3.52.64 \(\int \frac {6561+e^{4-24 x+\frac {x^2}{13122}} (6561-157464 x+x^2)}{6561} \, dx\)

Optimal. Leaf size=22 \[ 5+x+e^{4 (1-6 x)+\frac {x^2}{13122}} x \]

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Rubi [A]  time = 0.03, antiderivative size = 33, normalized size of antiderivative = 1.50, number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {12, 2288} \begin {gather*} \frac {e^{\frac {x^2}{13122}-24 x+4} \left (157464 x-x^2\right )}{157464-x}+x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(6561 + E^(4 - 24*x + x^2/13122)*(6561 - 157464*x + x^2))/6561,x]

[Out]

x + (E^(4 - 24*x + x^2/13122)*(157464*x - x^2))/(157464 - 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 2288

Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = (v*y)/(Log[F]*D[u, x])}, Simp[F^u*z, x] /; EqQ[D[z,
x], w*y]] /; FreeQ[F, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int \left (6561+e^{4-24 x+\frac {x^2}{13122}} \left (6561-157464 x+x^2\right )\right ) \, dx}{6561}\\ &=x+\frac {\int e^{4-24 x+\frac {x^2}{13122}} \left (6561-157464 x+x^2\right ) \, dx}{6561}\\ &=x+\frac {e^{4-24 x+\frac {x^2}{13122}} \left (157464 x-x^2\right )}{157464-x}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.03, size = 18, normalized size = 0.82 \begin {gather*} \left (1+e^{4-24 x+\frac {x^2}{13122}}\right ) x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(6561 + E^(4 - 24*x + x^2/13122)*(6561 - 157464*x + x^2))/6561,x]

[Out]

(1 + E^(4 - 24*x + x^2/13122))*x

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fricas [A]  time = 0.66, size = 15, normalized size = 0.68 \begin {gather*} x e^{\left (\frac {1}{13122} \, x^{2} - 24 \, x + 4\right )} + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/6561*(x^2-157464*x+6561)*exp(-24*x+4)*exp(1/13122*x^2)+1,x, algorithm="fricas")

[Out]

x*e^(1/13122*x^2 - 24*x + 4) + x

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giac [A]  time = 0.21, size = 15, normalized size = 0.68 \begin {gather*} x e^{\left (\frac {1}{13122} \, x^{2} - 24 \, x + 4\right )} + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/6561*(x^2-157464*x+6561)*exp(-24*x+4)*exp(1/13122*x^2)+1,x, algorithm="giac")

[Out]

x*e^(1/13122*x^2 - 24*x + 4) + x

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maple [A]  time = 0.04, size = 16, normalized size = 0.73

method	result	size
default	\(x +x \,{\mathrm e}^{-24 x +4+\frac {1}{13122} x^{2}}\)	\(16\)
risch	\(x +x \,{\mathrm e}^{-24 x +4+\frac {1}{13122} x^{2}}\)	\(16\)
norman	\(x +{\mathrm e}^{\frac {x^{2}}{13122}} {\mathrm e}^{-24 x +4} x\)	\(17\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/6561*(x^2-157464*x+6561)*exp(-24*x+4)*exp(1/13122*x^2)+1,x,method=_RETURNVERBOSE)

[Out]

x+x*exp(-24*x+4+1/13122*x^2)

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maxima [C]  time = 0.50, size = 158, normalized size = 7.18 \begin {gather*} -\frac {81}{2} i \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (\frac {1}{162} i \, \sqrt {2} x - 972 i \, \sqrt {2}\right ) e^{\left (-1889564\right )} - 81 \, \sqrt {2} {\left (\frac {{\left (x - 157464\right )}^{3} \Gamma \left (\frac {3}{2}, -\frac {1}{13122} \, {\left (x - 157464\right )}^{2}\right )}{\left (-{\left (x - 157464\right )}^{2}\right )^{\frac {3}{2}}} - \frac {1889568 \, \sqrt {\pi } {\left (x - 157464\right )} {\left (\operatorname {erf}\left (\frac {1}{81} \, \sqrt {\frac {1}{2}} \sqrt {-{\left (x - 157464\right )}^{2}}\right ) - 1\right )}}{\sqrt {-{\left (x - 157464\right )}^{2}}} - 1944 \, \sqrt {2} e^{\left (\frac {1}{13122} \, {\left (x - 157464\right )}^{2}\right )}\right )} e^{\left (-1889564\right )} - 78732 \, \sqrt {2} {\left (\frac {1944 \, \sqrt {\pi } {\left (x - 157464\right )} {\left (\operatorname {erf}\left (\frac {1}{81} \, \sqrt {\frac {1}{2}} \sqrt {-{\left (x - 157464\right )}^{2}}\right ) - 1\right )}}{\sqrt {-{\left (x - 157464\right )}^{2}}} + \sqrt {2} e^{\left (\frac {1}{13122} \, {\left (x - 157464\right )}^{2}\right )}\right )} e^{\left (-1889564\right )} + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/6561*(x^2-157464*x+6561)*exp(-24*x+4)*exp(1/13122*x^2)+1,x, algorithm="maxima")

[Out]

-81/2*I*sqrt(2)*sqrt(pi)*erf(1/162*I*sqrt(2)*x - 972*I*sqrt(2))*e^(-1889564) - 81*sqrt(2)*((x - 157464)^3*gamm
a(3/2, -1/13122*(x - 157464)^2)/(-(x - 157464)^2)^(3/2) - 1889568*sqrt(pi)*(x - 157464)*(erf(1/81*sqrt(1/2)*sq
rt(-(x - 157464)^2)) - 1)/sqrt(-(x - 157464)^2) - 1944*sqrt(2)*e^(1/13122*(x - 157464)^2))*e^(-1889564) - 7873
2*sqrt(2)*(1944*sqrt(pi)*(x - 157464)*(erf(1/81*sqrt(1/2)*sqrt(-(x - 157464)^2)) - 1)/sqrt(-(x - 157464)^2) +
sqrt(2)*e^(1/13122*(x - 157464)^2))*e^(-1889564) + x

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mupad [B]  time = 0.09, size = 15, normalized size = 0.68 \begin {gather*} x\,\left ({\mathrm {e}}^{\frac {x^2}{13122}-24\,x+4}+1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(4 - 24*x)*exp(x^2/13122)*(x^2 - 157464*x + 6561))/6561 + 1,x)

[Out]

x*(exp(x^2/13122 - 24*x + 4) + 1)

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sympy [A]  time = 0.29, size = 15, normalized size = 0.68 \begin {gather*} x e^{\frac {x^{2}}{13122}} e^{4 - 24 x} + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/6561*(x**2-157464*x+6561)*exp(-24*x+4)*exp(1/13122*x**2)+1,x)

[Out]

x*exp(x**2/13122)*exp(4 - 24*x) + x

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