3.87.50 \(\int e^{-x} (-1-20 e^{3 x}+4 e^{5 x}+e^x (8-2 x)+x) \, dx\)

Optimal. Leaf size=27 \[ \left (5-e^{2 x}\right )^2-(-4+x)^2-e^{-x} x \]

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Rubi [A]  time = 0.14, antiderivative size = 30, normalized size of antiderivative = 1.11, number of steps used = 7, number of rules used = 3, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.094, Rules used = {6742, 2194, 2176} \begin {gather*} -(4-x)^2-10 e^{2 x}+e^{4 x}-e^{-x} x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-1 - 20*E^(3*x) + 4*E^(5*x) + E^x*(8 - 2*x) + x)/E^x,x]

[Out]

-10*E^(2*x) + E^(4*x) - (4 - x)^2 - x/E^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 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-e^{-x}-20 e^{2 x}+4 e^{4 x}-2 (-4+x)+e^{-x} x\right ) \, dx\\ &=-(4-x)^2+4 \int e^{4 x} \, dx-20 \int e^{2 x} \, dx-\int e^{-x} \, dx+\int e^{-x} x \, dx\\ &=e^{-x}-10 e^{2 x}+e^{4 x}-(4-x)^2-e^{-x} x+\int e^{-x} \, dx\\ &=-10 e^{2 x}+e^{4 x}-(4-x)^2-e^{-x} x\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.10, size = 29, normalized size = 1.07 \begin {gather*} -10 e^{2 x}+e^{4 x}+8 x-e^{-x} x-x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-1 - 20*E^(3*x) + 4*E^(5*x) + E^x*(8 - 2*x) + x)/E^x,x]

[Out]

-10*E^(2*x) + E^(4*x) + 8*x - x/E^x - x^2

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fricas [A]  time = 0.65, size = 30, normalized size = 1.11 \begin {gather*} -{\left ({\left (x^{2} - 8 \, x\right )} e^{x} + x - e^{\left (5 \, x\right )} + 10 \, e^{\left (3 \, x\right )}\right )} e^{\left (-x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*exp(x)*exp(2*x)^2-20*exp(x)*exp(2*x)+(-2*x+8)*exp(x)+x-1)/exp(x),x, algorithm="fricas")

[Out]

-((x^2 - 8*x)*e^x + x - e^(5*x) + 10*e^(3*x))*e^(-x)

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giac [A]  time = 0.21, size = 26, normalized size = 0.96 \begin {gather*} -x^{2} - x e^{\left (-x\right )} + 8 \, x + e^{\left (4 \, x\right )} - 10 \, e^{\left (2 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*exp(x)*exp(2*x)^2-20*exp(x)*exp(2*x)+(-2*x+8)*exp(x)+x-1)/exp(x),x, algorithm="giac")

[Out]

-x^2 - x*e^(-x) + 8*x + e^(4*x) - 10*e^(2*x)

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

method	result	size
default	\(8 x -x \,{\mathrm e}^{-x}-x^{2}-10 \,{\mathrm e}^{2 x}+{\mathrm e}^{4 x}\)	\(27\)
risch	\(8 x -x \,{\mathrm e}^{-x}-x^{2}-10 \,{\mathrm e}^{2 x}+{\mathrm e}^{4 x}\)	\(27\)
norman	\(\left ({\mathrm e}^{5 x}-x -10 \,{\mathrm e}^{3 x}+8 \,{\mathrm e}^{x} x -{\mathrm e}^{x} x^{2}\right ) {\mathrm e}^{-x}\)	\(32\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*exp(x)*exp(2*x)^2-20*exp(x)*exp(2*x)+(-2*x+8)*exp(x)+x-1)/exp(x),x,method=_RETURNVERBOSE)

[Out]

8*x-x/exp(x)-x^2-10*exp(x)^2+exp(x)^4

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maxima [A]  time = 0.38, size = 32, normalized size = 1.19 \begin {gather*} -x^{2} - {\left (x + 1\right )} e^{\left (-x\right )} + 8 \, x + e^{\left (4 \, x\right )} - 10 \, e^{\left (2 \, x\right )} + e^{\left (-x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*exp(x)*exp(2*x)^2-20*exp(x)*exp(2*x)+(-2*x+8)*exp(x)+x-1)/exp(x),x, algorithm="maxima")

[Out]

-x^2 - (x + 1)*e^(-x) + 8*x + e^(4*x) - 10*e^(2*x) + e^(-x)

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mupad [B]  time = 0.07, size = 26, normalized size = 0.96 \begin {gather*} 8\,x-10\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^{4\,x}-x\,{\mathrm {e}}^{-x}-x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-exp(-x)*(20*exp(3*x) - x - 4*exp(5*x) + exp(x)*(2*x - 8) + 1),x)

[Out]

8*x - 10*exp(2*x) + exp(4*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} + 8 x - x e^{- x} + e^{4 x} - 10 e^{2 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*exp(x)*exp(2*x)**2-20*exp(x)*exp(2*x)+(-2*x+8)*exp(x)+x-1)/exp(x),x)

[Out]

-x**2 + 8*x - x*exp(-x) + exp(4*x) - 10*exp(2*x)

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