\(\int \frac {6561+e^{4-24 x+\frac {x^2}{13122}} (6561-157464 x+x^2)}{6561} \, dx\) [5164]

Optimal result

Integrand size = 29, antiderivative size = 22 \[ \int \frac {6561+e^{4-24 x+\frac {x^2}{13122}} \left (6561-157464 x+x^2\right )}{6561} \, dx=5+x+e^{4 (1-6 x)+\frac {x^2}{13122}} x \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.50, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {12, 2326} \[ \int \frac {6561+e^{4-24 x+\frac {x^2}{13122}} \left (6561-157464 x+x^2\right )}{6561} \, dx=\frac {e^{\frac {x^2}{13122}-24 x+4} \left (157464 x-x^2\right )}{157464-x}+x \]

[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 2326

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{align*} \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{align*}

Mathematica [A] (verified)

Time = 0.41 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.82 \[ \int \frac {6561+e^{4-24 x+\frac {x^2}{13122}} \left (6561-157464 x+x^2\right )}{6561} \, dx=\left (1+e^{4-24 x+\frac {x^2}{13122}}\right ) x \]

[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

Maple [A] (verified)

Time = 0.07 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.73

method	result	size
risch	\(x +x \,{\mathrm e}^{\frac {1}{13122} x^{2}-24 x +4}\)	\(16\)
norman	\(x +{\mathrm e}^{\frac {x^{2}}{13122}} {\mathrm e}^{-24 x +4} x\)	\(17\)
parallelrisch	\(x +{\mathrm e}^{\frac {x^{2}}{13122}} {\mathrm e}^{-24 x +4} x\)	\(17\)
default	\(x +\frac {{\mathrm e}^{4} \left (6561 x \,{\mathrm e}^{-24 x +\frac {1}{13122} x^{2}}[phone] \,{\mathrm e}^{-24 x +\frac {1}{13122} x^{2}}-\frac {2008387283535 i \sqrt {\pi }\, {\mathrm e}^{-1889568} \sqrt {2}\, \operatorname {erf}\left (\frac {i \sqrt {2}\, x}{162}-972 i \sqrt {2}\right )}{2}\right )}{6561}-\frac {81 i {\mathrm e}^{4} \sqrt {\pi }\, {\mathrm e}^{-1889568} \sqrt {2}\, \operatorname {erf}\left (\frac {i \sqrt {2}\, x}{162}-972 i \sqrt {2}\right )}{2}-24 \,{\mathrm e}^{4} \left (6561 \,{\mathrm e}^{-24 x +\frac {1}{13122} x^{2}}-6377292 i \sqrt {\pi }\, {\mathrm e}^{-1889568} \sqrt {2}\, \operatorname {erf}\left (\frac {i \sqrt {2}\, x}{162}-972 i \sqrt {2}\right )\right )\)	\(130\)
parts	\(x +\frac {{\mathrm e}^{4} \left (6561 x \,{\mathrm e}^{-24 x +\frac {1}{13122} x^{2}}[phone] \,{\mathrm e}^{-24 x +\frac {1}{13122} x^{2}}-\frac {2008387283535 i \sqrt {\pi }\, {\mathrm e}^{-1889568} \sqrt {2}\, \operatorname {erf}\left (\frac {i \sqrt {2}\, x}{162}-972 i \sqrt {2}\right )}{2}\right )}{6561}-\frac {81 i {\mathrm e}^{4} \sqrt {\pi }\, {\mathrm e}^{-1889568} \sqrt {2}\, \operatorname {erf}\left (\frac {i \sqrt {2}\, x}{162}-972 i \sqrt {2}\right )}{2}-24 \,{\mathrm e}^{4} \left (6561 \,{\mathrm e}^{-24 x +\frac {1}{13122} x^{2}}-6377292 i \sqrt {\pi }\, {\mathrm e}^{-1889568} \sqrt {2}\, \operatorname {erf}\left (\frac {i \sqrt {2}\, x}{162}-972 i \sqrt {2}\right )\right )\)	\(130\)

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.68 \[ \int \frac {6561+e^{4-24 x+\frac {x^2}{13122}} \left (6561-157464 x+x^2\right )}{6561} \, dx=x e^{\left (\frac {1}{13122} \, x^{2} - 24 \, x + 4\right )} + x \]

[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

Sympy [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.68 \[ \int \frac {6561+e^{4-24 x+\frac {x^2}{13122}} \left (6561-157464 x+x^2\right )}{6561} \, dx=x e^{\frac {x^{2}}{13122}} e^{4 - 24 x} + x \]

[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

Maxima [C] (verification not implemented)

Result contains higher order function than in optimal. Order 4 vs. order 3.

Time = 0.31 (sec) , antiderivative size = 158, normalized size of antiderivative = 7.18 \[ \int \frac {6561+e^{4-24 x+\frac {x^2}{13122}} \left (6561-157464 x+x^2\right )}{6561} \, dx=-\frac {81}{2} i \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (\frac {1}{162} i \, \sqrt {2} x - 972 i \, \sqrt {2}\right ) e^{\left (-1889564\right )} - 81 \, \sqrt {2} {\left (\frac {{\left (x - 157464\right )}^{3} \Gamma \left (\frac {3}{2}, -\frac {1}{13122} \, {\left (x - 157464\right )}^{2}\right )}{\left (-{\left (x - 157464\right )}^{2}\right )^{\frac {3}{2}}} - \frac {1889568 \, \sqrt {\pi } {\left (x - 157464\right )} {\left (\operatorname {erf}\left (\frac {1}{81} \, \sqrt {\frac {1}{2}} \sqrt {-{\left (x - 157464\right )}^{2}}\right ) - 1\right )}}{\sqrt {-{\left (x - 157464\right )}^{2}}} - 1944 \, \sqrt {2} e^{\left (\frac {1}{13122} \, {\left (x - 157464\right )}^{2}\right )}\right )} e^{\left (-1889564\right )} - 78732 \, \sqrt {2} {\left (\frac {1944 \, \sqrt {\pi } {\left (x - 157464\right )} {\left (\operatorname {erf}\left (\frac {1}{81} \, \sqrt {\frac {1}{2}} \sqrt {-{\left (x - 157464\right )}^{2}}\right ) - 1\right )}}{\sqrt {-{\left (x - 157464\right )}^{2}}} + \sqrt {2} e^{\left (\frac {1}{13122} \, {\left (x - 157464\right )}^{2}\right )}\right )} e^{\left (-1889564\right )} + x \]

[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

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.68 \[ \int \frac {6561+e^{4-24 x+\frac {x^2}{13122}} \left (6561-157464 x+x^2\right )}{6561} \, dx=x e^{\left (\frac {1}{13122} \, x^{2} - 24 \, x + 4\right )} + x \]

[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

Mupad [B] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.68 \[ \int \frac {6561+e^{4-24 x+\frac {x^2}{13122}} \left (6561-157464 x+x^2\right )}{6561} \, dx=x\,\left ({\mathrm {e}}^{\frac {x^2}{13122}-24\,x+4}+1\right ) \]

[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)