3.71.3 \(\int e^{-26-x} (-4 e^x+e^{26+x} (2 x-3 x^2)) \, dx\)

Optimal. Leaf size=16 \[ x^2-x \left (\frac {4}{e^{26}}+x^2\right ) \]

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Rubi [A]  time = 0.04, antiderivative size = 15, normalized size of antiderivative = 0.94, number of steps used = 4, number of rules used = 2, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {6688, 43} \begin {gather*} -x^3+x^2-\frac {4 x}{e^{26}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^(-26 - x)*(-4*E^x + E^(26 + x)*(2*x - 3*x^2)),x]

[Out]

(-4*x)/E^26 + x^2 - x^3

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 6688

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-\frac {4}{e^{26}}+(2-3 x) x\right ) \, dx\\ &=-\frac {4 x}{e^{26}}+\int (2-3 x) x \, dx\\ &=-\frac {4 x}{e^{26}}+\int \left (2 x-3 x^2\right ) \, dx\\ &=-\frac {4 x}{e^{26}}+x^2-x^3\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 15, normalized size = 0.94 \begin {gather*} -\frac {4 x}{e^{26}}+x^2-x^3 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^(-26 - x)*(-4*E^x + E^(26 + x)*(2*x - 3*x^2)),x]

[Out]

(-4*x)/E^26 + x^2 - x^3

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fricas [A]  time = 0.64, size = 20, normalized size = 1.25 \begin {gather*} -{\left ({\left (x^{3} - x^{2}\right )} e^{26} + 4 \, x\right )} e^{\left (-26\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3*x^2+2*x)*exp(x+26)-4*exp(x))/exp(x+26),x, algorithm="fricas")

[Out]

-((x^3 - x^2)*e^26 + 4*x)*e^(-26)

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giac [A]  time = 0.12, size = 21, normalized size = 1.31 \begin {gather*} -{\left (x^{3} e^{26} - x^{2} e^{26} + 4 \, x\right )} e^{\left (-26\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3*x^2+2*x)*exp(x+26)-4*exp(x))/exp(x+26),x, algorithm="giac")

[Out]

-(x^3*e^26 - x^2*e^26 + 4*x)*e^(-26)

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maple [A]  time = 0.05, size = 15, normalized size = 0.94

method	result	size
risch	\(-x^{3}+x^{2}-4 x \,{\mathrm e}^{-26}\)	\(15\)
default	\(-x^{3}+x^{2}-4 x \,{\mathrm e}^{-26}\)	\(17\)
norman	\(\left ({\mathrm e}^{x} x^{2}-{\mathrm e}^{x} x^{3}-4 \,{\mathrm e}^{-26} x \,{\mathrm e}^{x}\right ) {\mathrm e}^{-x}\)	\(29\)
meijerg	\(-\frac {3 \,{\mathrm e}^{x \,{\mathrm e}^{-26}-x +78} \left (2-\frac {\left (3 x^{2} {\mathrm e}^{-52} \left (-{\mathrm e}^{26}+1\right )^{2}+6 x \,{\mathrm e}^{-26} \left (-{\mathrm e}^{26}+1\right )+6\right ) {\mathrm e}^{-x \,{\mathrm e}^{-26} \left (-{\mathrm e}^{26}+1\right )}}{3}\right )}{\left (-{\mathrm e}^{26}+1\right )^{3}}+\frac {2 \,{\mathrm e}^{x \,{\mathrm e}^{-26}-x +52} \left (1-\frac {\left (2+2 x \,{\mathrm e}^{-26} \left (-{\mathrm e}^{26}+1\right )\right ) {\mathrm e}^{-x \,{\mathrm e}^{-26} \left (-{\mathrm e}^{26}+1\right )}}{2}\right )}{\left (-{\mathrm e}^{26}+1\right )^{2}}-\frac {4 \,{\mathrm e}^{x \,{\mathrm e}^{-26}-x} \left (1-{\mathrm e}^{-x \,{\mathrm e}^{-26} \left (-{\mathrm e}^{26}+1\right )}\right )}{-{\mathrm e}^{26}+1}\)	\(150\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-3*x^2+2*x)*exp(x+26)-4*exp(x))/exp(x+26),x,method=_RETURNVERBOSE)

[Out]

-x^3+x^2-4*x*exp(-26)

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maxima [A]  time = 0.39, size = 14, normalized size = 0.88 \begin {gather*} -x^{3} + x^{2} - 4 \, x e^{\left (-26\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3*x^2+2*x)*exp(x+26)-4*exp(x))/exp(x+26),x, algorithm="maxima")

[Out]

-x^3 + x^2 - 4*x*e^(-26)

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mupad [B]  time = 0.06, size = 14, normalized size = 0.88 \begin {gather*} -x\,\left (x^2-x+4\,{\mathrm {e}}^{-26}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-exp(- x - 26)*(4*exp(x) - exp(x + 26)*(2*x - 3*x^2)),x)

[Out]

-x*(4*exp(-26) - x + x^2)

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sympy [A]  time = 0.06, size = 12, normalized size = 0.75 \begin {gather*} - x^{3} + x^{2} - \frac {4 x}{e^{26}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3*x**2+2*x)*exp(x+26)-4*exp(x))/exp(x+26),x)

[Out]

-x**3 + x**2 - 4*x*exp(-26)

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