\(\int \frac {e^{\frac {8427+636 x+12 x^2+x^3}{3 x^2}} (-16854-636 x+x^3)}{3 x^3} \, dx\) [5219]

Optimal result

Integrand size = 37, antiderivative size = 19 \[ \int \frac {e^{\frac {8427+636 x+12 x^2+x^3}{3 x^2}} \left (-16854-636 x+x^3\right )}{3 x^3} \, dx=e^{\frac {(-53-2 x)^2}{x^2}+\frac {x}{3}} \]

[Out]

exp((-53-2*x)^2/x^2+1/3*x)

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.16, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.054, Rules used = {12, 6838} \[ \int \frac {e^{\frac {8427+636 x+12 x^2+x^3}{3 x^2}} \left (-16854-636 x+x^3\right )}{3 x^3} \, dx=e^{\frac {x^3+12 x^2+636 x+8427}{3 x^2}} \]

[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 6838

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{align*} \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{align*}

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \frac {e^{\frac {8427+636 x+12 x^2+x^3}{3 x^2}} \left (-16854-636 x+x^3\right )}{3 x^3} \, dx=e^{4+\frac {2809}{x^2}+\frac {212}{x}+\frac {x}{3}} \]

[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)

Maple [A] (verified)

Time = 0.08 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.05

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

[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)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \frac {e^{\frac {8427+636 x+12 x^2+x^3}{3 x^2}} \left (-16854-636 x+x^3\right )}{3 x^3} \, dx=e^{\left (\frac {x^{3} + 12 \, x^{2} + 636 \, x + 8427}{3 \, x^{2}}\right )} \]

[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)

Sympy [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \frac {e^{\frac {8427+636 x+12 x^2+x^3}{3 x^2}} \left (-16854-636 x+x^3\right )}{3 x^3} \, dx=e^{\frac {\frac {x^{3}}{3} + 4 x^{2} + 212 x + 2809}{x^{2}}} \]

[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)

Maxima [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.84 \[ \int \frac {e^{\frac {8427+636 x+12 x^2+x^3}{3 x^2}} \left (-16854-636 x+x^3\right )}{3 x^3} \, dx=e^{\left (\frac {1}{3} \, x + \frac {212}{x} + \frac {2809}{x^{2}} + 4\right )} \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.84 \[ \int \frac {e^{\frac {8427+636 x+12 x^2+x^3}{3 x^2}} \left (-16854-636 x+x^3\right )}{3 x^3} \, dx=e^{\left (\frac {1}{3} \, x + \frac {212}{x} + \frac {2809}{x^{2}} + 4\right )} \]

[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)

Mupad [B] (verification not implemented)

Time = 12.25 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \frac {e^{\frac {8427+636 x+12 x^2+x^3}{3 x^2}} \left (-16854-636 x+x^3\right )}{3 x^3} \, dx={\mathrm {e}}^{x/3}\,{\mathrm {e}}^4\,{\mathrm {e}}^{212/x}\,{\mathrm {e}}^{\frac {2809}{x^2}} \]

[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)