3.96.72 \(\int \frac {-9 e^{2/3}-9 e^3+e^{-4+x} (162-72 x+8 x^2)}{162-72 x+8 x^2} \, dx\)

Optimal. Leaf size=27 \[ e^{-4+x}+\frac {\left (e^{2/3}+e^3\right ) x}{4 \left (-\frac {9}{2}+x\right )} \]

________________________________________________________________________________________

Rubi [A]  time = 0.18, antiderivative size = 29, normalized size of antiderivative = 1.07, number of steps used = 6, number of rules used = 5, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.119, Rules used = {27, 12, 6741, 6742, 2194} \begin {gather*} e^{x-4}-\frac {9 e^{2/3} \left (1+e^{7/3}\right )}{4 (9-2 x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-9*E^(2/3) - 9*E^3 + E^(-4 + x)*(162 - 72*x + 8*x^2))/(162 - 72*x + 8*x^2),x]

[Out]

E^(-4 + x) - (9*E^(2/3)*(1 + E^(7/3)))/(4*(9 - 2*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 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 2194

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rule 6741

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

Rule 6742

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-9 e^{2/3}-9 e^3+e^{-4+x} \left (162-72 x+8 x^2\right )}{2 (-9+2 x)^2} \, dx\\ &=\frac {1}{2} \int \frac {-9 e^{2/3}-9 e^3+e^{-4+x} \left (162-72 x+8 x^2\right )}{(-9+2 x)^2} \, dx\\ &=\frac {1}{2} \int \frac {-9 e^{2/3} \left (1+e^{7/3}\right )+e^{-4+x} \left (162-72 x+8 x^2\right )}{(9-2 x)^2} \, dx\\ &=\frac {1}{2} \int \left (2 e^{-4+x}-\frac {9 e^{2/3} \left (1+e^{7/3}\right )}{(-9+2 x)^2}\right ) \, dx\\ &=-\frac {9 e^{2/3} \left (1+e^{7/3}\right )}{4 (9-2 x)}+\int e^{-4+x} \, dx\\ &=e^{-4+x}-\frac {9 e^{2/3} \left (1+e^{7/3}\right )}{4 (9-2 x)}\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.07, size = 31, normalized size = 1.15 \begin {gather*} \frac {2 e^x+\frac {9 \left (e^{14/3}+e^7\right )}{-18+4 x}}{2 e^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-9*E^(2/3) - 9*E^3 + E^(-4 + x)*(162 - 72*x + 8*x^2))/(162 - 72*x + 8*x^2),x]

[Out]

(2*E^x + (9*(E^(14/3) + E^7))/(-18 + 4*x))/(2*E^4)

________________________________________________________________________________________

fricas [A]  time = 0.55, size = 29, normalized size = 1.07 \begin {gather*} \frac {4 \, {\left (2 \, x - 9\right )} e^{\left (x - 4\right )} + 9 \, e^{3} + 9 \, e^{\frac {2}{3}}}{4 \, {\left (2 \, x - 9\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*x^2-72*x+162)*exp(x-4)-9*exp(3)-9*exp(2/3))/(8*x^2-72*x+162),x, algorithm="fricas")

[Out]

1/4*(4*(2*x - 9)*e^(x - 4) + 9*e^3 + 9*e^(2/3))/(2*x - 9)

________________________________________________________________________________________

giac [A]  time = 0.13, size = 32, normalized size = 1.19 \begin {gather*} \frac {8 \, x e^{x} + 9 \, e^{7} + 9 \, e^{\frac {14}{3}} - 36 \, e^{x}}{4 \, {\left (2 \, x e^{4} - 9 \, e^{4}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*x^2-72*x+162)*exp(x-4)-9*exp(3)-9*exp(2/3))/(8*x^2-72*x+162),x, algorithm="giac")

[Out]

1/4*(8*x*e^x + 9*e^7 + 9*e^(14/3) - 36*e^x)/(2*x*e^4 - 9*e^4)

________________________________________________________________________________________

maple [A]  time = 0.11, size = 24, normalized size = 0.89

method	result	size
risch	\(\frac {9 \,{\mathrm e}^{3}}{8 \left (x -\frac {9}{2}\right )}+\frac {9 \,{\mathrm e}^{\frac {2}{3}}}{8 \left (x -\frac {9}{2}\right )}+{\mathrm e}^{x -4}\)	\(24\)
derivativedivides	\(\frac {9 \,{\mathrm e}^{3}}{4 \left (2 x -9\right )}+\frac {9 \,{\mathrm e}^{\frac {2}{3}}}{4 \left (2 x -9\right )}+{\mathrm e}^{x -4}\)	\(28\)
default	\(\frac {9 \,{\mathrm e}^{3}}{4 \left (2 x -9\right )}+\frac {9 \,{\mathrm e}^{\frac {2}{3}}}{4 \left (2 x -9\right )}+{\mathrm e}^{x -4}\)	\(28\)
norman	\(\frac {2 x \,{\mathrm e}^{x -4}-9 \,{\mathrm e}^{x -4}+\frac {9 \,{\mathrm e}^{3}}{4}+\frac {9 \,{\mathrm e}^{\frac {2}{3}}}{4}}{2 x -9}\)	\(31\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((8*x^2-72*x+162)*exp(x-4)-9*exp(3)-9*exp(2/3))/(8*x^2-72*x+162),x,method=_RETURNVERBOSE)

[Out]

9/8/(x-9/2)*exp(3)+9/8/(x-9/2)*exp(2/3)+exp(x-4)

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {4 \, {\left (x^{2} - 9 \, x\right )} e^{x}}{4 \, x^{2} e^{4} - 36 \, x e^{4} + 81 \, e^{4}} - \frac {81 \, e^{\frac {1}{2}} E_{2}\left (-x + \frac {9}{2}\right )}{2 \, {\left (2 \, x - 9\right )}} + \frac {9 \, e^{3}}{4 \, {\left (2 \, x - 9\right )}} + \frac {9 \, e^{\frac {2}{3}}}{4 \, {\left (2 \, x - 9\right )}} - 324 \, \int \frac {e^{x}}{8 \, x^{3} e^{4} - 108 \, x^{2} e^{4} + 486 \, x e^{4} - 729 \, e^{4}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*x^2-72*x+162)*exp(x-4)-9*exp(3)-9*exp(2/3))/(8*x^2-72*x+162),x, algorithm="maxima")

[Out]

4*(x^2 - 9*x)*e^x/(4*x^2*e^4 - 36*x*e^4 + 81*e^4) - 81/2*e^(1/2)*exp_integral_e(2, -x + 9/2)/(2*x - 9) + 9/4*e
^3/(2*x - 9) + 9/4*e^(2/3)/(2*x - 9) - 324*integrate(e^x/(8*x^3*e^4 - 108*x^2*e^4 + 486*x*e^4 - 729*e^4), x)

________________________________________________________________________________________

mupad [B]  time = 0.19, size = 22, normalized size = 0.81 \begin {gather*} {\mathrm {e}}^{x-4}+\frac {\frac {9\,{\mathrm {e}}^3}{2}+\frac {9\,{\mathrm {e}}^{2/3}}{2}}{4\,x-18} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(9*exp(3) + 9*exp(2/3) - exp(x - 4)*(8*x^2 - 72*x + 162))/(8*x^2 - 72*x + 162),x)

[Out]

exp(x - 4) + ((9*exp(3))/2 + (9*exp(2/3))/2)/(4*x - 18)

________________________________________________________________________________________

sympy [A]  time = 0.14, size = 22, normalized size = 0.81 \begin {gather*} e^{x - 4} - \frac {- 9 e^{3} - 9 e^{\frac {2}{3}}}{8 x - 36} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*x**2-72*x+162)*exp(x-4)-9*exp(3)-9*exp(2/3))/(8*x**2-72*x+162),x)

[Out]

exp(x - 4) - (-9*exp(3) - 9*exp(2/3))/(8*x - 36)

________________________________________________________________________________________