\(\int \frac {e^{\frac {-12+28 x+11 x^2}{12 x}} (12+11 x^2)}{12 x^2} \, dx\) [6747]

Optimal result

Integrand size = 33, antiderivative size = 27 \[ \int \frac {e^{\frac {-12+28 x+11 x^2}{12 x}} \left (12+11 x^2\right )}{12 x^2} \, dx=e^{\frac {2 x+x^2+\frac {1}{3} \left (-3+x-\frac {x^2}{4}\right )}{x}} \]

[Out]

exp((7/3*x+11/12*x^2-1)/x)

Rubi [A] (verified)

Time = 0.12 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.70, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {12, 6838} \[ \int \frac {e^{\frac {-12+28 x+11 x^2}{12 x}} \left (12+11 x^2\right )}{12 x^2} \, dx=e^{-\frac {-11 x^2-28 x+12}{12 x}} \]

[In]

Int[(E^((-12 + 28*x + 11*x^2)/(12*x))*(12 + 11*x^2))/(12*x^2),x]

[Out]

E^(-1/12*(12 - 28*x - 11*x^2)/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 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}{12} \int \frac {e^{\frac {-12+28 x+11 x^2}{12 x}} \left (12+11 x^2\right )}{x^2} \, dx \\ & = e^{-\frac {12-28 x-11 x^2}{12 x}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.59 \[ \int \frac {e^{\frac {-12+28 x+11 x^2}{12 x}} \left (12+11 x^2\right )}{12 x^2} \, dx=e^{\frac {7}{3}-\frac {1}{x}+\frac {11 x}{12}} \]

[In]

Integrate[(E^((-12 + 28*x + 11*x^2)/(12*x))*(12 + 11*x^2))/(12*x^2),x]

[Out]

E^(7/3 - x^(-1) + (11*x)/12)

Maple [A] (verified)

Time = 0.06 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.63

method	result	size
gosper	\({\mathrm e}^{\frac {11 x^{2}+28 x -12}{12 x}}\)	\(17\)
derivativedivides	\({\mathrm e}^{\frac {11 x^{2}+28 x -12}{12 x}}\)	\(17\)
default	\({\mathrm e}^{\frac {11 x^{2}+28 x -12}{12 x}}\)	\(17\)
norman	\({\mathrm e}^{\frac {11 x^{2}+28 x -12}{12 x}}\)	\(17\)
risch	\({\mathrm e}^{\frac {11 x^{2}+28 x -12}{12 x}}\)	\(17\)
parallelrisch	\({\mathrm e}^{\frac {11 x^{2}+28 x -12}{12 x}}\)	\(17\)

[In]

int(1/12*(11*x^2+12)*exp(1/12*(11*x^2+28*x-12)/x)/x^2,x,method=_RETURNVERBOSE)

[Out]

exp(1/12*(11*x^2+28*x-12)/x)

Fricas [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.59 \[ \int \frac {e^{\frac {-12+28 x+11 x^2}{12 x}} \left (12+11 x^2\right )}{12 x^2} \, dx=e^{\left (\frac {11 \, x^{2} + 28 \, x - 12}{12 \, x}\right )} \]

[In]

integrate(1/12*(11*x^2+12)*exp(1/12*(11*x^2+28*x-12)/x)/x^2,x, algorithm="fricas")

[Out]

e^(1/12*(11*x^2 + 28*x - 12)/x)

Sympy [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.56 \[ \int \frac {e^{\frac {-12+28 x+11 x^2}{12 x}} \left (12+11 x^2\right )}{12 x^2} \, dx=e^{\frac {\frac {11 x^{2}}{12} + \frac {7 x}{3} - 1}{x}} \]

[In]

integrate(1/12*(11*x**2+12)*exp(1/12*(11*x**2+28*x-12)/x)/x**2,x)

[Out]

exp((11*x**2/12 + 7*x/3 - 1)/x)

Maxima [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.41 \[ \int \frac {e^{\frac {-12+28 x+11 x^2}{12 x}} \left (12+11 x^2\right )}{12 x^2} \, dx=e^{\left (\frac {11}{12} \, x - \frac {1}{x} + \frac {7}{3}\right )} \]

[In]

integrate(1/12*(11*x^2+12)*exp(1/12*(11*x^2+28*x-12)/x)/x^2,x, algorithm="maxima")

[Out]

e^(11/12*x - 1/x + 7/3)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.41 \[ \int \frac {e^{\frac {-12+28 x+11 x^2}{12 x}} \left (12+11 x^2\right )}{12 x^2} \, dx=e^{\left (\frac {11}{12} \, x - \frac {1}{x} + \frac {7}{3}\right )} \]

[In]

integrate(1/12*(11*x^2+12)*exp(1/12*(11*x^2+28*x-12)/x)/x^2,x, algorithm="giac")

[Out]

e^(11/12*x - 1/x + 7/3)

Mupad [B] (verification not implemented)

Time = 11.96 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.48 \[ \int \frac {e^{\frac {-12+28 x+11 x^2}{12 x}} \left (12+11 x^2\right )}{12 x^2} \, dx={\mathrm {e}}^{\frac {11\,x}{12}}\,{\mathrm {e}}^{7/3}\,{\mathrm {e}}^{-\frac {1}{x}} \]

[In]

int((exp(((7*x)/3 + (11*x^2)/12 - 1)/x)*(11*x^2 + 12))/(12*x^2),x)

[Out]

exp((11*x)/12)*exp(7/3)*exp(-1/x)