3.2.3 \(\int \frac {-x^2+e^{\frac {729-9 x+2 x^2+x^3}{x}} (729-2 x^2-2 x^3)}{3 e^{\frac {2 (729-9 x+2 x^2+x^3)}{x}} x^2+6 e^{\frac {729-9 x+2 x^2+x^3}{x}} x^3+3 x^4} \, dx\)

Optimal. Leaf size=25 \[ \frac {1}{3 \left (e^{-\frac {9 (-81+x)}{x}+2 x+x^2}+x\right )} \]

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Rubi [A]  time = 0.55, antiderivative size = 29, normalized size of antiderivative = 1.16, number of steps used = 3, number of rules used = 3, integrand size = 96, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.031, Rules used = {6688, 12, 6686} \begin {gather*} \frac {e^9}{3 \left (e^{x^2+2 x+\frac {729}{x}}+e^9 x\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-x^2 + E^((729 - 9*x + 2*x^2 + x^3)/x)*(729 - 2*x^2 - 2*x^3))/(3*E^((2*(729 - 9*x + 2*x^2 + x^3))/x)*x^2
+ 6*E^((729 - 9*x + 2*x^2 + x^3)/x)*x^3 + 3*x^4),x]

[Out]

E^9/(3*(E^(729/x + 2*x + x^2) + E^9*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 6686

Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[(q*y^(m + 1))/(m + 1), x] /;  !F
alseQ[q]] /; FreeQ[m, 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 {e^9 \left (-e^9 x^2-e^{\frac {729}{x}+2 x+x^2} \left (-729+2 x^2+2 x^3\right )\right )}{3 x^2 \left (e^{\frac {729}{x}+2 x+x^2}+e^9 x\right )^2} \, dx\\ &=\frac {1}{3} e^9 \int \frac {-e^9 x^2-e^{\frac {729}{x}+2 x+x^2} \left (-729+2 x^2+2 x^3\right )}{x^2 \left (e^{\frac {729}{x}+2 x+x^2}+e^9 x\right )^2} \, dx\\ &=\frac {e^9}{3 \left (e^{\frac {729}{x}+2 x+x^2}+e^9 x\right )}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.06, size = 29, normalized size = 1.16 \begin {gather*} \frac {e^9}{3 \left (e^{\frac {729}{x}+2 x+x^2}+e^9 x\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-x^2 + E^((729 - 9*x + 2*x^2 + x^3)/x)*(729 - 2*x^2 - 2*x^3))/(3*E^((2*(729 - 9*x + 2*x^2 + x^3))/x
)*x^2 + 6*E^((729 - 9*x + 2*x^2 + x^3)/x)*x^3 + 3*x^4),x]

[Out]

E^9/(3*(E^(729/x + 2*x + x^2) + E^9*x))

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fricas [A]  time = 1.42, size = 24, normalized size = 0.96 \begin {gather*} \frac {1}{3 \, {\left (x + e^{\left (\frac {x^{3} + 2 \, x^{2} - 9 \, x + 729}{x}\right )}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x^3-2*x^2+729)*exp((x^3+2*x^2-9*x+729)/x)-x^2)/(3*x^2*exp((x^3+2*x^2-9*x+729)/x)^2+6*x^3*exp((x
^3+2*x^2-9*x+729)/x)+3*x^4),x, algorithm="fricas")

[Out]

1/3/(x + e^((x^3 + 2*x^2 - 9*x + 729)/x))

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giac [A]  time = 0.36, size = 24, normalized size = 0.96 \begin {gather*} \frac {1}{3 \, {\left (x + e^{\left (\frac {x^{3} + 2 \, x^{2} - 9 \, x + 729}{x}\right )}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x^3-2*x^2+729)*exp((x^3+2*x^2-9*x+729)/x)-x^2)/(3*x^2*exp((x^3+2*x^2-9*x+729)/x)^2+6*x^3*exp((x
^3+2*x^2-9*x+729)/x)+3*x^4),x, algorithm="giac")

[Out]

1/3/(x + e^((x^3 + 2*x^2 - 9*x + 729)/x))

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maple [A]  time = 0.84, size = 25, normalized size = 1.00

method	result	size
norman	\(\frac {1}{3 x +3 \,{\mathrm e}^{\frac {x^{3}+2 x^{2}-9 x +729}{x}}}\)	\(25\)
risch	\(\frac {1}{3 x +3 \,{\mathrm e}^{\frac {x^{3}+2 x^{2}-9 x +729}{x}}}\)	\(25\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-2*x^3-2*x^2+729)*exp((x^3+2*x^2-9*x+729)/x)-x^2)/(3*x^2*exp((x^3+2*x^2-9*x+729)/x)^2+6*x^3*exp((x^3+2*x
^2-9*x+729)/x)+3*x^4),x,method=_RETURNVERBOSE)

[Out]

1/3/(x+exp((x^3+2*x^2-9*x+729)/x))

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maxima [A]  time = 0.54, size = 24, normalized size = 0.96 \begin {gather*} \frac {e^{9}}{3 \, {\left (x e^{9} + e^{\left (x^{2} + 2 \, x + \frac {729}{x}\right )}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x^3-2*x^2+729)*exp((x^3+2*x^2-9*x+729)/x)-x^2)/(3*x^2*exp((x^3+2*x^2-9*x+729)/x)^2+6*x^3*exp((x
^3+2*x^2-9*x+729)/x)+3*x^4),x, algorithm="maxima")

[Out]

1/3*e^9/(x*e^9 + e^(x^2 + 2*x + 729/x))

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mupad [B]  time = 0.44, size = 27, normalized size = 1.08 \begin {gather*} \frac {{\mathrm {e}}^9}{3\,x\,{\mathrm {e}}^9+3\,{\mathrm {e}}^{2\,x}\,{\mathrm {e}}^{x^2}\,{\mathrm {e}}^{729/x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp((2*x^2 - 9*x + x^3 + 729)/x)*(2*x^2 + 2*x^3 - 729) + x^2)/(6*x^3*exp((2*x^2 - 9*x + x^3 + 729)/x) +
3*x^2*exp((2*(2*x^2 - 9*x + x^3 + 729))/x) + 3*x^4),x)

[Out]

exp(9)/(3*x*exp(9) + 3*exp(2*x)*exp(x^2)*exp(729/x))

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sympy [A]  time = 0.17, size = 22, normalized size = 0.88 \begin {gather*} \frac {1}{3 x + 3 e^{\frac {x^{3} + 2 x^{2} - 9 x + 729}{x}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x**3-2*x**2+729)*exp((x**3+2*x**2-9*x+729)/x)-x**2)/(3*x**2*exp((x**3+2*x**2-9*x+729)/x)**2+6*x
**3*exp((x**3+2*x**2-9*x+729)/x)+3*x**4),x)

[Out]

1/(3*x + 3*exp((x**3 + 2*x**2 - 9*x + 729)/x))

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