3.81.36 \(\int (162+e^x (-2-2 x)-4 x) \, dx\)

Optimal. Leaf size=11 \[ -24-2 x \left (-81+e^x+x\right ) \]

________________________________________________________________________________________

Rubi [A]  time = 0.01, antiderivative size = 22, normalized size of antiderivative = 2.00, number of steps used = 3, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2176, 2194} \begin {gather*} -2 x^2+162 x+2 e^x-2 e^x (x+1) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[162 + E^x*(-2 - 2*x) - 4*x,x]

[Out]

2*E^x + 162*x - 2*x^2 - 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]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=162 x-2 x^2+\int e^x (-2-2 x) \, dx\\ &=162 x-2 x^2-2 e^x (1+x)+2 \int e^x \, dx\\ &=2 e^x+162 x-2 x^2-2 e^x (1+x)\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.00, size = 14, normalized size = 1.27 \begin {gather*} -2 \left (-81 x+e^x x+x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[162 + E^x*(-2 - 2*x) - 4*x,x]

[Out]

-2*(-81*x + E^x*x + x^2)

________________________________________________________________________________________

fricas [A]  time = 0.66, size = 14, normalized size = 1.27 \begin {gather*} -2 \, x^{2} - 2 \, x e^{x} + 162 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x-2)*exp(x)-4*x+162,x, algorithm="fricas")

[Out]

-2*x^2 - 2*x*e^x + 162*x

________________________________________________________________________________________

giac [A]  time = 0.19, size = 14, normalized size = 1.27 \begin {gather*} -2 \, x^{2} - 2 \, x e^{x} + 162 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x-2)*exp(x)-4*x+162,x, algorithm="giac")

[Out]

-2*x^2 - 2*x*e^x + 162*x

________________________________________________________________________________________

maple [A]  time = 0.02, size = 15, normalized size = 1.36

method	result	size
default	\(162 x -2 \,{\mathrm e}^{x} x -2 x^{2}\)	\(15\)
norman	\(162 x -2 \,{\mathrm e}^{x} x -2 x^{2}\)	\(15\)
risch	\(162 x -2 \,{\mathrm e}^{x} x -2 x^{2}\)	\(15\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-2*x-2)*exp(x)-4*x+162,x,method=_RETURNVERBOSE)

[Out]

162*x-2*exp(x)*x-2*x^2

________________________________________________________________________________________

maxima [A]  time = 0.35, size = 14, normalized size = 1.27 \begin {gather*} -2 \, x^{2} - 2 \, x e^{x} + 162 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x-2)*exp(x)-4*x+162,x, algorithm="maxima")

[Out]

-2*x^2 - 2*x*e^x + 162*x

________________________________________________________________________________________

mupad [B]  time = 0.05, size = 8, normalized size = 0.73 \begin {gather*} -2\,x\,\left (x+{\mathrm {e}}^x-81\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(162 - exp(x)*(2*x + 2) - 4*x,x)

[Out]

-2*x*(x + exp(x) - 81)

________________________________________________________________________________________

sympy [A]  time = 0.08, size = 14, normalized size = 1.27 \begin {gather*} - 2 x^{2} - 2 x e^{x} + 162 x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x-2)*exp(x)-4*x+162,x)

[Out]

-2*x**2 - 2*x*exp(x) + 162*x

________________________________________________________________________________________