3.89.91 \(\int (-1+e^{5+e (-1+x)+2 x} (2+e)+e^x (-1-x)-2 x) \, dx\)

Optimal. Leaf size=27 \[ e^{5-e (1-x)+2 x}-x-x \left (e^x+x\right ) \]

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Rubi [A]  time = 0.03, antiderivative size = 32, normalized size of antiderivative = 1.19, number of steps used = 5, number of rules used = 3, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2227, 2194, 2176} \begin {gather*} -x^2-x+e^x+e^{(2+e) x-e+5}-e^x (x+1) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[-1 + E^(5 + E*(-1 + x) + 2*x)*(2 + E) + E^x*(-1 - x) - 2*x,x]

[Out]

E^x + E^(5 - E + (2 + E)*x) - x - x^2 - E^x*(1 + x)

Rule 2176

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^m
*(b*F^(g*(e + f*x)))^n)/(f*g*n*Log[F]), x] - Dist[(d*m)/(f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*(b*F^(g*(e + f*x
)))^n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && GtQ[m, 0] && IntegerQ[2*m] &&  !$UseGamma === True

Rule 2194

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rule 2227

Int[(u_.)*(F_)^((a_.) + (b_.)*(v_)), x_Symbol] :> Int[u*F^(a + b*NormalizePowerOfLinear[v, x]), x] /; FreeQ[{F
, a, b}, x] && PolynomialQ[u, x] && PowerOfLinearQ[v, x] &&  !PowerOfLinearMatchQ[v, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=-x-x^2+(2+e) \int e^{5+e (-1+x)+2 x} \, dx+\int e^x (-1-x) \, dx\\ &=-x-x^2-e^x (1+x)+(2+e) \int e^{5-e+(2+e) x} \, dx+\int e^x \, dx\\ &=e^x+e^{5-e+(2+e) x}-x-x^2-e^x (1+x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.03, size = 25, normalized size = 0.93 \begin {gather*} e^{5+e (-1+x)+2 x}-e^x x-x (1+x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[-1 + E^(5 + E*(-1 + x) + 2*x)*(2 + E) + E^x*(-1 - x) - 2*x,x]

[Out]

E^(5 + E*(-1 + x) + 2*x) - E^x*x - x*(1 + x)

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fricas [A]  time = 0.45, size = 26, normalized size = 0.96 \begin {gather*} -x^{2} - x e^{x} - x + e^{\left ({\left (x - 1\right )} e + 2 \, x + 5\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(1)+2)*exp((x-1)*exp(1)+5+2*x)+(-x-1)*exp(x)-2*x-1,x, algorithm="fricas")

[Out]

-x^2 - x*e^x - x + e^((x - 1)*e + 2*x + 5)

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giac [A]  time = 0.14, size = 26, normalized size = 0.96 \begin {gather*} -x^{2} - x e^{x} - x + e^{\left ({\left (x - 1\right )} e + 2 \, x + 5\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(1)+2)*exp((x-1)*exp(1)+5+2*x)+(-x-1)*exp(x)-2*x-1,x, algorithm="giac")

[Out]

-x^2 - x*e^x - x + e^((x - 1)*e + 2*x + 5)

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maple [A]  time = 0.04, size = 27, normalized size = 1.00

method	result	size
default	\(-{\mathrm e}^{x} x -x +{\mathrm e}^{\left (x -1\right ) {\mathrm e}+5+2 x}-x^{2}\)	\(27\)
norman	\(-{\mathrm e}^{x} x -x +{\mathrm e}^{\left (x -1\right ) {\mathrm e}+5+2 x}-x^{2}\)	\(27\)
risch	\(-{\mathrm e}^{x} x -x +{\mathrm e}^{x \,{\mathrm e}-{\mathrm e}+2 x +5}-x^{2}\)	\(29\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(1)+2)*exp((x-1)*exp(1)+5+2*x)+(-x-1)*exp(x)-2*x-1,x,method=_RETURNVERBOSE)

[Out]

-exp(x)*x-x+exp((x-1)*exp(1)+5+2*x)-x^2

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maxima [A]  time = 0.36, size = 26, normalized size = 0.96 \begin {gather*} -x^{2} - x e^{x} - x + e^{\left ({\left (x - 1\right )} e + 2 \, x + 5\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(1)+2)*exp((x-1)*exp(1)+5+2*x)+(-x-1)*exp(x)-2*x-1,x, algorithm="maxima")

[Out]

-x^2 - x*e^x - x + e^((x - 1)*e + 2*x + 5)

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mupad [B]  time = 0.11, size = 28, normalized size = 1.04 \begin {gather*} {\mathrm {e}}^{2\,x-\mathrm {e}+x\,\mathrm {e}+5}-x-x\,{\mathrm {e}}^x-x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(2*x + exp(1)*(x - 1) + 5)*(exp(1) + 2) - exp(x)*(x + 1) - 2*x - 1,x)

[Out]

exp(2*x - exp(1) + x*exp(1) + 5) - x - x*exp(x) - x^2

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sympy [A]  time = 0.13, size = 22, normalized size = 0.81 \begin {gather*} - x^{2} - x e^{x} - x + e^{2 x + e \left (x - 1\right ) + 5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(1)+2)*exp((x-1)*exp(1)+5+2*x)+(-x-1)*exp(x)-2*x-1,x)

[Out]

-x**2 - x*exp(x) - x + exp(2*x + E*(x - 1) + 5)

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