\(\int \frac {e^{\frac {1}{9} (9-9 x+6 x^2-x^3-3750 x^4+1250 x^5-390625 x^7)} (-9-9 x+12 x^2-3 x^3-15000 x^4+6250 x^5-2734375 x^7)}{9 x^2} \, dx\) [8663]

Optimal result

Integrand size = 73, antiderivative size = 33 \[ \int \frac {e^{\frac {1}{9} \left (9-9 x+6 x^2-x^3-3750 x^4+1250 x^5-390625 x^7\right )} \left (-9-9 x+12 x^2-3 x^3-15000 x^4+6250 x^5-2734375 x^7\right )}{9 x^2} \, dx=\frac {e^{\frac {x-\frac {1}{9} x^2 \left (3-x \left (1-625 x^2\right )\right )^2}{x}}}{x} \]

[Out]

exp((x-1/9*x^2*(3-(-625*x^2+1)*x)^2)/x)/x

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(96\) vs. \(2(33)=66\).

Time = 0.20 (sec) , antiderivative size = 96, normalized size of antiderivative = 2.91, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.027, Rules used = {12, 2326} \[ \int \frac {e^{\frac {1}{9} \left (9-9 x+6 x^2-x^3-3750 x^4+1250 x^5-390625 x^7\right )} \left (-9-9 x+12 x^2-3 x^3-15000 x^4+6250 x^5-2734375 x^7\right )}{9 x^2} \, dx=\frac {\left (2734375 x^7-6250 x^5+15000 x^4+3 x^3-12 x^2+9 x\right ) \exp \left (\frac {1}{9} \left (-390625 x^7+1250 x^5-3750 x^4-x^3+6 x^2-9 x+9\right )\right )}{x^2 \left (2734375 x^6-6250 x^4+15000 x^3+3 x^2-12 x+9\right )} \]

[In]

Int[(E^((9 - 9*x + 6*x^2 - x^3 - 3750*x^4 + 1250*x^5 - 390625*x^7)/9)*(-9 - 9*x + 12*x^2 - 3*x^3 - 15000*x^4 +
6250*x^5 - 2734375*x^7))/(9*x^2),x]

[Out]

(E^((9 - 9*x + 6*x^2 - x^3 - 3750*x^4 + 1250*x^5 - 390625*x^7)/9)*(9*x - 12*x^2 + 3*x^3 + 15000*x^4 - 6250*x^5
+ 2734375*x^7))/(x^2*(9 - 12*x + 3*x^2 + 15000*x^3 - 6250*x^4 + 2734375*x^6))

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 {1}{9} \int \frac {\exp \left (\frac {1}{9} \left (9-9 x+6 x^2-x^3-3750 x^4+1250 x^5-390625 x^7\right )\right ) \left (-9-9 x+12 x^2-3 x^3-15000 x^4+6250 x^5-2734375 x^7\right )}{x^2} \, dx \\ & = \frac {\exp \left (\frac {1}{9} \left (9-9 x+6 x^2-x^3-3750 x^4+1250 x^5-390625 x^7\right )\right ) \left (9 x-12 x^2+3 x^3+15000 x^4-6250 x^5+2734375 x^7\right )}{x^2 \left (9-12 x+3 x^2+15000 x^3-6250 x^4+2734375 x^6\right )} \\ \end{align*}

Mathematica [A] (verified)

Time = 7.42 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.21 \[ \int \frac {e^{\frac {1}{9} \left (9-9 x+6 x^2-x^3-3750 x^4+1250 x^5-390625 x^7\right )} \left (-9-9 x+12 x^2-3 x^3-15000 x^4+6250 x^5-2734375 x^7\right )}{9 x^2} \, dx=\frac {e^{\frac {1}{9} \left (9-9 x+6 x^2-x^3-3750 x^4+1250 x^5-390625 x^7\right )}}{x} \]

[In]

Integrate[(E^((9 - 9*x + 6*x^2 - x^3 - 3750*x^4 + 1250*x^5 - 390625*x^7)/9)*(-9 - 9*x + 12*x^2 - 3*x^3 - 15000
*x^4 + 6250*x^5 - 2734375*x^7))/(9*x^2),x]

[Out]

E^((9 - 9*x + 6*x^2 - x^3 - 3750*x^4 + 1250*x^5 - 390625*x^7)/9)/x

Maple [A] (verified)

Time = 0.15 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.09

method	result	size
gosper	\(\frac {{\mathrm e}^{-\frac {390625}{9} x^{7}+\frac {1250}{9} x^{5}-\frac {1250}{3} x^{4}-\frac {1}{9} x^{3}+\frac {2}{3} x^{2}-x +1}}{x}\)	\(36\)
norman	\(\frac {{\mathrm e}^{-\frac {390625}{9} x^{7}+\frac {1250}{9} x^{5}-\frac {1250}{3} x^{4}-\frac {1}{9} x^{3}+\frac {2}{3} x^{2}-x +1}}{x}\)	\(36\)
parallelrisch	\(\frac {{\mathrm e}^{-\frac {390625}{9} x^{7}+\frac {1250}{9} x^{5}-\frac {1250}{3} x^{4}-\frac {1}{9} x^{3}+\frac {2}{3} x^{2}-x +1}}{x}\)	\(36\)
risch	\(\frac {{\mathrm e}^{-\frac {\left (25 x^{2}+x +1\right ) \left (15625 x^{5}-625 x^{4}-650 x^{3}+201 x^{2}+18 x -9\right )}{9}}}{x}\)	\(41\)

[In]

int(1/9*(-2734375*x^7+6250*x^5-15000*x^4-3*x^3+12*x^2-9*x-9)*exp(-390625/9*x^7+1250/9*x^5-1250/3*x^4-1/9*x^3+2
/3*x^2-x+1)/x^2,x,method=_RETURNVERBOSE)

[Out]

exp(-390625/9*x^7+1250/9*x^5-1250/3*x^4-1/9*x^3+2/3*x^2-x+1)/x

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.06 \[ \int \frac {e^{\frac {1}{9} \left (9-9 x+6 x^2-x^3-3750 x^4+1250 x^5-390625 x^7\right )} \left (-9-9 x+12 x^2-3 x^3-15000 x^4+6250 x^5-2734375 x^7\right )}{9 x^2} \, dx=\frac {e^{\left (-\frac {390625}{9} \, x^{7} + \frac {1250}{9} \, x^{5} - \frac {1250}{3} \, x^{4} - \frac {1}{9} \, x^{3} + \frac {2}{3} \, x^{2} - x + 1\right )}}{x} \]

[In]

integrate(1/9*(-2734375*x^7+6250*x^5-15000*x^4-3*x^3+12*x^2-9*x-9)*exp(-390625/9*x^7+1250/9*x^5-1250/3*x^4-1/9
*x^3+2/3*x^2-x+1)/x^2,x, algorithm="fricas")

[Out]

e^(-390625/9*x^7 + 1250/9*x^5 - 1250/3*x^4 - 1/9*x^3 + 2/3*x^2 - x + 1)/x

Sympy [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.12 \[ \int \frac {e^{\frac {1}{9} \left (9-9 x+6 x^2-x^3-3750 x^4+1250 x^5-390625 x^7\right )} \left (-9-9 x+12 x^2-3 x^3-15000 x^4+6250 x^5-2734375 x^7\right )}{9 x^2} \, dx=\frac {e^{- \frac {390625 x^{7}}{9} + \frac {1250 x^{5}}{9} - \frac {1250 x^{4}}{3} - \frac {x^{3}}{9} + \frac {2 x^{2}}{3} - x + 1}}{x} \]

[In]

integrate(1/9*(-2734375*x**7+6250*x**5-15000*x**4-3*x**3+12*x**2-9*x-9)*exp(-390625/9*x**7+1250/9*x**5-1250/3*
x**4-1/9*x**3+2/3*x**2-x+1)/x**2,x)

[Out]

exp(-390625*x**7/9 + 1250*x**5/9 - 1250*x**4/3 - x**3/9 + 2*x**2/3 - x + 1)/x

Maxima [A] (verification not implemented)

none

Time = 0.36 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.06 \[ \int \frac {e^{\frac {1}{9} \left (9-9 x+6 x^2-x^3-3750 x^4+1250 x^5-390625 x^7\right )} \left (-9-9 x+12 x^2-3 x^3-15000 x^4+6250 x^5-2734375 x^7\right )}{9 x^2} \, dx=\frac {e^{\left (-\frac {390625}{9} \, x^{7} + \frac {1250}{9} \, x^{5} - \frac {1250}{3} \, x^{4} - \frac {1}{9} \, x^{3} + \frac {2}{3} \, x^{2} - x + 1\right )}}{x} \]

[In]

integrate(1/9*(-2734375*x^7+6250*x^5-15000*x^4-3*x^3+12*x^2-9*x-9)*exp(-390625/9*x^7+1250/9*x^5-1250/3*x^4-1/9
*x^3+2/3*x^2-x+1)/x^2,x, algorithm="maxima")

[Out]

e^(-390625/9*x^7 + 1250/9*x^5 - 1250/3*x^4 - 1/9*x^3 + 2/3*x^2 - x + 1)/x

Giac [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.06 \[ \int \frac {e^{\frac {1}{9} \left (9-9 x+6 x^2-x^3-3750 x^4+1250 x^5-390625 x^7\right )} \left (-9-9 x+12 x^2-3 x^3-15000 x^4+6250 x^5-2734375 x^7\right )}{9 x^2} \, dx=\frac {e^{\left (-\frac {390625}{9} \, x^{7} + \frac {1250}{9} \, x^{5} - \frac {1250}{3} \, x^{4} - \frac {1}{9} \, x^{3} + \frac {2}{3} \, x^{2} - x + 1\right )}}{x} \]

[In]

integrate(1/9*(-2734375*x^7+6250*x^5-15000*x^4-3*x^3+12*x^2-9*x-9)*exp(-390625/9*x^7+1250/9*x^5-1250/3*x^4-1/9
*x^3+2/3*x^2-x+1)/x^2,x, algorithm="giac")

[Out]

e^(-390625/9*x^7 + 1250/9*x^5 - 1250/3*x^4 - 1/9*x^3 + 2/3*x^2 - x + 1)/x

Mupad [B] (verification not implemented)

Time = 14.19 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.21 \[ \int \frac {e^{\frac {1}{9} \left (9-9 x+6 x^2-x^3-3750 x^4+1250 x^5-390625 x^7\right )} \left (-9-9 x+12 x^2-3 x^3-15000 x^4+6250 x^5-2734375 x^7\right )}{9 x^2} \, dx=\frac {{\mathrm {e}}^{-x}\,\mathrm {e}\,{\mathrm {e}}^{\frac {2\,x^2}{3}}\,{\mathrm {e}}^{-\frac {x^3}{9}}\,{\mathrm {e}}^{-\frac {1250\,x^4}{3}}\,{\mathrm {e}}^{\frac {1250\,x^5}{9}}\,{\mathrm {e}}^{-\frac {390625\,x^7}{9}}}{x} \]

[In]

int(-(exp((2*x^2)/3 - x - x^3/9 - (1250*x^4)/3 + (1250*x^5)/9 - (390625*x^7)/9 + 1)*(9*x - 12*x^2 + 3*x^3 + 15
000*x^4 - 6250*x^5 + 2734375*x^7 + 9))/(9*x^2),x)

[Out]

(exp(-x)*exp(1)*exp((2*x^2)/3)*exp(-x^3/9)*exp(-(1250*x^4)/3)*exp((1250*x^5)/9)*exp(-(390625*x^7)/9))/x