\(\int -\frac {4 e^{3-\frac {-3+14 x}{x}}}{3 x^2} \, dx\) [7531]

Optimal result

Integrand size = 21, antiderivative size = 19 \[ \int -\frac {4 e^{3-\frac {-3+14 x}{x}}}{3 x^2} \, dx=\frac {1}{3} \left (5+\frac {4}{3} e^{-11+\frac {3}{x}}\right ) \]

[Out]

4/9*exp(3)/exp(14-3/x)+5/3

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.68, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {12, 2257, 2240} \[ \int -\frac {4 e^{3-\frac {-3+14 x}{x}}}{3 x^2} \, dx=\frac {4}{9} e^{\frac {3}{x}-11} \]

[In]

Int[(-4*E^(3 - (-3 + 14*x)/x))/(3*x^2),x]

[Out]

(4*E^(-11 + 3/x))/9

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 2240

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Simp[(e + f*x)^n*(
F^(a + b*(c + d*x)^n)/(b*f*n*(c + d*x)^n*Log[F])), x] /; FreeQ[{F, a, b, c, d, e, f, n}, x] && EqQ[m, n - 1] &
& EqQ[d*e - c*f, 0]

Rule 2257

Int[(F_)^(v_)*(u_)^(m_.), x_Symbol] :> Int[ExpandToSum[u, x]^m*F^ExpandToSum[v, x], x] /; FreeQ[{F, m}, x] &&
LinearQ[u, x] && BinomialQ[v, x] &&  !(LinearMatchQ[u, x] && BinomialMatchQ[v, x])

Rubi steps \begin{align*} \text {integral}& = -\left (\frac {4}{3} \int \frac {e^{3-\frac {-3+14 x}{x}}}{x^2} \, dx\right ) \\ & = -\left (\frac {4}{3} \int \frac {e^{-11+\frac {3}{x}}}{x^2} \, dx\right ) \\ & = \frac {4}{9} e^{-11+\frac {3}{x}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.68 \[ \int -\frac {4 e^{3-\frac {-3+14 x}{x}}}{3 x^2} \, dx=\frac {4}{9} e^{-11+\frac {3}{x}} \]

[In]

Integrate[(-4*E^(3 - (-3 + 14*x)/x))/(3*x^2),x]

[Out]

(4*E^(-11 + 3/x))/9

Maple [A] (verified)

Time = 0.04 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.74

method	result	size
risch	\(\frac {4 \,{\mathrm e}^{-\frac {11 x -3}{x}}}{9}\)	\(14\)
gosper	\(\frac {4 \,{\mathrm e}^{3} {\mathrm e}^{-\frac {14 x -3}{x}}}{9}\)	\(17\)
derivativedivides	\(\frac {4 \,{\mathrm e}^{3} {\mathrm e}^{-\frac {14 x -3}{x}}}{9}\)	\(17\)
default	\(\frac {4 \,{\mathrm e}^{3} {\mathrm e}^{-\frac {14 x -3}{x}}}{9}\)	\(17\)
norman	\(\frac {4 \,{\mathrm e}^{3} {\mathrm e}^{-\frac {14 x -3}{x}}}{9}\)	\(17\)
parallelrisch	\(\frac {4 \,{\mathrm e}^{3} {\mathrm e}^{-\frac {14 x -3}{x}}}{9}\)	\(17\)

[In]

int(-4/3*exp(3)/x^2/exp((14*x-3)/x),x,method=_RETURNVERBOSE)

[Out]

4/9*exp(-(11*x-3)/x)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.68 \[ \int -\frac {4 e^{3-\frac {-3+14 x}{x}}}{3 x^2} \, dx=\frac {4}{9} \, e^{\left (-\frac {11 \, x - 3}{x}\right )} \]

[In]

integrate(-4/3*exp(3)/x^2/exp((14*x-3)/x),x, algorithm="fricas")

[Out]

4/9*e^(-(11*x - 3)/x)

Sympy [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.74 \[ \int -\frac {4 e^{3-\frac {-3+14 x}{x}}}{3 x^2} \, dx=\frac {4 e^{3} e^{- \frac {14 x - 3}{x}}}{9} \]

[In]

integrate(-4/3*exp(3)/x**2/exp((14*x-3)/x),x)

[Out]

4*exp(3)*exp(-(14*x - 3)/x)/9

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.53 \[ \int -\frac {4 e^{3-\frac {-3+14 x}{x}}}{3 x^2} \, dx=\frac {4}{9} \, e^{\left (\frac {3}{x} - 11\right )} \]

[In]

integrate(-4/3*exp(3)/x^2/exp((14*x-3)/x),x, algorithm="maxima")

[Out]

4/9*e^(3/x - 11)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.53 \[ \int -\frac {4 e^{3-\frac {-3+14 x}{x}}}{3 x^2} \, dx=\frac {4}{9} \, e^{\left (\frac {3}{x} - 11\right )} \]

[In]

integrate(-4/3*exp(3)/x^2/exp((14*x-3)/x),x, algorithm="giac")

[Out]

4/9*e^(3/x - 11)

Mupad [B] (verification not implemented)

Time = 12.28 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.53 \[ \int -\frac {4 e^{3-\frac {-3+14 x}{x}}}{3 x^2} \, dx=\frac {4\,{\mathrm {e}}^{-11}\,{\mathrm {e}}^{3/x}}{9} \]

[In]

int(-(4*exp(3)*exp(-(14*x - 3)/x))/(3*x^2),x)

[Out]

(4*exp(-11)*exp(3/x))/9