x+\frac {-54-18 x+x^3}{9 (3+x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(-2 + E^x + x^3/(27 + 9*x))*(9*x^2 + 2*x^3 + E^x*(81 + 54*x + 9*x^2)))/(81 + 54*x + 9*x^2),x]

[Out]

E^(E^x + (-54 - 18*x + x^3)/(9*(3 + x)))

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fricas [A]  time = 0.57, size = 23, normalized size = 1.00 \begin {gather*} e^{\left (\frac {x^{3} + 9 \, {\left (x + 3\right )} e^{x} - 18 \, x - 54}{9 \, {\left (x + 3\right )}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((9*x^2+54*x+81)*exp(x)+2*x^3+9*x^2)*exp(x^3/(9*x+27))*exp(exp(x)-2)/(9*x^2+54*x+81),x, algorithm="f
ricas")

[Out]

e^(1/9*(x^3 + 9*(x + 3)*e^x - 18*x - 54)/(x + 3))

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giac [B]  time = 0.20, size = 45, normalized size = 1.96 \begin {gather*} e^{\left (\frac {x^{3}}{9 \, {\left (x + 3\right )}} + \frac {x e^{x}}{x + 3} - \frac {2 \, x}{x + 3} + \frac {3 \, e^{x}}{x + 3} - \frac {6}{x + 3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((9*x^2+54*x+81)*exp(x)+2*x^3+9*x^2)*exp(x^3/(9*x+27))*exp(exp(x)-2)/(9*x^2+54*x+81),x, algorithm="g
iac")

[Out]

e^(1/9*x^3/(x + 3) + x*e^x/(x + 3) - 2*x/(x + 3) + 3*e^x/(x + 3) - 6/(x + 3))

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maple [A]  time = 0.18, size = 26, normalized size = 1.13

method	result	size
risch	\({\mathrm e}^{\frac {x^{3}+9 \,{\mathrm e}^{x} x +27 \,{\mathrm e}^{x}-18 x -54}{9 x +27}}\)	\(26\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((9*x^2+54*x+81)*exp(x)+2*x^3+9*x^2)*exp(x^3/(9*x+27))*exp(exp(x)-2)/(9*x^2+54*x+81),x,method=_RETURNVERBO
SE)

[Out]

exp(1/9*(x^3+9*exp(x)*x+27*exp(x)-18*x-54)/(3+x))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {1}{9} \, \int \frac {{\left (2 \, x^{3} + 9 \, x^{2} + 9 \, {\left (x^{2} + 6 \, x + 9\right )} e^{x}\right )} e^{\left (\frac {x^{3}}{9 \, {\left (x + 3\right )}} + e^{x} - 2\right )}}{x^{2} + 6 \, x + 9}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((9*x^2+54*x+81)*exp(x)+2*x^3+9*x^2)*exp(x^3/(9*x+27))*exp(exp(x)-2)/(9*x^2+54*x+81),x, algorithm="m
axima")

[Out]

1/9*integrate((2*x^3 + 9*x^2 + 9*(x^2 + 6*x + 9)*e^x)*e^(1/9*x^3/(x + 3) + e^x - 2)/(x^2 + 6*x + 9), x)

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mupad [B]  time = 2.29, size = 18, normalized size = 0.78 \begin {gather*} {\mathrm {e}}^{\frac {x^3}{9\,x+27}}\,{\mathrm {e}}^{{\mathrm {e}}^x}\,{\mathrm {e}}^{-2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(x^3/(9*x + 27))*exp(exp(x) - 2)*(exp(x)*(54*x + 9*x^2 + 81) + 9*x^2 + 2*x^3))/(54*x + 9*x^2 + 81),x)

[Out]

exp(x^3/(9*x + 27))*exp(exp(x))*exp(-2)

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sympy [A]  time = 9.71, size = 15, normalized size = 0.65 \begin {gather*} e^{\frac {x^{3}}{9 x + 27}} e^{e^{x} - 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((9*x**2+54*x+81)*exp(x)+2*x**3+9*x**2)*exp(x**3/(9*x+27))*exp(exp(x)-2)/(9*x**2+54*x+81),x)

[Out]

exp(x**3/(9*x + 27))*exp(exp(x) - 2)

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