3.53.19 \(\int \frac {e^{\frac {8427+636 x+12 x^2+x^3}{3 x^2}} (-16854-636 x+x^3)}{3 x^3} \, dx\)

Optimal. Leaf size=19 \[ e^{\frac {(-53-2 x)^2}{x^2}+\frac {x}{3}} \]

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Rubi [A]  time = 0.17, antiderivative size = 22, normalized size of antiderivative = 1.16, number of steps used = 2, number of rules used = 2, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.054, Rules used = {12, 6706} \begin {gather*} e^{\frac {x^3+12 x^2+636 x+8427}{3 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^((8427 + 636*x + 12*x^2 + x^3)/(3*x^2))*(-16854 - 636*x + x^3))/(3*x^3),x]

[Out]

E^((8427 + 636*x + 12*x^2 + x^3)/(3*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 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} &=\frac {1}{3} \int \frac {e^{\frac {8427+636 x+12 x^2+x^3}{3 x^2}} \left (-16854-636 x+x^3\right )}{x^3} \, dx\\ &=e^{\frac {8427+636 x+12 x^2+x^3}{3 x^2}}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.07, size = 19, normalized size = 1.00 \begin {gather*} e^{4+\frac {2809}{x^2}+\frac {212}{x}+\frac {x}{3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((8427 + 636*x + 12*x^2 + x^3)/(3*x^2))*(-16854 - 636*x + x^3))/(3*x^3),x]

[Out]

E^(4 + 2809/x^2 + 212/x + x/3)

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fricas [A]  time = 0.84, size = 19, normalized size = 1.00 \begin {gather*} e^{\left (\frac {x^{3} + 12 \, x^{2} + 636 \, x + 8427}{3 \, x^{2}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(x^3-636*x-16854)*exp(1/3*(x^3+12*x^2+636*x+8427)/x^2)/x^3,x, algorithm="fricas")

[Out]

e^(1/3*(x^3 + 12*x^2 + 636*x + 8427)/x^2)

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giac [A]  time = 0.27, size = 16, normalized size = 0.84 \begin {gather*} e^{\left (\frac {1}{3} \, x + \frac {212}{x} + \frac {2809}{x^{2}} + 4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(x^3-636*x-16854)*exp(1/3*(x^3+12*x^2+636*x+8427)/x^2)/x^3,x, algorithm="giac")

[Out]

e^(1/3*x + 212/x + 2809/x^2 + 4)

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maple [A]  time = 0.17, size = 20, normalized size = 1.05

method	result	size
gosper	\({\mathrm e}^{\frac {x^{3}+12 x^{2}+636 x +8427}{3 x^{2}}}\)	\(20\)
norman	\({\mathrm e}^{\frac {x^{3}+12 x^{2}+636 x +8427}{3 x^{2}}}\)	\(20\)
risch	\({\mathrm e}^{\frac {x^{3}+12 x^{2}+636 x +8427}{3 x^{2}}}\)	\(20\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/3*(x^3-636*x-16854)*exp(1/3*(x^3+12*x^2+636*x+8427)/x^2)/x^3,x,method=_RETURNVERBOSE)

[Out]

exp(1/3*(x^3+12*x^2+636*x+8427)/x^2)

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maxima [A]  time = 0.51, size = 16, normalized size = 0.84 \begin {gather*} e^{\left (\frac {1}{3} \, x + \frac {212}{x} + \frac {2809}{x^{2}} + 4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(x^3-636*x-16854)*exp(1/3*(x^3+12*x^2+636*x+8427)/x^2)/x^3,x, algorithm="maxima")

[Out]

e^(1/3*x + 212/x + 2809/x^2 + 4)

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mupad [B]  time = 3.38, size = 19, normalized size = 1.00 \begin {gather*} {\mathrm {e}}^{x/3}\,{\mathrm {e}}^4\,{\mathrm {e}}^{212/x}\,{\mathrm {e}}^{\frac {2809}{x^2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp((212*x + 4*x^2 + x^3/3 + 2809)/x^2)*(636*x - x^3 + 16854))/(3*x^3),x)

[Out]

exp(x/3)*exp(4)*exp(212/x)*exp(2809/x^2)

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sympy [A]  time = 0.13, size = 19, normalized size = 1.00 \begin {gather*} e^{\frac {\frac {x^{3}}{3} + 4 x^{2} + 212 x + 2809}{x^{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(x**3-636*x-16854)*exp(1/3*(x**3+12*x**2+636*x+8427)/x**2)/x**3,x)

[Out]

exp((x**3/3 + 4*x**2 + 212*x + 2809)/x**2)

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