3.55.91 \(\int \frac {1}{3} e^{\frac {1}{3} (4-20 x+4 x^2)} (-77+12 x+8 x^2) \, dx\)

Optimal. Leaf size=21 \[ e^{-4 \left (-\frac {1}{3} (1-x)^2+x\right )} (4+x) \]

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Rubi [A]  time = 0.25, antiderivative size = 41, normalized size of antiderivative = 1.95, number of steps used = 17, number of rules used = 8, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {12, 6742, 2235, 2234, 2204, 2244, 2240, 2241} \begin {gather*} e^{\frac {4 x^2}{3}-\frac {20 x}{3}+\frac {4}{3}} x+4 e^{\frac {4 x^2}{3}-\frac {20 x}{3}+\frac {4}{3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^((4 - 20*x + 4*x^2)/3)*(-77 + 12*x + 8*x^2))/3,x]

[Out]

4*E^(4/3 - (20*x)/3 + (4*x^2)/3) + E^(4/3 - (20*x)/3 + (4*x^2)/3)*x

Rule 12

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

Rule 2204

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

Rule 2234

Int[(F_)^((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[F^(a - b^2/(4*c)), Int[F^((b + 2*c*x)^2/(4*c))
, x], x] /; FreeQ[{F, a, b, c}, x]

Rule 2235

Int[(F_)^(v_), x_Symbol] :> Int[F^ExpandToSum[v, x], x] /; FreeQ[F, x] && QuadraticQ[v, x] &&  !QuadraticMatch
Q[v, x]

Rule 2240

Int[(F_)^((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)*((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[(e*F^(a + b*x + c*x^2))/(
2*c*Log[F]), x] - Dist[(b*e - 2*c*d)/(2*c), Int[F^(a + b*x + c*x^2), x], x] /; FreeQ[{F, a, b, c, d, e}, x] &&
NeQ[b*e - 2*c*d, 0]

Rule 2241

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

Rule 2244

Int[(F_)^(v_)*(u_)^(m_.), x_Symbol] :> Int[ExpandToSum[u, x]^m*F^ExpandToSum[v, x], x] /; FreeQ[{F, m}, x] &&
LinearQ[u, x] && QuadraticQ[v, x] &&  !(LinearMatchQ[u, x] && QuadraticMatchQ[v, x])

Rule 6742

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{3} \int e^{\frac {1}{3} \left (4-20 x+4 x^2\right )} \left (-77+12 x+8 x^2\right ) \, dx\\ &=\frac {1}{3} \int \left (-77 e^{\frac {1}{3} \left (4-20 x+4 x^2\right )}+12 e^{\frac {1}{3} \left (4-20 x+4 x^2\right )} x+8 e^{\frac {1}{3} \left (4-20 x+4 x^2\right )} x^2\right ) \, dx\\ &=\frac {8}{3} \int e^{\frac {1}{3} \left (4-20 x+4 x^2\right )} x^2 \, dx+4 \int e^{\frac {1}{3} \left (4-20 x+4 x^2\right )} x \, dx-\frac {77}{3} \int e^{\frac {1}{3} \left (4-20 x+4 x^2\right )} \, dx\\ &=\frac {8}{3} \int e^{\frac {4}{3}-\frac {20 x}{3}+\frac {4 x^2}{3}} x^2 \, dx+4 \int e^{\frac {4}{3}-\frac {20 x}{3}+\frac {4 x^2}{3}} x \, dx-\frac {77}{3} \int e^{\frac {4}{3}-\frac {20 x}{3}+\frac {4 x^2}{3}} \, dx\\ &=\frac {3}{2} e^{\frac {4}{3}-\frac {20 x}{3}+\frac {4 x^2}{3}}+e^{\frac {4}{3}-\frac {20 x}{3}+\frac {4 x^2}{3}} x+\frac {20}{3} \int e^{\frac {4}{3}-\frac {20 x}{3}+\frac {4 x^2}{3}} x \, dx+10 \int e^{\frac {4}{3}-\frac {20 x}{3}+\frac {4 x^2}{3}} \, dx-\frac {77 \int e^{\frac {3}{16} \left (-\frac {20}{3}+\frac {8 x}{3}\right )^2} \, dx}{3 e^7}-\int e^{\frac {4}{3}-\frac {20 x}{3}+\frac {4 x^2}{3}} \, dx\\ &=4 e^{\frac {4}{3}-\frac {20 x}{3}+\frac {4 x^2}{3}}+e^{\frac {4}{3}-\frac {20 x}{3}+\frac {4 x^2}{3}} x+\frac {77 \sqrt {\frac {\pi }{3}} \text {erfi}\left (\frac {5-2 x}{\sqrt {3}}\right )}{4 e^7}+\frac {50}{3} \int e^{\frac {4}{3}-\frac {20 x}{3}+\frac {4 x^2}{3}} \, dx-\frac {\int e^{\frac {3}{16} \left (-\frac {20}{3}+\frac {8 x}{3}\right )^2} \, dx}{e^7}+\frac {10 \int e^{\frac {3}{16} \left (-\frac {20}{3}+\frac {8 x}{3}\right )^2} \, dx}{e^7}\\ &=4 e^{\frac {4}{3}-\frac {20 x}{3}+\frac {4 x^2}{3}}+e^{\frac {4}{3}-\frac {20 x}{3}+\frac {4 x^2}{3}} x+\frac {77 \sqrt {\frac {\pi }{3}} \text {erfi}\left (\frac {5-2 x}{\sqrt {3}}\right )}{4 e^7}-\frac {9 \sqrt {3 \pi } \text {erfi}\left (\frac {5-2 x}{\sqrt {3}}\right )}{4 e^7}+\frac {50 \int e^{\frac {3}{16} \left (-\frac {20}{3}+\frac {8 x}{3}\right )^2} \, dx}{3 e^7}\\ &=4 e^{\frac {4}{3}-\frac {20 x}{3}+\frac {4 x^2}{3}}+e^{\frac {4}{3}-\frac {20 x}{3}+\frac {4 x^2}{3}} x\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.06, size = 18, normalized size = 0.86 \begin {gather*} e^{\frac {4}{3} \left (1-5 x+x^2\right )} (4+x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((4 - 20*x + 4*x^2)/3)*(-77 + 12*x + 8*x^2))/3,x]

[Out]

E^((4*(1 - 5*x + x^2))/3)*(4 + x)

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fricas [A]  time = 2.56, size = 15, normalized size = 0.71 \begin {gather*} {\left (x + 4\right )} e^{\left (\frac {4}{3} \, x^{2} - \frac {20}{3} \, x + \frac {4}{3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(8*x^2+12*x-77)/exp(-4/3*x^2+20/3*x-4/3),x, algorithm="fricas")

[Out]

(x + 4)*e^(4/3*x^2 - 20/3*x + 4/3)

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giac [A]  time = 1.73, size = 15, normalized size = 0.71 \begin {gather*} {\left (x + 4\right )} e^{\left (\frac {4}{3} \, x^{2} - \frac {20}{3} \, x + \frac {4}{3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(8*x^2+12*x-77)/exp(-4/3*x^2+20/3*x-4/3),x, algorithm="giac")

[Out]

(x + 4)*e^(4/3*x^2 - 20/3*x + 4/3)

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maple [A]  time = 0.04, size = 18, normalized size = 0.86

method	result	size
gosper	\(\left (4+x \right ) {\mathrm e}^{\frac {4}{3} x^{2}-\frac {20}{3} x +\frac {4}{3}}\)	\(18\)
norman	\(\left (4+x \right ) {\mathrm e}^{\frac {4}{3} x^{2}-\frac {20}{3} x +\frac {4}{3}}\)	\(18\)
risch	\(\frac {\left (3 x +12\right ) {\mathrm e}^{\frac {4}{3} x^{2}-\frac {20}{3} x +\frac {4}{3}}}{3}\)	\(19\)
default	\(4 \,{\mathrm e}^{\frac {4}{3} x^{2}-\frac {20}{3} x +\frac {4}{3}}+x \,{\mathrm e}^{\frac {4}{3} x^{2}-\frac {20}{3} x +\frac {4}{3}}\)	\(28\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/3*(8*x^2+12*x-77)/exp(-4/3*x^2+20/3*x-4/3),x,method=_RETURNVERBOSE)

[Out]

(4+x)/exp(-4/3*x^2+20/3*x-4/3)

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maxima [C]  time = 0.52, size = 196, normalized size = 9.33 \begin {gather*} \frac {77}{12} i \, \sqrt {3} \sqrt {\pi } \operatorname {erf}\left (\frac {2}{3} i \, \sqrt {3} x - \frac {5}{3} i \, \sqrt {3}\right ) e^{\left (-7\right )} - \frac {1}{6} \, \sqrt {3} {\left (\frac {3 \, \sqrt {3} \sqrt {\frac {1}{3}} {\left (2 \, x - 5\right )}^{3} \Gamma \left (\frac {3}{2}, -\frac {1}{3} \, {\left (2 \, x - 5\right )}^{2}\right )}{\left (-{\left (2 \, x - 5\right )}^{2}\right )^{\frac {3}{2}}} - \frac {25 \, \sqrt {3} \sqrt {\frac {1}{3}} \sqrt {\pi } {\left (2 \, x - 5\right )} {\left (\operatorname {erf}\left (\sqrt {\frac {1}{3}} \sqrt {-{\left (2 \, x - 5\right )}^{2}}\right ) - 1\right )}}{\sqrt {-{\left (2 \, x - 5\right )}^{2}}} - 10 \, \sqrt {3} e^{\left (\frac {1}{3} \, {\left (2 \, x - 5\right )}^{2}\right )}\right )} e^{\left (-7\right )} + \frac {1}{2} \, \sqrt {3} {\left (\frac {5 \, \sqrt {3} \sqrt {\frac {1}{3}} \sqrt {\pi } {\left (2 \, x - 5\right )} {\left (\operatorname {erf}\left (\sqrt {\frac {1}{3}} \sqrt {-{\left (2 \, x - 5\right )}^{2}}\right ) - 1\right )}}{\sqrt {-{\left (2 \, x - 5\right )}^{2}}} + \sqrt {3} e^{\left (\frac {1}{3} \, {\left (2 \, x - 5\right )}^{2}\right )}\right )} e^{\left (-7\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(8*x^2+12*x-77)/exp(-4/3*x^2+20/3*x-4/3),x, algorithm="maxima")

[Out]

77/12*I*sqrt(3)*sqrt(pi)*erf(2/3*I*sqrt(3)*x - 5/3*I*sqrt(3))*e^(-7) - 1/6*sqrt(3)*(3*sqrt(3)*sqrt(1/3)*(2*x -
5)^3*gamma(3/2, -1/3*(2*x - 5)^2)/(-(2*x - 5)^2)^(3/2) - 25*sqrt(3)*sqrt(1/3)*sqrt(pi)*(2*x - 5)*(erf(sqrt(1/
3)*sqrt(-(2*x - 5)^2)) - 1)/sqrt(-(2*x - 5)^2) - 10*sqrt(3)*e^(1/3*(2*x - 5)^2))*e^(-7) + 1/2*sqrt(3)*(5*sqrt(
3)*sqrt(1/3)*sqrt(pi)*(2*x - 5)*(erf(sqrt(1/3)*sqrt(-(2*x - 5)^2)) - 1)/sqrt(-(2*x - 5)^2) + sqrt(3)*e^(1/3*(2
*x - 5)^2))*e^(-7)

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mupad [B]  time = 3.41, size = 15, normalized size = 0.71 \begin {gather*} {\mathrm {e}}^{\frac {4\,x^2}{3}-\frac {20\,x}{3}+\frac {4}{3}}\,\left (x+4\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp((4*x^2)/3 - (20*x)/3 + 4/3)*(4*x + (8*x^2)/3 - 77/3),x)

[Out]

exp((4*x^2)/3 - (20*x)/3 + 4/3)*(x + 4)

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sympy [A]  time = 0.10, size = 19, normalized size = 0.90 \begin {gather*} \left (x + 4\right ) e^{\frac {4 x^{2}}{3} - \frac {20 x}{3} + \frac {4}{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(8*x**2+12*x-77)/exp(-4/3*x**2+20/3*x-4/3),x)

[Out]

(x + 4)*exp(4*x**2/3 - 20*x/3 + 4/3)

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