3.34.71 \(\int \frac {e^{\frac {x^2}{1+x+x^2}} (2 x+x^2)}{1+2 x+3 x^2+2 x^3+x^4} \, dx\)

Optimal. Leaf size=17 \[ e^{\frac {x}{x+\frac {x+x^2}{x^2}}} \]

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Rubi [A]  time = 0.32, antiderivative size = 14, normalized size of antiderivative = 0.82, number of steps used = 3, number of rules used = 3, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {1593, 6688, 6706} \begin {gather*} e^{\frac {x^2}{x^2+x+1}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^(x^2/(1 + x + x^2))*(2*x + x^2))/(1 + 2*x + 3*x^2 + 2*x^3 + x^4),x]

[Out]

E^(x^2/(1 + x + x^2))

Rule 1593

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 6688

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

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^{\frac {x^2}{1+x+x^2}} x (2+x)}{1+2 x+3 x^2+2 x^3+x^4} \, dx\\ &=\int \frac {e^{\frac {x^2}{1+x+x^2}} x (2+x)}{\left (1+x+x^2\right )^2} \, dx\\ &=e^{\frac {x^2}{1+x+x^2}}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.26, size = 14, normalized size = 0.82 \begin {gather*} e^{\frac {x^2}{1+x+x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(x^2/(1 + x + x^2))*(2*x + x^2))/(1 + 2*x + 3*x^2 + 2*x^3 + x^4),x]

[Out]

E^(x^2/(1 + x + x^2))

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fricas [A]  time = 0.83, size = 13, normalized size = 0.76 \begin {gather*} e^{\left (\frac {x^{2}}{x^{2} + x + 1}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2+2*x)*exp(x^2/(x^2+x+1))/(x^4+2*x^3+3*x^2+2*x+1),x, algorithm="fricas")

[Out]

e^(x^2/(x^2 + x + 1))

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giac [A]  time = 4.84, size = 13, normalized size = 0.76 \begin {gather*} e^{\left (\frac {x^{2}}{x^{2} + x + 1}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2+2*x)*exp(x^2/(x^2+x+1))/(x^4+2*x^3+3*x^2+2*x+1),x, algorithm="giac")

[Out]

e^(x^2/(x^2 + x + 1))

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maple [A]  time = 0.06, size = 14, normalized size = 0.82

method	result	size
gosper	\({\mathrm e}^{\frac {x^{2}}{x^{2}+x +1}}\)	\(14\)
risch	\({\mathrm e}^{\frac {x^{2}}{x^{2}+x +1}}\)	\(14\)
norman	\(\frac {x \,{\mathrm e}^{\frac {x^{2}}{x^{2}+x +1}}+x^{2} {\mathrm e}^{\frac {x^{2}}{x^{2}+x +1}}+{\mathrm e}^{\frac {x^{2}}{x^{2}+x +1}}}{x^{2}+x +1}\)	\(56\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2+2*x)*exp(x^2/(x^2+x+1))/(x^4+2*x^3+3*x^2+2*x+1),x,method=_RETURNVERBOSE)

[Out]

exp(x^2/(x^2+x+1))

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maxima [A]  time = 0.84, size = 24, normalized size = 1.41 \begin {gather*} e^{\left (-\frac {x}{x^{2} + x + 1} - \frac {1}{x^{2} + x + 1} + 1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2+2*x)*exp(x^2/(x^2+x+1))/(x^4+2*x^3+3*x^2+2*x+1),x, algorithm="maxima")

[Out]

e^(-x/(x^2 + x + 1) - 1/(x^2 + x + 1) + 1)

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mupad [B]  time = 0.19, size = 13, normalized size = 0.76 \begin {gather*} {\mathrm {e}}^{\frac {x^2}{x^2+x+1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(x^2/(x + x^2 + 1))*(2*x + x^2))/(2*x + 3*x^2 + 2*x^3 + x^4 + 1),x)

[Out]

exp(x^2/(x + x^2 + 1))

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sympy [A]  time = 0.16, size = 10, normalized size = 0.59 \begin {gather*} e^{\frac {x^{2}}{x^{2} + x + 1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**2+2*x)*exp(x**2/(x**2+x+1))/(x**4+2*x**3+3*x**2+2*x+1),x)

[Out]

exp(x**2/(x**2 + x + 1))

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