3.54.26 \(\int \frac {1}{3} (60+40 x+e^{3-e-x^2} (-20+40 x^2)) \, dx\)

Optimal. Leaf size=22 \[ \frac {20}{3} x \left (3-e^{3-e-x^2}+x\right ) \]

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Rubi [A]  time = 0.06, antiderivative size = 28, normalized size of antiderivative = 1.27, number of steps used = 7, number of rules used = 4, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {12, 2226, 2205, 2212} \begin {gather*} \frac {20 x^2}{3}-\frac {20}{3} e^{-x^2-e+3} x+20 x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(60 + 40*x + E^(3 - E - x^2)*(-20 + 40*x^2))/3,x]

[Out]

20*x - (20*E^(3 - E - x^2)*x)/3 + (20*x^2)/3

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 2205

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[(F^a*Sqrt[Pi]*Erf[(c + d*x)*Rt[-(b*Log[F]),
2]])/(2*d*Rt[-(b*Log[F]), 2]), x] /; FreeQ[{F, a, b, c, d}, x] && NegQ[b]

Rule 2212

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^(m
- n + 1)*F^(a + b*(c + d*x)^n))/(b*d*n*Log[F]), x] - Dist[(m - n + 1)/(b*n*Log[F]), Int[(c + d*x)^(m - n)*F^(
a + b*(c + d*x)^n), x], x] /; FreeQ[{F, a, b, c, d}, x] && IntegerQ[(2*(m + 1))/n] && LtQ[0, (m + 1)/n, 5] &&
IntegerQ[n] && (LtQ[0, n, m + 1] || LtQ[m, n, 0])

Rule 2226

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*(u_), x_Symbol] :> Int[ExpandLinearProduct[F^(a + b*(c + d*
x)^n), u, c, d, x], x] /; FreeQ[{F, a, b, c, d, n}, x] && PolynomialQ[u, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{3} \int \left (60+40 x+e^{3-e-x^2} \left (-20+40 x^2\right )\right ) \, dx\\ &=20 x+\frac {20 x^2}{3}+\frac {1}{3} \int e^{3-e-x^2} \left (-20+40 x^2\right ) \, dx\\ &=20 x+\frac {20 x^2}{3}+\frac {1}{3} \int \left (-20 e^{3-e-x^2}+40 e^{3-e-x^2} x^2\right ) \, dx\\ &=20 x+\frac {20 x^2}{3}-\frac {20}{3} \int e^{3-e-x^2} \, dx+\frac {40}{3} \int e^{3-e-x^2} x^2 \, dx\\ &=20 x-\frac {20}{3} e^{3-e-x^2} x+\frac {20 x^2}{3}-\frac {10}{3} e^{3-e} \sqrt {\pi } \text {erf}(x)+\frac {20}{3} \int e^{3-e-x^2} \, dx\\ &=20 x-\frac {20}{3} e^{3-e-x^2} x+\frac {20 x^2}{3}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.04, size = 26, normalized size = 1.18 \begin {gather*} \frac {20}{3} \left (3 x-e^{3-e-x^2} x+x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(60 + 40*x + E^(3 - E - x^2)*(-20 + 40*x^2))/3,x]

[Out]

(20*(3*x - E^(3 - E - x^2)*x + x^2))/3

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fricas [A]  time = 0.95, size = 24, normalized size = 1.09 \begin {gather*} \frac {20}{3} \, x^{2} - \frac {20}{3} \, x e^{\left (-x^{2} - e + 3\right )} + 20 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(40*x^2-20)*exp(-exp(1/2)^2-x^2+3)+40/3*x+20,x, algorithm="fricas")

[Out]

20/3*x^2 - 20/3*x*e^(-x^2 - e + 3) + 20*x

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giac [A]  time = 0.16, size = 24, normalized size = 1.09 \begin {gather*} \frac {20}{3} \, x^{2} - \frac {20}{3} \, x e^{\left (-x^{2} - e + 3\right )} + 20 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(40*x^2-20)*exp(-exp(1/2)^2-x^2+3)+40/3*x+20,x, algorithm="giac")

[Out]

20/3*x^2 - 20/3*x*e^(-x^2 - e + 3) + 20*x

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maple [A]  time = 0.03, size = 25, normalized size = 1.14

method	result	size
risch	\(20 x +\frac {20 x^{2}}{3}-\frac {20 x \,{\mathrm e}^{-{\mathrm e}-x^{2}+3}}{3}\)	\(25\)
norman	\(20 x +\frac {20 x^{2}}{3}-\frac {20 x \,{\mathrm e}^{-{\mathrm e}-x^{2}+3}}{3}\)	\(27\)
default	\(20 x +\frac {20 x^{2}}{3}-\frac {10 \,{\mathrm e}^{-{\mathrm e}} {\mathrm e}^{3} \sqrt {\pi }\, \erf \relax (x )}{3}+\frac {40 \,{\mathrm e}^{-{\mathrm e}} {\mathrm e}^{3} \left (-\frac {x \,{\mathrm e}^{-x^{2}}}{2}+\frac {\sqrt {\pi }\, \erf \relax (x )}{4}\right )}{3}\)	\(54\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/3*(40*x^2-20)*exp(-exp(1/2)^2-x^2+3)+40/3*x+20,x,method=_RETURNVERBOSE)

[Out]

20*x+20/3*x^2-20/3*x*exp(-exp(1)-x^2+3)

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maxima [A]  time = 0.52, size = 24, normalized size = 1.09 \begin {gather*} \frac {20}{3} \, x^{2} - \frac {20}{3} \, x e^{\left (-x^{2} - e + 3\right )} + 20 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(40*x^2-20)*exp(-exp(1/2)^2-x^2+3)+40/3*x+20,x, algorithm="maxima")

[Out]

20/3*x^2 - 20/3*x*e^(-x^2 - e + 3) + 20*x

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mupad [B]  time = 0.12, size = 25, normalized size = 1.14 \begin {gather*} 20\,x+\frac {20\,x^2}{3}-\frac {20\,x\,{\mathrm {e}}^{-\mathrm {e}}\,{\mathrm {e}}^3\,{\mathrm {e}}^{-x^2}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((40*x)/3 + (exp(3 - x^2 - exp(1))*(40*x^2 - 20))/3 + 20,x)

[Out]

20*x + (20*x^2)/3 - (20*x*exp(-exp(1))*exp(3)*exp(-x^2))/3

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sympy [A]  time = 0.10, size = 24, normalized size = 1.09 \begin {gather*} \frac {20 x^{2}}{3} - \frac {20 x e^{- x^{2} - e + 3}}{3} + 20 x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(40*x**2-20)*exp(-exp(1/2)**2-x**2+3)+40/3*x+20,x)

[Out]

20*x**2/3 - 20*x*exp(-x**2 - E + 3)/3 + 20*x

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