\(\int \frac {1}{3} e^{\frac {1}{3} (12-43 x+12 x^2)} (-172+96 x) \, dx\) [4374]

Optimal result

Integrand size = 25, antiderivative size = 33 \[ \int \frac {1}{3} e^{\frac {1}{3} \left (12-43 x+12 x^2\right )} (-172+96 x) \, dx=4 e^{-x+\left (\frac {4}{3}-4 x\right ) \left (-2 x+\frac {-x+x (4+x)}{x}\right )} \]

[Out]

4/exp(x-(((4+x)*x-x)/x-2*x)*(4/3-4*x))

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.48, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {12, 2276, 2268} \[ \int \frac {1}{3} e^{\frac {1}{3} \left (12-43 x+12 x^2\right )} (-172+96 x) \, dx=4 e^{4 x^2-\frac {43 x}{3}+4} \]

[In]

Int[(E^((12 - 43*x + 12*x^2)/3)*(-172 + 96*x))/3,x]

[Out]

4*E^(4 - (43*x)/3 + 4*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 2268

Int[(F_)^((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)*((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[e*(F^(a + b*x + c*x^2)/(2
*c*Log[F])), x] /; FreeQ[{F, a, b, c, d, e}, x] && EqQ[b*e - 2*c*d, 0]

Rule 2276

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

Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \int e^{\frac {1}{3} \left (12-43 x+12 x^2\right )} (-172+96 x) \, dx \\ & = \frac {1}{3} \int e^{4-\frac {43 x}{3}+4 x^2} (-172+96 x) \, dx \\ & = 4 e^{4-\frac {43 x}{3}+4 x^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.48 \[ \int \frac {1}{3} e^{\frac {1}{3} \left (12-43 x+12 x^2\right )} (-172+96 x) \, dx=4 e^{4-\frac {43 x}{3}+4 x^2} \]

[In]

Integrate[(E^((12 - 43*x + 12*x^2)/3)*(-172 + 96*x))/3,x]

[Out]

4*E^(4 - (43*x)/3 + 4*x^2)

Maple [A] (verified)

Time = 0.02 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.42

method	result	size
risch	\(4 \,{\mathrm e}^{4 x^{2}-\frac {43}{3} x +4}\)	\(14\)
gosper	\(4 \,{\mathrm e}^{4 x^{2}-\frac {43}{3} x +4}\)	\(16\)
norman	\(4 \,{\mathrm e}^{4 x^{2}-\frac {43}{3} x +4}\)	\(16\)
parallelrisch	\(4 \,{\mathrm e}^{4 x^{2}-\frac {43}{3} x +4}\)	\(16\)
default	\(\frac {43 i {\mathrm e}^{4} \sqrt {\pi }\, {\mathrm e}^{-\frac {1849}{144}} \operatorname {erf}\left (2 i x -\frac {43}{12} i\right )}{3}+32 \,{\mathrm e}^{4} \left (\frac {{\mathrm e}^{4 x^{2}-\frac {43}{3} x}}{8}-\frac {43 i \sqrt {\pi }\, {\mathrm e}^{-\frac {1849}{144}} \operatorname {erf}\left (2 i x -\frac {43}{12} i\right )}{96}\right )\)	\(57\)

[In]

int(1/3*(96*x-172)/exp(-4*x^2+43/3*x-4),x,method=_RETURNVERBOSE)

[Out]

4*exp(4*x^2-43/3*x+4)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.39 \[ \int \frac {1}{3} e^{\frac {1}{3} \left (12-43 x+12 x^2\right )} (-172+96 x) \, dx=4 \, e^{\left (4 \, x^{2} - \frac {43}{3} \, x + 4\right )} \]

[In]

integrate(1/3*(96*x-172)/exp(-4*x^2+43/3*x-4),x, algorithm="fricas")

[Out]

4*e^(4*x^2 - 43/3*x + 4)

Sympy [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.42 \[ \int \frac {1}{3} e^{\frac {1}{3} \left (12-43 x+12 x^2\right )} (-172+96 x) \, dx=4 e^{4 x^{2} - \frac {43 x}{3} + 4} \]

[In]

integrate(1/3*(96*x-172)/exp(-4*x**2+43/3*x-4),x)

[Out]

4*exp(4*x**2 - 43*x/3 + 4)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.39 \[ \int \frac {1}{3} e^{\frac {1}{3} \left (12-43 x+12 x^2\right )} (-172+96 x) \, dx=4 \, e^{\left (4 \, x^{2} - \frac {43}{3} \, x + 4\right )} \]

[In]

integrate(1/3*(96*x-172)/exp(-4*x^2+43/3*x-4),x, algorithm="maxima")

[Out]

4*e^(4*x^2 - 43/3*x + 4)

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.39 \[ \int \frac {1}{3} e^{\frac {1}{3} \left (12-43 x+12 x^2\right )} (-172+96 x) \, dx=4 \, e^{\left (4 \, x^{2} - \frac {43}{3} \, x + 4\right )} \]

[In]

integrate(1/3*(96*x-172)/exp(-4*x^2+43/3*x-4),x, algorithm="giac")

[Out]

4*e^(4*x^2 - 43/3*x + 4)

Mupad [B] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.42 \[ \int \frac {1}{3} e^{\frac {1}{3} \left (12-43 x+12 x^2\right )} (-172+96 x) \, dx=4\,{\mathrm {e}}^{-\frac {43\,x}{3}}\,{\mathrm {e}}^4\,{\mathrm {e}}^{4\,x^2} \]

[In]

int(exp(4*x^2 - (43*x)/3 + 4)*(32*x - 172/3),x)

[Out]

4*exp(-(43*x)/3)*exp(4)*exp(4*x^2)