3.11.37 \(\int (-1+e^{\frac {1}{2} (-3-2 x+3 x^2)} (-1+3 x)) \, dx\)

Optimal. Leaf size=22 \[ e^{-x+x^2+\frac {1}{2} \left (-3+x^2\right )}-x \]

________________________________________________________________________________________

Rubi [A]  time = 0.07, antiderivative size = 20, normalized size of antiderivative = 0.91, number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {2244, 2236} \begin {gather*} e^{\frac {3 x^2}{2}-x-\frac {3}{2}}-x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[-1 + E^((-3 - 2*x + 3*x^2)/2)*(-1 + 3*x),x]

[Out]

E^(-3/2 - x + (3*x^2)/2) - x

Rule 2236

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 2244

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 {gather*} \begin {aligned} \text {integral} &=-x+\int e^{\frac {1}{2} \left (-3-2 x+3 x^2\right )} (-1+3 x) \, dx\\ &=-x+\int e^{-\frac {3}{2}-x+\frac {3 x^2}{2}} (-1+3 x) \, dx\\ &=e^{-\frac {3}{2}-x+\frac {3 x^2}{2}}-x\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.04, size = 20, normalized size = 0.91 \begin {gather*} e^{-\frac {3}{2}-x+\frac {3 x^2}{2}}-x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[-1 + E^((-3 - 2*x + 3*x^2)/2)*(-1 + 3*x),x]

[Out]

E^(-3/2 - x + (3*x^2)/2) - x

________________________________________________________________________________________

fricas [A]  time = 0.96, size = 15, normalized size = 0.68 \begin {gather*} -x + e^{\left (\frac {3}{2} \, x^{2} - x - \frac {3}{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x-1)*exp(3/2*x^2-x-3/2)-1,x, algorithm="fricas")

[Out]

-x + e^(3/2*x^2 - x - 3/2)

________________________________________________________________________________________

giac [A]  time = 0.61, size = 15, normalized size = 0.68 \begin {gather*} -x + e^{\left (\frac {3}{2} \, x^{2} - x - \frac {3}{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x-1)*exp(3/2*x^2-x-3/2)-1,x, algorithm="giac")

[Out]

-x + e^(3/2*x^2 - x - 3/2)

________________________________________________________________________________________

maple [A]  time = 0.01, size = 16, normalized size = 0.73

method	result	size
default	\({\mathrm e}^{\frac {3}{2} x^{2}-x -\frac {3}{2}}-x\)	\(16\)
norman	\({\mathrm e}^{\frac {3}{2} x^{2}-x -\frac {3}{2}}-x\)	\(16\)
risch	\({\mathrm e}^{\frac {3}{2} x^{2}-x -\frac {3}{2}}-x\)	\(16\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*x-1)*exp(3/2*x^2-x-3/2)-1,x,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.55, size = 15, normalized size = 0.68 \begin {gather*} -x + e^{\left (\frac {3}{2} \, x^{2} - x - \frac {3}{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x-1)*exp(3/2*x^2-x-3/2)-1,x, algorithm="maxima")

[Out]

-x + e^(3/2*x^2 - x - 3/2)

________________________________________________________________________________________

mupad [B]  time = 0.10, size = 15, normalized size = 0.68 \begin {gather*} {\mathrm {e}}^{\frac {3\,x^2}{2}-x-\frac {3}{2}}-x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp((3*x^2)/2 - x - 3/2)*(3*x - 1) - 1,x)

[Out]

exp((3*x^2)/2 - x - 3/2) - x

________________________________________________________________________________________

sympy [A]  time = 0.09, size = 14, normalized size = 0.64 \begin {gather*} - x + e^{\frac {3 x^{2}}{2} - x - \frac {3}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x-1)*exp(3/2*x**2-x-3/2)-1,x)

[Out]

-x + exp(3*x**2/2 - x - 3/2)

________________________________________________________________________________________