\(\int \frac {e^{\frac {8-3 x+e^2 x}{x}} (e^2 x-x^2)+e^{\frac {8-3 x+e^2 x}{x}} (e^2 (-8-3 x)+8 x+4 x^2) \log (x)}{e^4 x^5-2 e^2 x^6+x^7} \, dx\) [4440]

Optimal result

Integrand size = 88, antiderivative size = 27 \[ \int \frac {e^{\frac {8-3 x+e^2 x}{x}} \left (e^2 x-x^2\right )+e^{\frac {8-3 x+e^2 x}{x}} \left (e^2 (-8-3 x)+8 x+4 x^2\right ) \log (x)}{e^4 x^5-2 e^2 x^6+x^7} \, dx=\frac {e^{-3+e^2+\frac {8}{x}} \log (x)}{\left (e^2-x\right ) x^3} \]

[Out]

exp(exp(2)+8/x-3)/x^3*ln(x)/(exp(2)-x)

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(272\) vs. \(2(27)=54\).

Time = 2.27 (sec) , antiderivative size = 272, normalized size of antiderivative = 10.07, number of steps used = 32, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {1608, 27, 6820, 6874, 2254, 2241, 2260, 2209, 2243, 2240, 2255, 2634} \[ \int \frac {e^{\frac {8-3 x+e^2 x}{x}} \left (e^2 x-x^2\right )+e^{\frac {8-3 x+e^2 x}{x}} \left (e^2 (-8-3 x)+8 x+4 x^2\right ) \log (x)}{e^4 x^5-2 e^2 x^6+x^7} \, dx=\frac {e^{\frac {8}{x}+e^2-5} \log (x)}{x^3}-\frac {3 e^{\frac {8}{x}+e^2-5} \log (x)}{8 x^2}+\frac {\left (8+3 e^2\right ) e^{\frac {8}{x}+e^2-7} \log (x)}{8 x^2}-e^{\frac {8}{x}+e^2-11} \log (x)-\frac {3}{256} e^{\frac {8}{x}+e^2-5} \log (x)+\frac {e^{\frac {8}{x}+e^2-9} \log (x)}{e^2-x}+\frac {3 e^{\frac {8}{x}+e^2-5} \log (x)}{32 x}-\frac {\left (8+3 e^2\right ) e^{\frac {8}{x}+e^2-7} \log (x)}{32 x}+\frac {\left (4+e^2\right ) e^{\frac {8}{x}+e^2-9} \log (x)}{4 x}+\frac {1}{256} \left (8+3 e^2\right ) e^{\frac {8}{x}+e^2-7} \log (x)+\frac {1}{8} \left (8+e^2\right ) e^{\frac {8}{x}+e^2-11} \log (x)-\frac {1}{32} \left (4+e^2\right ) e^{\frac {8}{x}+e^2-9} \log (x) \]

[In]

Int[(E^((8 - 3*x + E^2*x)/x)*(E^2*x - x^2) + E^((8 - 3*x + E^2*x)/x)*(E^2*(-8 - 3*x) + 8*x + 4*x^2)*Log[x])/(E
^4*x^5 - 2*E^2*x^6 + x^7),x]

[Out]

-(E^(-11 + E^2 + 8/x)*Log[x]) - (3*E^(-5 + E^2 + 8/x)*Log[x])/256 - (E^(-9 + E^2 + 8/x)*(4 + E^2)*Log[x])/32 +
(E^(-11 + E^2 + 8/x)*(8 + E^2)*Log[x])/8 + (E^(-7 + E^2 + 8/x)*(8 + 3*E^2)*Log[x])/256 + (E^(-9 + E^2 + 8/x)*
Log[x])/(E^2 - x) + (E^(-5 + E^2 + 8/x)*Log[x])/x^3 - (3*E^(-5 + E^2 + 8/x)*Log[x])/(8*x^2) + (E^(-7 + E^2 + 8
/x)*(8 + 3*E^2)*Log[x])/(8*x^2) + (3*E^(-5 + E^2 + 8/x)*Log[x])/(32*x) + (E^(-9 + E^2 + 8/x)*(4 + E^2)*Log[x])
/(4*x) - (E^(-7 + E^2 + 8/x)*(8 + 3*E^2)*Log[x])/(32*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 1608

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.) + (c_.)*(x_)^(r_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^
(q - p) + c*x^(r - p))^n, x] /; FreeQ[{a, b, c, p, q, r}, x] && IntegerQ[n] && PosQ[q - p] && PosQ[r - p]

Rule 2209

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(F^(g*(e - c*(f/d)))/d)*ExpInteg
ralEi[f*g*(c + d*x)*(Log[F]/d)], x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !TrueQ[$UseGamma]

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 2243

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(c + d*x)^(m
- n + 1)*(F^(a + b*(c + d*x)^n)/(b*d*n*Log[F])), x] - Dist[(m - n + 1)/(b*n*Log[F]), Int[(c + d*x)^(m - n)*F^(
a + b*(c + d*x)^n), x], x] /; FreeQ[{F, a, b, c, d}, x] && IntegerQ[2*((m + 1)/n)] && LtQ[0, (m + 1)/n, 5] &&
IntegerQ[n] && (LtQ[0, n, m + 1] || LtQ[m, n, 0])

Rule 2254

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

Rule 2255

Int[(F_)^((a_.) + (b_.)/((c_.) + (d_.)*(x_)))*((e_.) + (f_.)*(x_))^(m_), x_Symbol] :> Simp[(e + f*x)^(m + 1)*(
F^(a + b/(c + d*x))/(f*(m + 1))), x] + Dist[b*d*(Log[F]/(f*(m + 1))), Int[(e + f*x)^(m + 1)*(F^(a + b/(c + d*x
))/(c + d*x)^2), x], x] /; FreeQ[{F, a, b, c, d, e, f}, x] && NeQ[d*e - c*f, 0] && ILtQ[m, -1]

Rule 2260

Int[(F_)^((a_.) + (b_.)/((c_.) + (d_.)*(x_)))/(((e_.) + (f_.)*(x_))*((g_.) + (h_.)*(x_))), x_Symbol] :> Dist[-
d/(f*(d*g - c*h)), Subst[Int[F^(a - b*(h/(d*g - c*h)) + d*b*(x/(d*g - c*h)))/x, x], x, (g + h*x)/(c + d*x)], x
] /; FreeQ[{F, a, b, c, d, e, f}, x] && EqQ[d*e - c*f, 0]

Rule 2634

Int[Log[u_]*(v_), x_Symbol] :> With[{w = IntHide[v, x]}, Dist[Log[u], w, x] - Int[SimplifyIntegrand[w*(D[u, x]
/u), x], x] /; InverseFunctionFreeQ[w, x]] /; InverseFunctionFreeQ[u, x]

Rule 6820

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

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 {8-3 x+e^2 x}{x}} \left (e^2 x-x^2\right )+e^{\frac {8-3 x+e^2 x}{x}} \left (e^2 (-8-3 x)+8 x+4 x^2\right ) \log (x)}{x^5 \left (e^4-2 e^2 x+x^2\right )} \, dx \\ & = \int \frac {e^{\frac {8-3 x+e^2 x}{x}} \left (e^2 x-x^2\right )+e^{\frac {8-3 x+e^2 x}{x}} \left (e^2 (-8-3 x)+8 x+4 x^2\right ) \log (x)}{x^5 \left (-e^2+x\right )^2} \, dx \\ & = \int \frac {e^{-3+e^2+\frac {8}{x}} \left (\left (e^2-x\right ) x+\left (4 x (2+x)-e^2 (8+3 x)\right ) \log (x)\right )}{\left (e^2-x\right )^2 x^5} \, dx \\ & = \int \left (\frac {e^{-3+e^2+\frac {8}{x}}}{\left (e^2-x\right ) x^4}+\frac {e^{-3+e^2+\frac {8}{x}} \left (-8 e^2+\left (8-3 e^2\right ) x+4 x^2\right ) \log (x)}{\left (e^2-x\right )^2 x^5}\right ) \, dx \\ & = \int \frac {e^{-3+e^2+\frac {8}{x}}}{\left (e^2-x\right ) x^4} \, dx+\int \frac {e^{-3+e^2+\frac {8}{x}} \left (-8 e^2+\left (8-3 e^2\right ) x+4 x^2\right ) \log (x)}{\left (e^2-x\right )^2 x^5} \, dx \\ & = -e^{-11+e^2+\frac {8}{x}} \log (x)-\frac {3}{256} e^{-5+e^2+\frac {8}{x}} \log (x)-\frac {1}{32} e^{-9+e^2+\frac {8}{x}} \left (4+e^2\right ) \log (x)+\frac {1}{8} e^{-11+e^2+\frac {8}{x}} \left (8+e^2\right ) \log (x)+\frac {1}{256} e^{-7+e^2+\frac {8}{x}} \left (8+3 e^2\right ) \log (x)+\frac {e^{-9+e^2+\frac {8}{x}} \log (x)}{e^2-x}+\frac {e^{-5+e^2+\frac {8}{x}} \log (x)}{x^3}-\frac {3 e^{-5+e^2+\frac {8}{x}} \log (x)}{8 x^2}+\frac {e^{-7+e^2+\frac {8}{x}} \left (8+3 e^2\right ) \log (x)}{8 x^2}+\frac {3 e^{-5+e^2+\frac {8}{x}} \log (x)}{32 x}+\frac {e^{-9+e^2+\frac {8}{x}} \left (4+e^2\right ) \log (x)}{4 x}-\frac {e^{-7+e^2+\frac {8}{x}} \left (8+3 e^2\right ) \log (x)}{32 x}+\int \left (\frac {e^{-11+e^2+\frac {8}{x}}}{e^2-x}+\frac {e^{-5+e^2+\frac {8}{x}}}{x^4}+\frac {e^{-7+e^2+\frac {8}{x}}}{x^3}+\frac {e^{-9+e^2+\frac {8}{x}}}{x^2}+\frac {e^{-11+e^2+\frac {8}{x}}}{x}\right ) \, dx-\int \frac {e^{-3+e^2+\frac {8}{x}}}{\left (e^2-x\right ) x^4} \, dx \\ & = -e^{-11+e^2+\frac {8}{x}} \log (x)-\frac {3}{256} e^{-5+e^2+\frac {8}{x}} \log (x)-\frac {1}{32} e^{-9+e^2+\frac {8}{x}} \left (4+e^2\right ) \log (x)+\frac {1}{8} e^{-11+e^2+\frac {8}{x}} \left (8+e^2\right ) \log (x)+\frac {1}{256} e^{-7+e^2+\frac {8}{x}} \left (8+3 e^2\right ) \log (x)+\frac {e^{-9+e^2+\frac {8}{x}} \log (x)}{e^2-x}+\frac {e^{-5+e^2+\frac {8}{x}} \log (x)}{x^3}-\frac {3 e^{-5+e^2+\frac {8}{x}} \log (x)}{8 x^2}+\frac {e^{-7+e^2+\frac {8}{x}} \left (8+3 e^2\right ) \log (x)}{8 x^2}+\frac {3 e^{-5+e^2+\frac {8}{x}} \log (x)}{32 x}+\frac {e^{-9+e^2+\frac {8}{x}} \left (4+e^2\right ) \log (x)}{4 x}-\frac {e^{-7+e^2+\frac {8}{x}} \left (8+3 e^2\right ) \log (x)}{32 x}-\int \left (\frac {e^{-11+e^2+\frac {8}{x}}}{e^2-x}+\frac {e^{-5+e^2+\frac {8}{x}}}{x^4}+\frac {e^{-7+e^2+\frac {8}{x}}}{x^3}+\frac {e^{-9+e^2+\frac {8}{x}}}{x^2}+\frac {e^{-11+e^2+\frac {8}{x}}}{x}\right ) \, dx+\int \frac {e^{-11+e^2+\frac {8}{x}}}{e^2-x} \, dx+\int \frac {e^{-5+e^2+\frac {8}{x}}}{x^4} \, dx+\int \frac {e^{-7+e^2+\frac {8}{x}}}{x^3} \, dx+\int \frac {e^{-9+e^2+\frac {8}{x}}}{x^2} \, dx+\int \frac {e^{-11+e^2+\frac {8}{x}}}{x} \, dx \\ & = -\frac {1}{8} e^{-9+e^2+\frac {8}{x}}-\frac {e^{-5+e^2+\frac {8}{x}}}{8 x^2}-\frac {e^{-7+e^2+\frac {8}{x}}}{8 x}-e^{-11+e^2} \text {Ei}\left (\frac {8}{x}\right )-e^{-11+e^2+\frac {8}{x}} \log (x)-\frac {3}{256} e^{-5+e^2+\frac {8}{x}} \log (x)-\frac {1}{32} e^{-9+e^2+\frac {8}{x}} \left (4+e^2\right ) \log (x)+\frac {1}{8} e^{-11+e^2+\frac {8}{x}} \left (8+e^2\right ) \log (x)+\frac {1}{256} e^{-7+e^2+\frac {8}{x}} \left (8+3 e^2\right ) \log (x)+\frac {e^{-9+e^2+\frac {8}{x}} \log (x)}{e^2-x}+\frac {e^{-5+e^2+\frac {8}{x}} \log (x)}{x^3}-\frac {3 e^{-5+e^2+\frac {8}{x}} \log (x)}{8 x^2}+\frac {e^{-7+e^2+\frac {8}{x}} \left (8+3 e^2\right ) \log (x)}{8 x^2}+\frac {3 e^{-5+e^2+\frac {8}{x}} \log (x)}{32 x}+\frac {e^{-9+e^2+\frac {8}{x}} \left (4+e^2\right ) \log (x)}{4 x}-\frac {e^{-7+e^2+\frac {8}{x}} \left (8+3 e^2\right ) \log (x)}{32 x}-\frac {1}{8} \int \frac {e^{-7+e^2+\frac {8}{x}}}{x^2} \, dx-\frac {1}{4} \int \frac {e^{-5+e^2+\frac {8}{x}}}{x^3} \, dx+e^2 \int \frac {e^{-11+e^2+\frac {8}{x}}}{\left (e^2-x\right ) x} \, dx-\int \frac {e^{-11+e^2+\frac {8}{x}}}{e^2-x} \, dx-\int \frac {e^{-5+e^2+\frac {8}{x}}}{x^4} \, dx-\int \frac {e^{-7+e^2+\frac {8}{x}}}{x^3} \, dx-\int \frac {e^{-9+e^2+\frac {8}{x}}}{x^2} \, dx-2 \int \frac {e^{-11+e^2+\frac {8}{x}}}{x} \, dx \\ & = \frac {1}{64} e^{-7+e^2+\frac {8}{x}}+\frac {e^{-5+e^2+\frac {8}{x}}}{32 x}+e^{-11+e^2} \text {Ei}\left (\frac {8}{x}\right )-e^{-11+e^2+\frac {8}{x}} \log (x)-\frac {3}{256} e^{-5+e^2+\frac {8}{x}} \log (x)-\frac {1}{32} e^{-9+e^2+\frac {8}{x}} \left (4+e^2\right ) \log (x)+\frac {1}{8} e^{-11+e^2+\frac {8}{x}} \left (8+e^2\right ) \log (x)+\frac {1}{256} e^{-7+e^2+\frac {8}{x}} \left (8+3 e^2\right ) \log (x)+\frac {e^{-9+e^2+\frac {8}{x}} \log (x)}{e^2-x}+\frac {e^{-5+e^2+\frac {8}{x}} \log (x)}{x^3}-\frac {3 e^{-5+e^2+\frac {8}{x}} \log (x)}{8 x^2}+\frac {e^{-7+e^2+\frac {8}{x}} \left (8+3 e^2\right ) \log (x)}{8 x^2}+\frac {3 e^{-5+e^2+\frac {8}{x}} \log (x)}{32 x}+\frac {e^{-9+e^2+\frac {8}{x}} \left (4+e^2\right ) \log (x)}{4 x}-\frac {e^{-7+e^2+\frac {8}{x}} \left (8+3 e^2\right ) \log (x)}{32 x}+\frac {1}{32} \int \frac {e^{-5+e^2+\frac {8}{x}}}{x^2} \, dx+\frac {1}{8} \int \frac {e^{-7+e^2+\frac {8}{x}}}{x^2} \, dx+\frac {1}{4} \int \frac {e^{-5+e^2+\frac {8}{x}}}{x^3} \, dx-e^2 \int \frac {e^{-11+e^2+\frac {8}{x}}}{\left (e^2-x\right ) x} \, dx+\int \frac {e^{-11+e^2+\frac {8}{x}}}{x} \, dx-\text {Subst}\left (\int \frac {e^{-11+\frac {8}{e^2}+e^2+\frac {8 x}{e^2}}}{x} \, dx,x,\frac {e^2-x}{x}\right ) \\ & = -\frac {1}{256} e^{-5+e^2+\frac {8}{x}}-e^{-11+\frac {8}{e^2}+e^2} \text {Ei}\left (-\frac {8}{e^2}+\frac {8}{x}\right )-e^{-11+e^2+\frac {8}{x}} \log (x)-\frac {3}{256} e^{-5+e^2+\frac {8}{x}} \log (x)-\frac {1}{32} e^{-9+e^2+\frac {8}{x}} \left (4+e^2\right ) \log (x)+\frac {1}{8} e^{-11+e^2+\frac {8}{x}} \left (8+e^2\right ) \log (x)+\frac {1}{256} e^{-7+e^2+\frac {8}{x}} \left (8+3 e^2\right ) \log (x)+\frac {e^{-9+e^2+\frac {8}{x}} \log (x)}{e^2-x}+\frac {e^{-5+e^2+\frac {8}{x}} \log (x)}{x^3}-\frac {3 e^{-5+e^2+\frac {8}{x}} \log (x)}{8 x^2}+\frac {e^{-7+e^2+\frac {8}{x}} \left (8+3 e^2\right ) \log (x)}{8 x^2}+\frac {3 e^{-5+e^2+\frac {8}{x}} \log (x)}{32 x}+\frac {e^{-9+e^2+\frac {8}{x}} \left (4+e^2\right ) \log (x)}{4 x}-\frac {e^{-7+e^2+\frac {8}{x}} \left (8+3 e^2\right ) \log (x)}{32 x}-\frac {1}{32} \int \frac {e^{-5+e^2+\frac {8}{x}}}{x^2} \, dx+\text {Subst}\left (\int \frac {e^{-11+\frac {8}{e^2}+e^2+\frac {8 x}{e^2}}}{x} \, dx,x,\frac {e^2-x}{x}\right ) \\ & = -e^{-11+e^2+\frac {8}{x}} \log (x)-\frac {3}{256} e^{-5+e^2+\frac {8}{x}} \log (x)-\frac {1}{32} e^{-9+e^2+\frac {8}{x}} \left (4+e^2\right ) \log (x)+\frac {1}{8} e^{-11+e^2+\frac {8}{x}} \left (8+e^2\right ) \log (x)+\frac {1}{256} e^{-7+e^2+\frac {8}{x}} \left (8+3 e^2\right ) \log (x)+\frac {e^{-9+e^2+\frac {8}{x}} \log (x)}{e^2-x}+\frac {e^{-5+e^2+\frac {8}{x}} \log (x)}{x^3}-\frac {3 e^{-5+e^2+\frac {8}{x}} \log (x)}{8 x^2}+\frac {e^{-7+e^2+\frac {8}{x}} \left (8+3 e^2\right ) \log (x)}{8 x^2}+\frac {3 e^{-5+e^2+\frac {8}{x}} \log (x)}{32 x}+\frac {e^{-9+e^2+\frac {8}{x}} \left (4+e^2\right ) \log (x)}{4 x}-\frac {e^{-7+e^2+\frac {8}{x}} \left (8+3 e^2\right ) \log (x)}{32 x} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.30 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \frac {e^{\frac {8-3 x+e^2 x}{x}} \left (e^2 x-x^2\right )+e^{\frac {8-3 x+e^2 x}{x}} \left (e^2 (-8-3 x)+8 x+4 x^2\right ) \log (x)}{e^4 x^5-2 e^2 x^6+x^7} \, dx=\frac {e^{-3+e^2+\frac {8}{x}} \log (x)}{\left (e^2-x\right ) x^3} \]

[In]

Integrate[(E^((8 - 3*x + E^2*x)/x)*(E^2*x - x^2) + E^((8 - 3*x + E^2*x)/x)*(E^2*(-8 - 3*x) + 8*x + 4*x^2)*Log[
x])/(E^4*x^5 - 2*E^2*x^6 + x^7),x]

[Out]

(E^(-3 + E^2 + 8/x)*Log[x])/((E^2 - x)*x^3)

Maple [A] (verified)

Time = 1.17 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.07

method	result	size
parallelrisch	\(\frac {\ln \left (x \right ) {\mathrm e}^{\frac {{\mathrm e}^{2} x -3 x +8}{x}}}{x^{3} \left ({\mathrm e}^{2}-x \right )}\)	\(29\)

[In]

int((((-3*x-8)*exp(2)+4*x^2+8*x)*exp((exp(2)*x-3*x+8)/x)*ln(x)+(exp(2)*x-x^2)*exp((exp(2)*x-3*x+8)/x))/(x^5*ex
p(2)^2-2*x^6*exp(2)+x^7),x,method=_RETURNVERBOSE)

[Out]

1/x^3*ln(x)*exp((exp(2)*x-3*x+8)/x)/(exp(2)-x)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.15 \[ \int \frac {e^{\frac {8-3 x+e^2 x}{x}} \left (e^2 x-x^2\right )+e^{\frac {8-3 x+e^2 x}{x}} \left (e^2 (-8-3 x)+8 x+4 x^2\right ) \log (x)}{e^4 x^5-2 e^2 x^6+x^7} \, dx=-\frac {e^{\left (\frac {x e^{2} - 3 \, x + 8}{x}\right )} \log \left (x\right )}{x^{4} - x^{3} e^{2}} \]

[In]

integrate((((-3*x-8)*exp(2)+4*x^2+8*x)*exp((exp(2)*x-3*x+8)/x)*log(x)+(exp(2)*x-x^2)*exp((exp(2)*x-3*x+8)/x))/
(x^5*exp(2)^2-2*x^6*exp(2)+x^7),x, algorithm="fricas")

[Out]

-e^((x*e^2 - 3*x + 8)/x)*log(x)/(x^4 - x^3*e^2)

Sympy [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \frac {e^{\frac {8-3 x+e^2 x}{x}} \left (e^2 x-x^2\right )+e^{\frac {8-3 x+e^2 x}{x}} \left (e^2 (-8-3 x)+8 x+4 x^2\right ) \log (x)}{e^4 x^5-2 e^2 x^6+x^7} \, dx=- \frac {e^{\frac {- 3 x + x e^{2} + 8}{x}} \log {\left (x \right )}}{x^{4} - x^{3} e^{2}} \]

[In]

integrate((((-3*x-8)*exp(2)+4*x**2+8*x)*exp((exp(2)*x-3*x+8)/x)*ln(x)+(exp(2)*x-x**2)*exp((exp(2)*x-3*x+8)/x))
/(x**5*exp(2)**2-2*x**6*exp(2)+x**7),x)

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.07 \[ \int \frac {e^{\frac {8-3 x+e^2 x}{x}} \left (e^2 x-x^2\right )+e^{\frac {8-3 x+e^2 x}{x}} \left (e^2 (-8-3 x)+8 x+4 x^2\right ) \log (x)}{e^4 x^5-2 e^2 x^6+x^7} \, dx=-\frac {e^{\left (\frac {8}{x} + e^{2}\right )} \log \left (x\right )}{x^{4} e^{3} - x^{3} e^{5}} \]

[In]

integrate((((-3*x-8)*exp(2)+4*x^2+8*x)*exp((exp(2)*x-3*x+8)/x)*log(x)+(exp(2)*x-x^2)*exp((exp(2)*x-3*x+8)/x))/
(x^5*exp(2)^2-2*x^6*exp(2)+x^7),x, algorithm="maxima")

[Out]

-e^(8/x + e^2)*log(x)/(x^4*e^3 - x^3*e^5)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.26 \[ \int \frac {e^{\frac {8-3 x+e^2 x}{x}} \left (e^2 x-x^2\right )+e^{\frac {8-3 x+e^2 x}{x}} \left (e^2 (-8-3 x)+8 x+4 x^2\right ) \log (x)}{e^4 x^5-2 e^2 x^6+x^7} \, dx=-\frac {e^{\left (\frac {x e^{2} - x + 8}{x}\right )} \log \left (x\right )}{x^{4} e^{2} - x^{3} e^{4}} \]

[In]

integrate((((-3*x-8)*exp(2)+4*x^2+8*x)*exp((exp(2)*x-3*x+8)/x)*log(x)+(exp(2)*x-x^2)*exp((exp(2)*x-3*x+8)/x))/
(x^5*exp(2)^2-2*x^6*exp(2)+x^7),x, algorithm="giac")

[Out]

-e^((x*e^2 - x + 8)/x)*log(x)/(x^4*e^2 - x^3*e^4)

Mupad [F(-1)]

Timed out. \[ \int \frac {e^{\frac {8-3 x+e^2 x}{x}} \left (e^2 x-x^2\right )+e^{\frac {8-3 x+e^2 x}{x}} \left (e^2 (-8-3 x)+8 x+4 x^2\right ) \log (x)}{e^4 x^5-2 e^2 x^6+x^7} \, dx=\int \frac {{\mathrm {e}}^{\frac {x\,{\mathrm {e}}^2-3\,x+8}{x}}\,\left (x\,{\mathrm {e}}^2-x^2\right )+{\mathrm {e}}^{\frac {x\,{\mathrm {e}}^2-3\,x+8}{x}}\,\ln \left (x\right )\,\left (8\,x+4\,x^2-{\mathrm {e}}^2\,\left (3\,x+8\right )\right )}{x^7-2\,{\mathrm {e}}^2\,x^6+{\mathrm {e}}^4\,x^5} \,d x \]

[In]

int((exp((x*exp(2) - 3*x + 8)/x)*(x*exp(2) - x^2) + exp((x*exp(2) - 3*x + 8)/x)*log(x)*(8*x + 4*x^2 - exp(2)*(
3*x + 8)))/(x^5*exp(4) - 2*x^6*exp(2) + x^7),x)

[Out]

int((exp((x*exp(2) - 3*x + 8)/x)*(x*exp(2) - x^2) + exp((x*exp(2) - 3*x + 8)/x)*log(x)*(8*x + 4*x^2 - exp(2)*(
3*x + 8)))/(x^5*exp(4) - 2*x^6*exp(2) + x^7), x)