3.16.93 \(\int (4 x-e^x x+e^{14+e+x} (1+x)) \, dx\)

Optimal. Leaf size=23 \[ e^x (1-x)+x^2+x \left (e^{14+e+x}+x\right ) \]

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Rubi [A]  time = 0.02, antiderivative size = 33, normalized size of antiderivative = 1.43, number of steps used = 5, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2176, 2194} \begin {gather*} 2 x^2-e^x x+e^x-e^{x+e+14}+e^{x+e+14} (x+1) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[4*x - E^x*x + E^(14 + E + x)*(1 + x),x]

[Out]

E^x - E^(14 + E + x) - E^x*x + 2*x^2 + E^(14 + 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]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=2 x^2-\int e^x x \, dx+\int e^{14+e+x} (1+x) \, dx\\ &=-e^x x+2 x^2+e^{14+e+x} (1+x)+\int e^x \, dx-\int e^{14+e+x} \, dx\\ &=e^x-e^{14+e+x}-e^x x+2 x^2+e^{14+e+x} (1+x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 22, normalized size = 0.96 \begin {gather*} -e^x (-1+x)+e^{14+e+x} x+2 x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[4*x - E^x*x + E^(14 + E + x)*(1 + x),x]

[Out]

-(E^x*(-1 + x)) + E^(14 + E + x)*x + 2*x^2

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fricas [A]  time = 0.64, size = 38, normalized size = 1.65 \begin {gather*} {\left (2 \, x^{2} e^{\left (e + 14\right )} + {\left (x e^{\left (e + 14\right )} - x + 1\right )} e^{\left (x + e + 14\right )}\right )} e^{\left (-e - 14\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x+1)*exp(exp(1)+x+14)-exp(x)*x+4*x,x, algorithm="fricas")

[Out]

(2*x^2*e^(e + 14) + (x*e^(e + 14) - x + 1)*e^(x + e + 14))*e^(-e - 14)

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giac [A]  time = 0.25, size = 21, normalized size = 0.91 \begin {gather*} 2 \, x^{2} + x e^{\left (x + e + 14\right )} - {\left (x - 1\right )} e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x+1)*exp(exp(1)+x+14)-exp(x)*x+4*x,x, algorithm="giac")

[Out]

2*x^2 + x*e^(x + e + 14) - (x - 1)*e^x

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maple [A]  time = 0.04, size = 21, normalized size = 0.91

method	result	size
norman	\(\left ({\mathrm e}^{{\mathrm e}} {\mathrm e}^{14}-1\right ) x \,{\mathrm e}^{x}+2 x^{2}+{\mathrm e}^{x}\)	\(21\)
risch	\({\mathrm e}^{{\mathrm e}+x +14} x -\left (x -1\right ) {\mathrm e}^{x}+2 x^{2}\)	\(22\)
default	\({\mathrm e}^{{\mathrm e}+x +14} \left ({\mathrm e}+x +14\right )-14 \,{\mathrm e}^{{\mathrm e}+x +14}-{\mathrm e}^{{\mathrm e}+x +14} {\mathrm e}+2 x^{2}-{\mathrm e}^{x} x +{\mathrm e}^{x}\)	\(44\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x+1)*exp(exp(1)+x+14)-exp(x)*x+4*x,x,method=_RETURNVERBOSE)

[Out]

(exp(exp(1))*exp(14)-1)*x*exp(x)+2*x^2+exp(x)

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maxima [A]  time = 0.79, size = 21, normalized size = 0.91 \begin {gather*} 2 \, x^{2} + x e^{\left (x + e + 14\right )} - {\left (x - 1\right )} e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x+1)*exp(exp(1)+x+14)-exp(x)*x+4*x,x, algorithm="maxima")

[Out]

2*x^2 + x*e^(x + e + 14) - (x - 1)*e^x

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mupad [B]  time = 1.32, size = 19, normalized size = 0.83 \begin {gather*} {\mathrm {e}}^x+2\,x^2+x\,{\mathrm {e}}^x\,\left ({\mathrm {e}}^{\mathrm {e}+14}-1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(4*x - x*exp(x) + exp(x + exp(1) + 14)*(x + 1),x)

[Out]

exp(x) + 2*x^2 + x*exp(x)*(exp(exp(1) + 14) - 1)

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sympy [A]  time = 0.11, size = 20, normalized size = 0.87 \begin {gather*} 2 x^{2} + \left (- x + x e^{14} e^{e} + 1\right ) e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x+1)*exp(exp(1)+x+14)-exp(x)*x+4*x,x)

[Out]

2*x**2 + (-x + x*exp(14)*exp(E) + 1)*exp(x)

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