\(\int \frac {e^{-\frac {2}{-82+3 x}} (26896-1944 x+36 x^2)}{6724-492 x+9 x^2} \, dx\) [7928]

Optimal result

Integrand size = 34, antiderivative size = 14 \[ \int \frac {e^{-\frac {2}{-82+3 x}} \left (26896-1944 x+36 x^2\right )}{6724-492 x+9 x^2} \, dx=4 e^{-\frac {2}{-82+3 x}} x \]

[Out]

4*x/exp(2/(3*x-82))

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(36\) vs. \(2(14)=28\).

Time = 0.13 (sec) , antiderivative size = 36, normalized size of antiderivative = 2.57, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.147, Rules used = {27, 6874, 2237, 2241, 2240} \[ \int \frac {e^{-\frac {2}{-82+3 x}} \left (26896-1944 x+36 x^2\right )}{6724-492 x+9 x^2} \, dx=\frac {328}{3} e^{\frac {2}{82-3 x}}-\frac {4}{3} e^{\frac {2}{82-3 x}} (82-3 x) \]

[In]

Int[(26896 - 1944*x + 36*x^2)/(E^(2/(-82 + 3*x))*(6724 - 492*x + 9*x^2)),x]

[Out]

(328*E^(2/(82 - 3*x)))/3 - (4*E^(2/(82 - 3*x))*(82 - 3*x))/3

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 2237

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

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 2241

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

Rule 6874

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

Rubi steps \begin{align*} \text {integral}& = \int \frac {e^{-\frac {2}{-82+3 x}} \left (26896-1944 x+36 x^2\right )}{(-82+3 x)^2} \, dx \\ & = \int \left (4 e^{-\frac {2}{-82+3 x}}+\frac {656 e^{-\frac {2}{-82+3 x}}}{(-82+3 x)^2}+\frac {8 e^{-\frac {2}{-82+3 x}}}{-82+3 x}\right ) \, dx \\ & = 4 \int e^{-\frac {2}{-82+3 x}} \, dx+8 \int \frac {e^{-\frac {2}{-82+3 x}}}{-82+3 x} \, dx+656 \int \frac {e^{-\frac {2}{-82+3 x}}}{(-82+3 x)^2} \, dx \\ & = \frac {328}{3} e^{\frac {2}{82-3 x}}-\frac {4}{3} e^{\frac {2}{82-3 x}} (82-3 x)-\frac {8}{3} \operatorname {ExpIntegralEi}\left (\frac {2}{82-3 x}\right )-8 \int \frac {e^{-\frac {2}{-82+3 x}}}{-82+3 x} \, dx \\ & = \frac {328}{3} e^{\frac {2}{82-3 x}}-\frac {4}{3} e^{\frac {2}{82-3 x}} (82-3 x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int \frac {e^{-\frac {2}{-82+3 x}} \left (26896-1944 x+36 x^2\right )}{6724-492 x+9 x^2} \, dx=4 e^{\frac {2}{82-3 x}} x \]

[In]

Integrate[(26896 - 1944*x + 36*x^2)/(E^(2/(-82 + 3*x))*(6724 - 492*x + 9*x^2)),x]

[Out]

4*E^(2/(82 - 3*x))*x

Maple [A] (verified)

Time = 0.37 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00

method	result	size
risch	\(4 x \,{\mathrm e}^{-\frac {2}{3 x -82}}\)	\(14\)
gosper	\(4 x \,{\mathrm e}^{-\frac {2}{3 x -82}}\)	\(16\)
norman	\(\frac {\left (12 x^{2}-328 x \right ) {\mathrm e}^{-\frac {2}{3 x -82}}}{3 x -82}\)	\(30\)
parallelrisch	\(\frac {\left (972 x^{2}-26568 x \right ) {\mathrm e}^{-\frac {2}{3 x -82}}}{243 x -6642}\)	\(31\)
derivativedivides	\(\frac {4 \,{\mathrm e}^{-\frac {2}{3 x -82}} \left (3 x -82\right )}{3}+\frac {328 \,{\mathrm e}^{-\frac {2}{3 x -82}}}{3}\)	\(35\)
default	\(\frac {4 \,{\mathrm e}^{-\frac {2}{3 x -82}} \left (3 x -82\right )}{3}+\frac {328 \,{\mathrm e}^{-\frac {2}{3 x -82}}}{3}\)	\(35\)

[In]

int((36*x^2-1944*x+26896)/(9*x^2-492*x+6724)/exp(2/(3*x-82)),x,method=_RETURNVERBOSE)

[Out]

4*x*exp(-2/(3*x-82))

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.93 \[ \int \frac {e^{-\frac {2}{-82+3 x}} \left (26896-1944 x+36 x^2\right )}{6724-492 x+9 x^2} \, dx=4 \, x e^{\left (-\frac {2}{3 \, x - 82}\right )} \]

[In]

integrate((36*x^2-1944*x+26896)/(9*x^2-492*x+6724)/exp(2/(3*x-82)),x, algorithm="fricas")

[Out]

4*x*e^(-2/(3*x - 82))

Sympy [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.71 \[ \int \frac {e^{-\frac {2}{-82+3 x}} \left (26896-1944 x+36 x^2\right )}{6724-492 x+9 x^2} \, dx=4 x e^{- \frac {2}{3 x - 82}} \]

[In]

integrate((36*x**2-1944*x+26896)/(9*x**2-492*x+6724)/exp(2/(3*x-82)),x)

[Out]

4*x*exp(-2/(3*x - 82))

Maxima [F]

\[ \int \frac {e^{-\frac {2}{-82+3 x}} \left (26896-1944 x+36 x^2\right )}{6724-492 x+9 x^2} \, dx=\int { \frac {4 \, {\left (9 \, x^{2} - 486 \, x + 6724\right )} e^{\left (-\frac {2}{3 \, x - 82}\right )}}{9 \, x^{2} - 492 \, x + 6724} \,d x } \]

[In]

integrate((36*x^2-1944*x+26896)/(9*x^2-492*x+6724)/exp(2/(3*x-82)),x, algorithm="maxima")

[Out]

4*x*e^(-2/(3*x - 82)) + 13448/3*e^(-2/(3*x - 82)) - 26896*integrate(e^(-2/(3*x - 82))/(9*x^2 - 492*x + 6724),
x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 37 vs. \(2 (13) = 26\).

Time = 0.27 (sec) , antiderivative size = 37, normalized size of antiderivative = 2.64 \[ \int \frac {e^{-\frac {2}{-82+3 x}} \left (26896-1944 x+36 x^2\right )}{6724-492 x+9 x^2} \, dx=\frac {4}{3} \, {\left (3 \, x - 82\right )} {\left (\frac {82 \, e^{\left (-\frac {2}{3 \, x - 82}\right )}}{3 \, x - 82} + e^{\left (-\frac {2}{3 \, x - 82}\right )}\right )} \]

[In]

integrate((36*x^2-1944*x+26896)/(9*x^2-492*x+6724)/exp(2/(3*x-82)),x, algorithm="giac")

[Out]

4/3*(3*x - 82)*(82*e^(-2/(3*x - 82))/(3*x - 82) + e^(-2/(3*x - 82)))

Mupad [B] (verification not implemented)

Time = 14.20 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.93 \[ \int \frac {e^{-\frac {2}{-82+3 x}} \left (26896-1944 x+36 x^2\right )}{6724-492 x+9 x^2} \, dx=4\,x\,{\mathrm {e}}^{-\frac {2}{3\,x-82}} \]

[In]

int((exp(-2/(3*x - 82))*(36*x^2 - 1944*x + 26896))/(9*x^2 - 492*x + 6724),x)

[Out]

4*x*exp(-2/(3*x - 82))