3.13.60 \(\int e^{-x^2} (8-62 x-13 x^2+50 x^3-2 x^4-6 x^5) \, dx\)

Optimal. Leaf size=30 \[ e^{-x^2} \left (-4+3 \left (\frac {4}{x}-x\right )\right ) x \left (1+x-x^2\right ) \]

________________________________________________________________________________________

Rubi [A]  time = 0.20, antiderivative size = 55, normalized size of antiderivative = 1.83, number of steps used = 14, number of rules used = 4, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {2226, 2205, 2209, 2212} \begin {gather*} -19 e^{-x^2} x^2+8 e^{-x^2} x+12 e^{-x^2}+3 e^{-x^2} x^4+e^{-x^2} x^3 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(8 - 62*x - 13*x^2 + 50*x^3 - 2*x^4 - 6*x^5)/E^x^2,x]

[Out]

12/E^x^2 + (8*x)/E^x^2 - (19*x^2)/E^x^2 + x^3/E^x^2 + (3*x^4)/E^x^2

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 2209

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Simp[((e + f*x)^n*
F^(a + b*(c + d*x)^n))/(b*f*n*(c + d*x)^n*Log[F]), x] /; FreeQ[{F, a, b, c, d, e, f, n}, x] && EqQ[m, n - 1] &
& EqQ[d*e - c*f, 0]

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} &=\int \left (8 e^{-x^2}-62 e^{-x^2} x-13 e^{-x^2} x^2+50 e^{-x^2} x^3-2 e^{-x^2} x^4-6 e^{-x^2} x^5\right ) \, dx\\ &=-\left (2 \int e^{-x^2} x^4 \, dx\right )-6 \int e^{-x^2} x^5 \, dx+8 \int e^{-x^2} \, dx-13 \int e^{-x^2} x^2 \, dx+50 \int e^{-x^2} x^3 \, dx-62 \int e^{-x^2} x \, dx\\ &=31 e^{-x^2}+\frac {13}{2} e^{-x^2} x-25 e^{-x^2} x^2+e^{-x^2} x^3+3 e^{-x^2} x^4+4 \sqrt {\pi } \text {erf}(x)-3 \int e^{-x^2} x^2 \, dx-\frac {13}{2} \int e^{-x^2} \, dx-12 \int e^{-x^2} x^3 \, dx+50 \int e^{-x^2} x \, dx\\ &=6 e^{-x^2}+8 e^{-x^2} x-19 e^{-x^2} x^2+e^{-x^2} x^3+3 e^{-x^2} x^4+\frac {3}{4} \sqrt {\pi } \text {erf}(x)-\frac {3}{2} \int e^{-x^2} \, dx-12 \int e^{-x^2} x \, dx\\ &=12 e^{-x^2}+8 e^{-x^2} x-19 e^{-x^2} x^2+e^{-x^2} x^3+3 e^{-x^2} x^4\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.02, size = 26, normalized size = 0.87 \begin {gather*} e^{-x^2} \left (12+8 x-19 x^2+x^3+3 x^4\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(8 - 62*x - 13*x^2 + 50*x^3 - 2*x^4 - 6*x^5)/E^x^2,x]

[Out]

(12 + 8*x - 19*x^2 + x^3 + 3*x^4)/E^x^2

________________________________________________________________________________________

fricas [A]  time = 0.54, size = 25, normalized size = 0.83 \begin {gather*} {\left (3 \, x^{4} + x^{3} - 19 \, x^{2} + 8 \, x + 12\right )} e^{\left (-x^{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-6*x^5-2*x^4+50*x^3-13*x^2-62*x+8)/exp(x^2),x, algorithm="fricas")

[Out]

(3*x^4 + x^3 - 19*x^2 + 8*x + 12)*e^(-x^2)

________________________________________________________________________________________

giac [A]  time = 0.42, size = 25, normalized size = 0.83 \begin {gather*} {\left (3 \, x^{4} + x^{3} - 19 \, x^{2} + 8 \, x + 12\right )} e^{\left (-x^{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-6*x^5-2*x^4+50*x^3-13*x^2-62*x+8)/exp(x^2),x, algorithm="giac")

[Out]

(3*x^4 + x^3 - 19*x^2 + 8*x + 12)*e^(-x^2)

________________________________________________________________________________________

maple [A]  time = 0.04, size = 26, normalized size = 0.87

method	result	size
gosper	\(\left (3 x^{4}+x^{3}-19 x^{2}+8 x +12\right ) {\mathrm e}^{-x^{2}}\)	\(26\)
norman	\(\left (3 x^{4}+x^{3}-19 x^{2}+8 x +12\right ) {\mathrm e}^{-x^{2}}\)	\(26\)
risch	\(\left (3 x^{4}+x^{3}-19 x^{2}+8 x +12\right ) {\mathrm e}^{-x^{2}}\)	\(26\)
default	\(12 \,{\mathrm e}^{-x^{2}}+8 x \,{\mathrm e}^{-x^{2}}-19 \,{\mathrm e}^{-x^{2}} x^{2}+{\mathrm e}^{-x^{2}} x^{3}+3 \,{\mathrm e}^{-x^{2}} x^{4}\)	\(51\)
meijerg	\(-12+\left (3 x^{4}+6 x^{2}+6\right ) {\mathrm e}^{-x^{2}}+\frac {x \left (10 x^{2}+15\right ) {\mathrm e}^{-x^{2}}}{10}-\frac {25 \left (2 x^{2}+2\right ) {\mathrm e}^{-x^{2}}}{2}+\frac {13 x \,{\mathrm e}^{-x^{2}}}{2}+31 \,{\mathrm e}^{-x^{2}}\)	\(70\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-6*x^5-2*x^4+50*x^3-13*x^2-62*x+8)/exp(x^2),x,method=_RETURNVERBOSE)

[Out]

(3*x^4+x^3-19*x^2+8*x+12)/exp(x^2)

________________________________________________________________________________________

maxima [B]  time = 0.44, size = 66, normalized size = 2.20 \begin {gather*} 3 \, {\left (x^{4} + 2 \, x^{2} + 2\right )} e^{\left (-x^{2}\right )} + \frac {1}{2} \, {\left (2 \, x^{3} + 3 \, x\right )} e^{\left (-x^{2}\right )} - 25 \, {\left (x^{2} + 1\right )} e^{\left (-x^{2}\right )} + \frac {13}{2} \, x e^{\left (-x^{2}\right )} + 31 \, e^{\left (-x^{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-6*x^5-2*x^4+50*x^3-13*x^2-62*x+8)/exp(x^2),x, algorithm="maxima")

[Out]

3*(x^4 + 2*x^2 + 2)*e^(-x^2) + 1/2*(2*x^3 + 3*x)*e^(-x^2) - 25*(x^2 + 1)*e^(-x^2) + 13/2*x*e^(-x^2) + 31*e^(-x
^2)

________________________________________________________________________________________

mupad [B]  time = 0.07, size = 25, normalized size = 0.83 \begin {gather*} {\mathrm {e}}^{-x^2}\,\left (3\,x^4+x^3-19\,x^2+8\,x+12\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-exp(-x^2)*(62*x + 13*x^2 - 50*x^3 + 2*x^4 + 6*x^5 - 8),x)

[Out]

exp(-x^2)*(8*x - 19*x^2 + x^3 + 3*x^4 + 12)

________________________________________________________________________________________

sympy [A]  time = 0.10, size = 22, normalized size = 0.73 \begin {gather*} \left (3 x^{4} + x^{3} - 19 x^{2} + 8 x + 12\right ) e^{- x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-6*x**5-2*x**4+50*x**3-13*x**2-62*x+8)/exp(x**2),x)

[Out]

(3*x**4 + x**3 - 19*x**2 + 8*x + 12)*exp(-x**2)

________________________________________________________________________________________