3.83.12 \(\int e^{-225 x^2} (-1+e^{2+x+225 x^2}-1800 x+450 x^2) \, dx\) [8212]

Optimal. Leaf size=20 \[ 3+e^{2+x}+e^{-225 x^2} (4-x) \]

[Out]

3+(4-x)/exp(225*x^2)+exp(2+x)

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Rubi [A]
time = 0.10, antiderivative size = 25, normalized size of antiderivative = 1.25, number of steps used = 7, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {6874, 2236, 2225, 2240, 2243} \begin {gather*} -e^{-225 x^2} x+4 e^{-225 x^2}+e^{x+2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-1 + E^(2 + x + 225*x^2) - 1800*x + 450*x^2)/E^(225*x^2),x]

[Out]

4/E^(225*x^2) + E^(2 + x) - x/E^(225*x^2)

Rule 2225

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 2236

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 2240

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 2243

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 6874

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^{-225 x^2}+e^{2+x}-1800 e^{-225 x^2} x+450 e^{-225 x^2} x^2\right ) \, dx\\ &=450 \int e^{-225 x^2} x^2 \, dx-1800 \int e^{-225 x^2} x \, dx-\int e^{-225 x^2} \, dx+\int e^{2+x} \, dx\\ &=4 e^{-225 x^2}+e^{2+x}-e^{-225 x^2} x-\frac {1}{30} \sqrt {\pi } \text {erf}(15 x)+\int e^{-225 x^2} \, dx\\ &=4 e^{-225 x^2}+e^{2+x}-e^{-225 x^2} x\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.42, size = 19, normalized size = 0.95 \begin {gather*} e^{2+x}+e^{-225 x^2} (4-x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-1 + E^(2 + x + 225*x^2) - 1800*x + 450*x^2)/E^(225*x^2),x]

[Out]

E^(2 + x) + (4 - x)/E^(225*x^2)

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Maple [A]
time = 0.16, size = 24, normalized size = 1.20

method	result	size
risch	\({\mathrm e}^{2+x}+\left (-x +4\right ) {\mathrm e}^{-225 x^{2}}\)	\(18\)
default	\({\mathrm e}^{2} {\mathrm e}^{x}+4 \,{\mathrm e}^{-225 x^{2}}-x \,{\mathrm e}^{-225 x^{2}}\)	\(24\)
norman	\(\left (4+{\mathrm e}^{2+x} {\mathrm e}^{225 x^{2}}-x \right ) {\mathrm e}^{-225 x^{2}}\)	\(26\)
meijerg	\(-{\mathrm e}^{2} \left (1-{\mathrm e}^{x}\right )-x \,{\mathrm e}^{-225 x^{2}}-4+4 \,{\mathrm e}^{-225 x^{2}}\)	\(30\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(2+x)*exp(225*x^2)+450*x^2-1800*x-1)/exp(225*x^2),x,method=_RETURNVERBOSE)

[Out]

exp(2)*exp(x)+4/exp(x^2)^225-1/exp(x^2)^225*x

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Maxima [A]
time = 0.49, size = 22, normalized size = 1.10 \begin {gather*} -x e^{\left (-225 \, x^{2}\right )} + 4 \, e^{\left (-225 \, x^{2}\right )} + e^{\left (x + 2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(2+x)*exp(225*x^2)+450*x^2-1800*x-1)/exp(225*x^2),x, algorithm="maxima")

[Out]

-x*e^(-225*x^2) + 4*e^(-225*x^2) + e^(x + 2)

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Fricas [A]
time = 0.43, size = 16, normalized size = 0.80 \begin {gather*} -{\left (x - 4\right )} e^{\left (-225 \, x^{2}\right )} + e^{\left (x + 2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(2+x)*exp(225*x^2)+450*x^2-1800*x-1)/exp(225*x^2),x, algorithm="fricas")

[Out]

-(x - 4)*e^(-225*x^2) + e^(x + 2)

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Sympy [A]
time = 0.08, size = 14, normalized size = 0.70 \begin {gather*} \left (4 - x\right ) e^{- 225 x^{2}} + e^{x + 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(2+x)*exp(225*x**2)+450*x**2-1800*x-1)/exp(225*x**2),x)

[Out]

(4 - x)*exp(-225*x**2) + exp(x + 2)

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Giac [A]
time = 0.42, size = 16, normalized size = 0.80 \begin {gather*} -{\left (x - 4\right )} e^{\left (-225 \, x^{2}\right )} + e^{\left (x + 2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(2+x)*exp(225*x^2)+450*x^2-1800*x-1)/exp(225*x^2),x, algorithm="giac")

[Out]

-(x - 4)*e^(-225*x^2) + e^(x + 2)

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Mupad [B]
time = 5.30, size = 22, normalized size = 1.10 \begin {gather*} {\mathrm {e}}^{x+2}+4\,{\mathrm {e}}^{-225\,x^2}-x\,{\mathrm {e}}^{-225\,x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-exp(-225*x^2)*(1800*x - exp(x + 2)*exp(225*x^2) - 450*x^2 + 1),x)

[Out]

exp(x + 2) + 4*exp(-225*x^2) - x*exp(-225*x^2)

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