3.88.77 \(\int (-180-40 e^{2+2 x}-40 x+e^{1+x} (240+40 x)) \, dx\)

Optimal. Leaf size=22 \[ 20 \left (25+e^5+x-\left (5-e^{1+x}+x\right )^2\right ) \]

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Rubi [A]  time = 0.02, antiderivative size = 35, normalized size of antiderivative = 1.59, number of steps used = 4, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {2194, 2176} \begin {gather*} -20 x^2-180 x-40 e^{x+1}-20 e^{2 x+2}+40 e^{x+1} (x+6) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[-180 - 40*E^(2 + 2*x) - 40*x + E^(1 + x)*(240 + 40*x),x]

[Out]

-40*E^(1 + x) - 20*E^(2 + 2*x) - 180*x - 20*x^2 + 40*E^(1 + x)*(6 + 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} &=-180 x-20 x^2-40 \int e^{2+2 x} \, dx+\int e^{1+x} (240+40 x) \, dx\\ &=-20 e^{2+2 x}-180 x-20 x^2+40 e^{1+x} (6+x)-40 \int e^{1+x} \, dx\\ &=-40 e^{1+x}-20 e^{2+2 x}-180 x-20 x^2+40 e^{1+x} (6+x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 32, normalized size = 1.45 \begin {gather*} 20 \left (-e^{2+2 x}-9 x-x^2+2 e^x (5 e+e x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[-180 - 40*E^(2 + 2*x) - 40*x + E^(1 + x)*(240 + 40*x),x]

[Out]

20*(-E^(2 + 2*x) - 9*x - x^2 + 2*E^x*(5*E + E*x))

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fricas [A]  time = 0.55, size = 26, normalized size = 1.18 \begin {gather*} -20 \, x^{2} + 40 \, {\left (x + 5\right )} e^{\left (x + 1\right )} - 180 \, x - 20 \, e^{\left (2 \, x + 2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-40*exp(x+1)^2+(40*x+240)*exp(x+1)-40*x-180,x, algorithm="fricas")

[Out]

-20*x^2 + 40*(x + 5)*e^(x + 1) - 180*x - 20*e^(2*x + 2)

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giac [A]  time = 0.19, size = 26, normalized size = 1.18 \begin {gather*} -20 \, x^{2} + 40 \, {\left (x + 5\right )} e^{\left (x + 1\right )} - 180 \, x - 20 \, e^{\left (2 \, x + 2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-40*exp(x+1)^2+(40*x+240)*exp(x+1)-40*x-180,x, algorithm="giac")

[Out]

-20*x^2 + 40*(x + 5)*e^(x + 1) - 180*x - 20*e^(2*x + 2)

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maple [A]  time = 0.06, size = 28, normalized size = 1.27

method	result	size
risch	\(-20 \,{\mathrm e}^{2 x +2}+\left (200+40 x \right ) {\mathrm e}^{x +1}-20 x^{2}-180 x\)	\(28\)
norman	\(-180 x -20 x^{2}-20 \,{\mathrm e}^{2 x +2}+40 x \,{\mathrm e}^{x +1}+200 \,{\mathrm e}^{x +1}\)	\(31\)
default	\(-180 x +40 \,{\mathrm e}^{x +1} \left (x +1\right )+160 \,{\mathrm e}^{x +1}-20 x^{2}-20 \,{\mathrm e}^{2 x +2}\)	\(33\)
derivativedivides	\(-140 x -140+40 \,{\mathrm e}^{x +1} \left (x +1\right )+160 \,{\mathrm e}^{x +1}-20 \left (x +1\right )^{2}-20 \,{\mathrm e}^{2 x +2}\)	\(36\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-40*exp(x+1)^2+(40*x+240)*exp(x+1)-40*x-180,x,method=_RETURNVERBOSE)

[Out]

-20*exp(2*x+2)+(200+40*x)*exp(x+1)-20*x^2-180*x

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maxima [A]  time = 0.35, size = 30, normalized size = 1.36 \begin {gather*} -20 \, x^{2} + 40 \, {\left (x e + 5 \, e\right )} e^{x} - 180 \, x - 20 \, e^{\left (2 \, x + 2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-40*exp(x+1)^2+(40*x+240)*exp(x+1)-40*x-180,x, algorithm="maxima")

[Out]

-20*x^2 + 40*(x*e + 5*e)*e^x - 180*x - 20*e^(2*x + 2)

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mupad [B]  time = 5.35, size = 30, normalized size = 1.36 \begin {gather*} 200\,{\mathrm {e}}^{x+1}-180\,x-20\,{\mathrm {e}}^{2\,x+2}+40\,x\,{\mathrm {e}}^{x+1}-20\,x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(x + 1)*(40*x + 240) - 40*exp(2*x + 2) - 40*x - 180,x)

[Out]

200*exp(x + 1) - 180*x - 20*exp(2*x + 2) + 40*x*exp(x + 1) - 20*x^2

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sympy [A]  time = 0.09, size = 26, normalized size = 1.18 \begin {gather*} - 20 x^{2} - 180 x + \left (40 x + 200\right ) e^{x + 1} - 20 e^{2 x + 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-40*exp(x+1)**2+(40*x+240)*exp(x+1)-40*x-180,x)

[Out]

-20*x**2 - 180*x + (40*x + 200)*exp(x + 1) - 20*exp(2*x + 2)

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