3.60.66 \(\int \frac {e^{4/x} (420 x+24 x^2-12 x^3) \log (\frac {5+x}{x})+\log (x) (e^{4/x} (-420 x+60 x^2)+e^{4/x} (-1680+324 x+12 x^2-24 x^3) \log (\frac {5+x}{x}))}{5 x+x^2} \, dx\)

Optimal. Leaf size=25 \[ 12 e^{4/x} (7-x) x \log \left (1+\frac {5}{x}\right ) \log (x) \]

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Rubi [A]  time = 16.85, antiderivative size = 43, normalized size of antiderivative = 1.72, number of steps used = 175, number of rules used = 16, integrand size = 93, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {1593, 6742, 6688, 2226, 2206, 2210, 2214, 2554, 6483, 6475, 12, 14, 2222, 2228, 2178, 2557} \begin {gather*} 84 e^{4/x} x \log \left (\frac {5}{x}+1\right ) \log (x)-12 e^{4/x} x^2 \log \left (\frac {5}{x}+1\right ) \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^(4/x)*(420*x + 24*x^2 - 12*x^3)*Log[(5 + x)/x] + Log[x]*(E^(4/x)*(-420*x + 60*x^2) + E^(4/x)*(-1680 + 3
24*x + 12*x^2 - 24*x^3)*Log[(5 + x)/x]))/(5*x + x^2),x]

[Out]

84*E^(4/x)*x*Log[1 + 5/x]*Log[x] - 12*E^(4/x)*x^2*Log[1 + 5/x]*Log[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 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 1593

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

Rule 2178

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

Rule 2206

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 2210

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 2214

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^(m
+ 1)*F^(a + b*(c + d*x)^n))/(d*(m + 1)), x] - Dist[(b*n*Log[F])/(m + 1), 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[-4, (m + 1)/n, 5] && IntegerQ[n
] && ((GtQ[n, 0] && LtQ[m, -1]) || (GtQ[-n, 0] && LeQ[-n, m + 1]))

Rule 2222

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 2226

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*(u_), x_Symbol] :> Int[ExpandLinearProduct[F^(a + b*(c + d*
x)^n), u, c, d, x], x] /; FreeQ[{F, a, b, c, d, n}, x] && PolynomialQ[u, x]

Rule 2228

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 2554

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 2557

Int[Log[v_]*Log[w_]*(u_), x_Symbol] :> With[{z = IntHide[u, x]}, Dist[Log[v]*Log[w], z, x] + (-Int[SimplifyInt
egrand[(z*Log[w]*D[v, x])/v, x], x] - Int[SimplifyIntegrand[(z*Log[v]*D[w, x])/w, x], x]) /; InverseFunctionFr
eeQ[z, x]] /; InverseFunctionFreeQ[v, x] && InverseFunctionFreeQ[w, x]

Rule 6475

Int[ExpIntegralE[1, (b_.)*(x_)]/(x_), x_Symbol] :> Simp[b*x*HypergeometricPFQ[{1, 1, 1}, {2, 2, 2}, -(b*x)], x
] + (-Simp[EulerGamma*Log[x], x] - Simp[(1*Log[b*x]^2)/2, x]) /; FreeQ[b, x]

Rule 6483

Int[ExpIntegralEi[(b_.)*(x_)]/(x_), x_Symbol] :> Simp[Log[x]*(ExpIntegralEi[b*x] + ExpIntegralE[1, -(b*x)]), x
] - Int[ExpIntegralE[1, -(b*x)]/x, x] /; FreeQ[b, x]

Rule 6688

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

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 {e^{4/x} \left (420 x+24 x^2-12 x^3\right ) \log \left (\frac {5+x}{x}\right )+\log (x) \left (e^{4/x} \left (-420 x+60 x^2\right )+e^{4/x} \left (-1680+324 x+12 x^2-24 x^3\right ) \log \left (\frac {5+x}{x}\right )\right )}{x (5+x)} \, dx\\ &=\int \left (-\frac {12 e^{4/x} \left (35 x \log (x)-5 x^2 \log (x)-35 x \log \left (\frac {5+x}{x}\right )-2 x^2 \log \left (\frac {5+x}{x}\right )+x^3 \log \left (\frac {5+x}{x}\right )+140 \log (x) \log \left (\frac {5+x}{x}\right )-27 x \log (x) \log \left (\frac {5+x}{x}\right )-x^2 \log (x) \log \left (\frac {5+x}{x}\right )+2 x^3 \log (x) \log \left (\frac {5+x}{x}\right )\right )}{5 x}+\frac {12 e^{4/x} \left (35 x \log (x)-5 x^2 \log (x)-35 x \log \left (\frac {5+x}{x}\right )-2 x^2 \log \left (\frac {5+x}{x}\right )+x^3 \log \left (\frac {5+x}{x}\right )+140 \log (x) \log \left (\frac {5+x}{x}\right )-27 x \log (x) \log \left (\frac {5+x}{x}\right )-x^2 \log (x) \log \left (\frac {5+x}{x}\right )+2 x^3 \log (x) \log \left (\frac {5+x}{x}\right )\right )}{5 (5+x)}\right ) \, dx\\ &=-\left (\frac {12}{5} \int \frac {e^{4/x} \left (35 x \log (x)-5 x^2 \log (x)-35 x \log \left (\frac {5+x}{x}\right )-2 x^2 \log \left (\frac {5+x}{x}\right )+x^3 \log \left (\frac {5+x}{x}\right )+140 \log (x) \log \left (\frac {5+x}{x}\right )-27 x \log (x) \log \left (\frac {5+x}{x}\right )-x^2 \log (x) \log \left (\frac {5+x}{x}\right )+2 x^3 \log (x) \log \left (\frac {5+x}{x}\right )\right )}{x} \, dx\right )+\frac {12}{5} \int \frac {e^{4/x} \left (35 x \log (x)-5 x^2 \log (x)-35 x \log \left (\frac {5+x}{x}\right )-2 x^2 \log \left (\frac {5+x}{x}\right )+x^3 \log \left (\frac {5+x}{x}\right )+140 \log (x) \log \left (\frac {5+x}{x}\right )-27 x \log (x) \log \left (\frac {5+x}{x}\right )-x^2 \log (x) \log \left (\frac {5+x}{x}\right )+2 x^3 \log (x) \log \left (\frac {5+x}{x}\right )\right )}{5+x} \, dx\\ &=-\left (\frac {12}{5} \int \frac {e^{4/x} \left (x \left (-35-2 x+x^2\right ) \log \left (\frac {5+x}{x}\right )+\log (x) \left (-5 (-7+x) x+\left (140-27 x-x^2+2 x^3\right ) \log \left (\frac {5+x}{x}\right )\right )\right )}{x} \, dx\right )+\frac {12}{5} \int \frac {e^{4/x} \left (x \left (-35-2 x+x^2\right ) \log \left (\frac {5+x}{x}\right )+\log (x) \left (-5 (-7+x) x+\left (140-27 x-x^2+2 x^3\right ) \log \left (\frac {5+x}{x}\right )\right )\right )}{5+x} \, dx\\ &=\frac {12}{5} \int \left (-\frac {5 e^{4/x} (-7+x) x \log (x)}{5+x}+e^{4/x} \log \left (1+\frac {5}{x}\right ) \left (-7 x+x^2+28 \log (x)-11 x \log (x)+2 x^2 \log (x)\right )\right ) \, dx-\frac {12}{5} \int \left (-5 e^{4/x} (-7+x) \log (x)+\frac {e^{4/x} (5+x) \log \left (1+\frac {5}{x}\right ) \left (-7 x+x^2+28 \log (x)-11 x \log (x)+2 x^2 \log (x)\right )}{x}\right ) \, dx\\ &=\frac {12}{5} \int e^{4/x} \log \left (1+\frac {5}{x}\right ) \left (-7 x+x^2+28 \log (x)-11 x \log (x)+2 x^2 \log (x)\right ) \, dx-\frac {12}{5} \int \frac {e^{4/x} (5+x) \log \left (1+\frac {5}{x}\right ) \left (-7 x+x^2+28 \log (x)-11 x \log (x)+2 x^2 \log (x)\right )}{x} \, dx+12 \int e^{4/x} (-7+x) \log (x) \, dx-12 \int \frac {e^{4/x} (-7+x) x \log (x)}{5+x} \, dx\\ &=60 e^{4/x} x \log (x)-\frac {720 \text {Ei}\left (\frac {4}{5}+\frac {4}{x}\right ) \log (x)}{e^{4/5}}+480 \text {Ei}\left (\frac {4}{x}\right ) \log (x)+\frac {12}{5} \int \left (e^{4/x} (-7+x) x \log \left (1+\frac {5}{x}\right )+e^{4/x} \left (28-11 x+2 x^2\right ) \log \left (1+\frac {5}{x}\right ) \log (x)\right ) \, dx-\frac {12}{5} \int \left (e^{4/x} \left (-35-2 x+x^2\right ) \log \left (1+\frac {5}{x}\right )+\frac {e^{4/x} \left (140-27 x-x^2+2 x^3\right ) \log \left (1+\frac {5}{x}\right ) \log (x)}{x}\right ) \, dx+12 \int \frac {\frac {1}{2} e^{4/x} (-20+x) x+\frac {60 \text {Ei}\left (\frac {4}{5}+\frac {4}{x}\right )}{e^{4/5}}-20 \text {Ei}\left (\frac {4}{x}\right )}{x} \, dx-12 \int \left (\frac {1}{2} e^{4/x} (-10+x)+\frac {20 \text {Ei}\left (\frac {4}{x}\right )}{x}\right ) \, dx\\ &=60 e^{4/x} x \log (x)-\frac {720 \text {Ei}\left (\frac {4}{5}+\frac {4}{x}\right ) \log (x)}{e^{4/5}}+480 \text {Ei}\left (\frac {4}{x}\right ) \log (x)+\frac {12}{5} \int e^{4/x} (-7+x) x \log \left (1+\frac {5}{x}\right ) \, dx-\frac {12}{5} \int e^{4/x} \left (-35-2 x+x^2\right ) \log \left (1+\frac {5}{x}\right ) \, dx+\frac {12}{5} \int e^{4/x} \left (28-11 x+2 x^2\right ) \log \left (1+\frac {5}{x}\right ) \log (x) \, dx-\frac {12}{5} \int \frac {e^{4/x} \left (140-27 x-x^2+2 x^3\right ) \log \left (1+\frac {5}{x}\right ) \log (x)}{x} \, dx-6 \int e^{4/x} (-10+x) \, dx+12 \int \left (\frac {1}{2} e^{4/x} (-20+x)-\frac {20 \left (-3 \text {Ei}\left (\frac {4}{5}+\frac {4}{x}\right )+e^{4/5} \text {Ei}\left (\frac {4}{x}\right )\right )}{e^{4/5} x}\right ) \, dx-240 \int \frac {\text {Ei}\left (\frac {4}{x}\right )}{x} \, dx\\ &=60 e^{4/x} x \log \left (1+\frac {5}{x}\right )-6 e^{4/x} x^2 \log \left (1+\frac {5}{x}\right )-240 \text {Ei}\left (\frac {4}{x}\right ) \log \left (1+\frac {5}{x}\right )+60 e^{4/x} x \log (x)-\frac {720 \text {Ei}\left (\frac {4}{5}+\frac {4}{x}\right ) \log (x)}{e^{4/5}}+480 \text {Ei}\left (\frac {4}{x}\right ) \log (x)+84 e^{4/x} x \log \left (1+\frac {5}{x}\right ) \log (x)-12 e^{4/x} x^2 \log \left (1+\frac {5}{x}\right ) \log (x)+\frac {12}{5} \int \frac {-5 e^{4/x} x \left (-109-x+x^2\right )-2180 \text {Ei}\left (\frac {4}{x}\right )}{3 x (5+x)} \, dx-\frac {12}{5} \int \frac {5 \left (-e^{4/x} x \left (-68-17 x+2 x^2\right )-272 \text {Ei}\left (\frac {4}{x}\right )\right )}{6 x (5+x)} \, dx-\frac {12}{5} \int \frac {\left (e^{4/x} x \left (68-25 x+4 x^2\right )-272 \text {Ei}\left (\frac {4}{x}\right )\right ) \log \left (1+\frac {5}{x}\right )}{6 x} \, dx+\frac {12}{5} \int \frac {\left (e^{4/x} x \left (-142+5 x+4 x^2\right )-272 \text {Ei}\left (\frac {4}{x}\right )\right ) \log \left (1+\frac {5}{x}\right )}{6 x} \, dx-\frac {12}{5} \int \frac {5 \left (-e^{4/x} x \left (68-25 x+4 x^2\right )+272 \text {Ei}\left (\frac {4}{x}\right )\right ) \log (x)}{6 x (5+x)} \, dx+\frac {12}{5} \int \frac {5 \left (-e^{4/x} x \left (-142+5 x+4 x^2\right )+272 \text {Ei}\left (\frac {4}{x}\right )\right ) \log (x)}{6 x (5+x)} \, dx+6 \int e^{4/x} (-20+x) \, dx-6 \int \left (-10 e^{4/x}+e^{4/x} x\right ) \, dx+240 \operatorname {Subst}\left (\int \frac {\text {Ei}(4 x)}{x} \, dx,x,\frac {1}{x}\right )-\frac {240 \int \frac {-3 \text {Ei}\left (\frac {4}{5}+\frac {4}{x}\right )+e^{4/5} \text {Ei}\left (\frac {4}{x}\right )}{x} \, dx}{e^{4/5}}\\ &=60 e^{4/x} x \log \left (1+\frac {5}{x}\right )-6 e^{4/x} x^2 \log \left (1+\frac {5}{x}\right )-240 \text {Ei}\left (\frac {4}{x}\right ) \log \left (1+\frac {5}{x}\right )+240 \left (E_1\left (-\frac {4}{x}\right )+\text {Ei}\left (\frac {4}{x}\right )\right ) \log \left (\frac {1}{x}\right )+60 e^{4/x} x \log (x)-\frac {720 \text {Ei}\left (\frac {4}{5}+\frac {4}{x}\right ) \log (x)}{e^{4/5}}+480 \text {Ei}\left (\frac {4}{x}\right ) \log (x)+84 e^{4/x} x \log \left (1+\frac {5}{x}\right ) \log (x)-12 e^{4/x} x^2 \log \left (1+\frac {5}{x}\right ) \log (x)-\frac {2}{5} \int \frac {\left (e^{4/x} x \left (68-25 x+4 x^2\right )-272 \text {Ei}\left (\frac {4}{x}\right )\right ) \log \left (1+\frac {5}{x}\right )}{x} \, dx+\frac {2}{5} \int \frac {\left (e^{4/x} x \left (-142+5 x+4 x^2\right )-272 \text {Ei}\left (\frac {4}{x}\right )\right ) \log \left (1+\frac {5}{x}\right )}{x} \, dx+\frac {4}{5} \int \frac {-5 e^{4/x} x \left (-109-x+x^2\right )-2180 \text {Ei}\left (\frac {4}{x}\right )}{x (5+x)} \, dx-2 \int \frac {-e^{4/x} x \left (-68-17 x+2 x^2\right )-272 \text {Ei}\left (\frac {4}{x}\right )}{x (5+x)} \, dx-2 \int \frac {\left (-e^{4/x} x \left (68-25 x+4 x^2\right )+272 \text {Ei}\left (\frac {4}{x}\right )\right ) \log (x)}{x (5+x)} \, dx+2 \int \frac {\left (-e^{4/x} x \left (-142+5 x+4 x^2\right )+272 \text {Ei}\left (\frac {4}{x}\right )\right ) \log (x)}{x (5+x)} \, dx-6 \int e^{4/x} x \, dx+6 \int \left (-20 e^{4/x}+e^{4/x} x\right ) \, dx+60 \int e^{4/x} \, dx-240 \operatorname {Subst}\left (\int \frac {E_1(-4 x)}{x} \, dx,x,\frac {1}{x}\right )-\frac {240 \int \left (-\frac {3 \text {Ei}\left (\frac {4}{5}+\frac {4}{x}\right )}{x}+\frac {e^{4/5} \text {Ei}\left (\frac {4}{x}\right )}{x}\right ) \, dx}{e^{4/5}}\\ &=60 e^{4/x} x-3 e^{4/x} x^2+\frac {960 \, _3F_3\left (1,1,1;2,2,2;\frac {4}{x}\right )}{x}+60 e^{4/x} x \log \left (1+\frac {5}{x}\right )-6 e^{4/x} x^2 \log \left (1+\frac {5}{x}\right )-240 \text {Ei}\left (\frac {4}{x}\right ) \log \left (1+\frac {5}{x}\right )+120 \log ^2\left (-\frac {4}{x}\right )+240 \left (E_1\left (-\frac {4}{x}\right )+\text {Ei}\left (\frac {4}{x}\right )\right ) \log \left (\frac {1}{x}\right )-240 \gamma \log (x)+60 e^{4/x} x \log (x)-\frac {720 \text {Ei}\left (\frac {4}{5}+\frac {4}{x}\right ) \log (x)}{e^{4/5}}+480 \text {Ei}\left (\frac {4}{x}\right ) \log (x)+84 e^{4/x} x \log \left (1+\frac {5}{x}\right ) \log (x)-12 e^{4/x} x^2 \log \left (1+\frac {5}{x}\right ) \log (x)-\frac {2}{5} \int \left (e^{4/x} \left (68-25 x+4 x^2\right ) \log \left (1+\frac {5}{x}\right )-\frac {272 \text {Ei}\left (\frac {4}{x}\right ) \log \left (1+\frac {5}{x}\right )}{x}\right ) \, dx+\frac {2}{5} \int \left (e^{4/x} \left (-142+5 x+4 x^2\right ) \log \left (1+\frac {5}{x}\right )-\frac {272 \text {Ei}\left (\frac {4}{x}\right ) \log \left (1+\frac {5}{x}\right )}{x}\right ) \, dx+\frac {4}{5} \int \left (-\frac {5 e^{4/x} \left (-109-x+x^2\right )}{5+x}-\frac {2180 \text {Ei}\left (\frac {4}{x}\right )}{x (5+x)}\right ) \, dx-2 \int \left (-\frac {e^{4/x} \left (-68-17 x+2 x^2\right )}{5+x}-\frac {272 \text {Ei}\left (\frac {4}{x}\right )}{x (5+x)}\right ) \, dx-2 \int \left (-\frac {e^{4/x} \left (68-25 x+4 x^2\right ) \log (x)}{5+x}+\frac {272 \text {Ei}\left (\frac {4}{x}\right ) \log (x)}{x (5+x)}\right ) \, dx+2 \int \left (-\frac {e^{4/x} \left (-142+5 x+4 x^2\right ) \log (x)}{5+x}+\frac {272 \text {Ei}\left (\frac {4}{x}\right ) \log (x)}{x (5+x)}\right ) \, dx+6 \int e^{4/x} x \, dx-12 \int e^{4/x} \, dx-120 \int e^{4/x} \, dx+240 \int \frac {e^{4/x}}{x} \, dx-240 \int \frac {\text {Ei}\left (\frac {4}{x}\right )}{x} \, dx+\frac {720 \int \frac {\text {Ei}\left (\frac {4}{5}+\frac {4}{x}\right )}{x} \, dx}{e^{4/5}}\\ &=-72 e^{4/x} x-240 \text {Ei}\left (\frac {4}{x}\right )+\frac {960 \, _3F_3\left (1,1,1;2,2,2;\frac {4}{x}\right )}{x}+60 e^{4/x} x \log \left (1+\frac {5}{x}\right )-6 e^{4/x} x^2 \log \left (1+\frac {5}{x}\right )-240 \text {Ei}\left (\frac {4}{x}\right ) \log \left (1+\frac {5}{x}\right )+120 \log ^2\left (-\frac {4}{x}\right )+240 \left (E_1\left (-\frac {4}{x}\right )+\text {Ei}\left (\frac {4}{x}\right )\right ) \log \left (\frac {1}{x}\right )-240 \gamma \log (x)+60 e^{4/x} x \log (x)-\frac {720 \text {Ei}\left (\frac {4}{5}+\frac {4}{x}\right ) \log (x)}{e^{4/5}}+480 \text {Ei}\left (\frac {4}{x}\right ) \log (x)+84 e^{4/x} x \log \left (1+\frac {5}{x}\right ) \log (x)-12 e^{4/x} x^2 \log \left (1+\frac {5}{x}\right ) \log (x)-\frac {2}{5} \int e^{4/x} \left (68-25 x+4 x^2\right ) \log \left (1+\frac {5}{x}\right ) \, dx+\frac {2}{5} \int e^{4/x} \left (-142+5 x+4 x^2\right ) \log \left (1+\frac {5}{x}\right ) \, dx+2 \int \frac {e^{4/x} \left (-68-17 x+2 x^2\right )}{5+x} \, dx+2 \int \frac {e^{4/x} \left (68-25 x+4 x^2\right ) \log (x)}{5+x} \, dx-2 \int \frac {e^{4/x} \left (-142+5 x+4 x^2\right ) \log (x)}{5+x} \, dx-4 \int \frac {e^{4/x} \left (-109-x+x^2\right )}{5+x} \, dx+12 \int e^{4/x} \, dx-48 \int \frac {e^{4/x}}{x} \, dx+240 \operatorname {Subst}\left (\int \frac {\text {Ei}(4 x)}{x} \, dx,x,\frac {1}{x}\right )-480 \int \frac {e^{4/x}}{x} \, dx+544 \int \frac {\text {Ei}\left (\frac {4}{x}\right )}{x (5+x)} \, dx-1744 \int \frac {\text {Ei}\left (\frac {4}{x}\right )}{x (5+x)} \, dx-\frac {720 \operatorname {Subst}\left (\int \frac {\text {Ei}\left (\frac {4}{5}+4 x\right )}{x} \, dx,x,\frac {1}{x}\right )}{e^{4/5}}\\ &=-60 e^{4/x} x+288 \text {Ei}\left (\frac {4}{x}\right )+\frac {960 \, _3F_3\left (1,1,1;2,2,2;\frac {4}{x}\right )}{x}+120 \log ^2\left (-\frac {4}{x}\right )+480 \left (E_1\left (-\frac {4}{x}\right )+\text {Ei}\left (\frac {4}{x}\right )\right ) \log \left (\frac {1}{x}\right )-240 \gamma \log (x)+84 e^{4/x} x \log \left (1+\frac {5}{x}\right ) \log (x)-12 e^{4/x} x^2 \log \left (1+\frac {5}{x}\right ) \log (x)-\frac {2}{5} \int \frac {5 \left (-e^{4/x} x \left (-728+31 x+8 x^2\right )-2912 \text {Ei}\left (\frac {4}{x}\right )\right )}{6 x (5+x)} \, dx+\frac {2}{5} \int \frac {-5 e^{4/x} x \left (172-59 x+8 x^2\right )+3440 \text {Ei}\left (\frac {4}{x}\right )}{6 x (5+x)} \, dx+2 \int \left (-27 e^{4/x}+2 e^{4/x} x+\frac {67 e^{4/x}}{5+x}\right ) \, dx-2 \int \frac {e^{4/x} x (-37+2 x)+\frac {293 \text {Ei}\left (\frac {4}{5}+\frac {4}{x}\right )}{e^{4/5}}-145 \text {Ei}\left (\frac {4}{x}\right )}{x} \, dx+2 \int \frac {e^{4/x} x (-7+2 x)-\frac {67 \text {Ei}\left (\frac {4}{5}+\frac {4}{x}\right )}{e^{4/5}}+95 \text {Ei}\left (\frac {4}{x}\right )}{x} \, dx-4 \int \left (-6 e^{4/x}+e^{4/x} x-\frac {79 e^{4/x}}{5+x}\right ) \, dx+48 \int \frac {e^{4/x}}{x} \, dx-240 \operatorname {Subst}\left (\int \frac {E_1(-4 x)}{x} \, dx,x,\frac {1}{x}\right )+544 \int \left (\frac {\text {Ei}\left (\frac {4}{x}\right )}{5 x}-\frac {\text {Ei}\left (\frac {4}{x}\right )}{5 (5+x)}\right ) \, dx-1744 \int \left (\frac {\text {Ei}\left (\frac {4}{x}\right )}{5 x}-\frac {\text {Ei}\left (\frac {4}{x}\right )}{5 (5+x)}\right ) \, dx-\frac {720 \operatorname {Subst}\left (\int \frac {\text {Ei}\left (\frac {4}{5}+4 x\right )}{x} \, dx,x,\frac {1}{x}\right )}{e^{4/5}}\\ &=-60 e^{4/x} x+240 \text {Ei}\left (\frac {4}{x}\right )+\frac {1920 \, _3F_3\left (1,1,1;2,2,2;\frac {4}{x}\right )}{x}+240 \log ^2\left (-\frac {4}{x}\right )+480 \left (E_1\left (-\frac {4}{x}\right )+\text {Ei}\left (\frac {4}{x}\right )\right ) \log \left (\frac {1}{x}\right )-480 \gamma \log (x)+84 e^{4/x} x \log \left (1+\frac {5}{x}\right ) \log (x)-12 e^{4/x} x^2 \log \left (1+\frac {5}{x}\right ) \log (x)+\frac {1}{15} \int \frac {-5 e^{4/x} x \left (172-59 x+8 x^2\right )+3440 \text {Ei}\left (\frac {4}{x}\right )}{x (5+x)} \, dx-\frac {1}{3} \int \frac {-e^{4/x} x \left (-728+31 x+8 x^2\right )-2912 \text {Ei}\left (\frac {4}{x}\right )}{x (5+x)} \, dx-2 \int \left (e^{4/x} (-37+2 x)+\frac {293 \text {Ei}\left (\frac {4}{5}+\frac {4}{x}\right )-145 e^{4/5} \text {Ei}\left (\frac {4}{x}\right )}{e^{4/5} x}\right ) \, dx+2 \int \left (e^{4/x} (-7+2 x)+\frac {-67 \text {Ei}\left (\frac {4}{5}+\frac {4}{x}\right )+95 e^{4/5} \text {Ei}\left (\frac {4}{x}\right )}{e^{4/5} x}\right ) \, dx+24 \int e^{4/x} \, dx-54 \int e^{4/x} \, dx+\frac {544}{5} \int \frac {\text {Ei}\left (\frac {4}{x}\right )}{x} \, dx-\frac {544}{5} \int \frac {\text {Ei}\left (\frac {4}{x}\right )}{5+x} \, dx+134 \int \frac {e^{4/x}}{5+x} \, dx+316 \int \frac {e^{4/x}}{5+x} \, dx-\frac {1744}{5} \int \frac {\text {Ei}\left (\frac {4}{x}\right )}{x} \, dx+\frac {1744}{5} \int \frac {\text {Ei}\left (\frac {4}{x}\right )}{5+x} \, dx-\frac {720 \operatorname {Subst}\left (\int \frac {\text {Ei}\left (\frac {4}{5}+4 x\right )}{x} \, dx,x,\frac {1}{x}\right )}{e^{4/5}}\\ &=-90 e^{4/x} x+240 \text {Ei}\left (\frac {4}{x}\right )+\frac {1920 \, _3F_3\left (1,1,1;2,2,2;\frac {4}{x}\right )}{x}+240 \log ^2\left (-\frac {4}{x}\right )+480 \left (E_1\left (-\frac {4}{x}\right )+\text {Ei}\left (\frac {4}{x}\right )\right ) \log \left (\frac {1}{x}\right )-480 \gamma \log (x)+84 e^{4/x} x \log \left (1+\frac {5}{x}\right ) \log (x)-12 e^{4/x} x^2 \log \left (1+\frac {5}{x}\right ) \log (x)+\frac {1}{15} \int \left (-\frac {5 e^{4/x} \left (172-59 x+8 x^2\right )}{5+x}+\frac {3440 \text {Ei}\left (\frac {4}{x}\right )}{x (5+x)}\right ) \, dx-\frac {1}{3} \int \left (-\frac {e^{4/x} \left (-728+31 x+8 x^2\right )}{5+x}-\frac {2912 \text {Ei}\left (\frac {4}{x}\right )}{x (5+x)}\right ) \, dx-2 \int e^{4/x} (-37+2 x) \, dx+2 \int e^{4/x} (-7+2 x) \, dx+96 \int \frac {e^{4/x}}{x} \, dx-\frac {544}{5} \int \frac {\text {Ei}\left (\frac {4}{x}\right )}{5+x} \, dx-\frac {544}{5} \operatorname {Subst}\left (\int \frac {\text {Ei}(4 x)}{x} \, dx,x,\frac {1}{x}\right )+134 \int \frac {e^{4/x}}{x} \, dx-216 \int \frac {e^{4/x}}{x} \, dx+316 \int \frac {e^{4/x}}{x} \, dx+\frac {1744}{5} \int \frac {\text {Ei}\left (\frac {4}{x}\right )}{5+x} \, dx+\frac {1744}{5} \operatorname {Subst}\left (\int \frac {\text {Ei}(4 x)}{x} \, dx,x,\frac {1}{x}\right )-670 \int \frac {e^{4/x}}{x (5+x)} \, dx-1580 \int \frac {e^{4/x}}{x (5+x)} \, dx-\frac {2 \int \frac {293 \text {Ei}\left (\frac {4}{5}+\frac {4}{x}\right )-145 e^{4/5} \text {Ei}\left (\frac {4}{x}\right )}{x} \, dx}{e^{4/5}}+\frac {2 \int \frac {-67 \text {Ei}\left (\frac {4}{5}+\frac {4}{x}\right )+95 e^{4/5} \text {Ei}\left (\frac {4}{x}\right )}{x} \, dx}{e^{4/5}}-\frac {720 \operatorname {Subst}\left (\int \frac {\text {Ei}\left (\frac {4}{5}+4 x\right )}{x} \, dx,x,\frac {1}{x}\right )}{e^{4/5}}\\ &=-90 e^{4/x} x-90 \text {Ei}\left (\frac {4}{x}\right )+\frac {1920 \, _3F_3\left (1,1,1;2,2,2;\frac {4}{x}\right )}{x}+240 \log ^2\left (-\frac {4}{x}\right )+720 \left (E_1\left (-\frac {4}{x}\right )+\text {Ei}\left (\frac {4}{x}\right )\right ) \log \left (\frac {1}{x}\right )-480 \gamma \log (x)+84 e^{4/x} x \log \left (1+\frac {5}{x}\right ) \log (x)-12 e^{4/x} x^2 \log \left (1+\frac {5}{x}\right ) \log (x)-\frac {1}{3} \int \frac {e^{4/x} \left (172-59 x+8 x^2\right )}{5+x} \, dx+\frac {1}{3} \int \frac {e^{4/x} \left (-728+31 x+8 x^2\right )}{5+x} \, dx-2 \int \left (-37 e^{4/x}+2 e^{4/x} x\right ) \, dx+2 \int \left (-7 e^{4/x}+2 e^{4/x} x\right ) \, dx-\frac {544}{5} \int \frac {\text {Ei}\left (\frac {4}{x}\right )}{5+x} \, dx+\frac {544}{5} \operatorname {Subst}\left (\int \frac {E_1(-4 x)}{x} \, dx,x,\frac {1}{x}\right )+134 \operatorname {Subst}\left (\int \frac {e^{-\frac {4}{5}+\frac {4 x}{5}}}{x} \, dx,x,\frac {5+x}{x}\right )+\frac {688}{3} \int \frac {\text {Ei}\left (\frac {4}{x}\right )}{x (5+x)} \, dx+316 \operatorname {Subst}\left (\int \frac {e^{-\frac {4}{5}+\frac {4 x}{5}}}{x} \, dx,x,\frac {5+x}{x}\right )+\frac {1744}{5} \int \frac {\text {Ei}\left (\frac {4}{x}\right )}{5+x} \, dx-\frac {1744}{5} \operatorname {Subst}\left (\int \frac {E_1(-4 x)}{x} \, dx,x,\frac {1}{x}\right )+\frac {2912}{3} \int \frac {\text {Ei}\left (\frac {4}{x}\right )}{x (5+x)} \, dx-\frac {2 \int \left (\frac {293 \text {Ei}\left (\frac {4}{5}+\frac {4}{x}\right )}{x}-\frac {145 e^{4/5} \text {Ei}\left (\frac {4}{x}\right )}{x}\right ) \, dx}{e^{4/5}}+\frac {2 \int \left (-\frac {67 \text {Ei}\left (\frac {4}{5}+\frac {4}{x}\right )}{x}+\frac {95 e^{4/5} \text {Ei}\left (\frac {4}{x}\right )}{x}\right ) \, dx}{e^{4/5}}-\frac {720 \operatorname {Subst}\left (\int \frac {\text {Ei}\left (\frac {4}{5}+4 x\right )}{x} \, dx,x,\frac {1}{x}\right )}{e^{4/5}}\\ &=-90 e^{4/x} x+\frac {450 \text {Ei}\left (\frac {4}{5}+\frac {4}{x}\right )}{e^{4/5}}-90 \text {Ei}\left (\frac {4}{x}\right )+\frac {2880 \, _3F_3\left (1,1,1;2,2,2;\frac {4}{x}\right )}{x}+360 \log ^2\left (-\frac {4}{x}\right )+720 \left (E_1\left (-\frac {4}{x}\right )+\text {Ei}\left (\frac {4}{x}\right )\right ) \log \left (\frac {1}{x}\right )-720 \gamma \log (x)+84 e^{4/x} x \log \left (1+\frac {5}{x}\right ) \log (x)-12 e^{4/x} x^2 \log \left (1+\frac {5}{x}\right ) \log (x)+\frac {1}{3} \int \left (-9 e^{4/x}+8 e^{4/x} x-\frac {683 e^{4/x}}{5+x}\right ) \, dx-\frac {1}{3} \int \left (-99 e^{4/x}+8 e^{4/x} x+\frac {667 e^{4/x}}{5+x}\right ) \, dx-14 \int e^{4/x} \, dx+74 \int e^{4/x} \, dx-\frac {544}{5} \int \frac {\text {Ei}\left (\frac {4}{x}\right )}{5+x} \, dx+190 \int \frac {\text {Ei}\left (\frac {4}{x}\right )}{x} \, dx+\frac {688}{3} \int \left (\frac {\text {Ei}\left (\frac {4}{x}\right )}{5 x}-\frac {\text {Ei}\left (\frac {4}{x}\right )}{5 (5+x)}\right ) \, dx+290 \int \frac {\text {Ei}\left (\frac {4}{x}\right )}{x} \, dx+\frac {1744}{5} \int \frac {\text {Ei}\left (\frac {4}{x}\right )}{5+x} \, dx+\frac {2912}{3} \int \left (\frac {\text {Ei}\left (\frac {4}{x}\right )}{5 x}-\frac {\text {Ei}\left (\frac {4}{x}\right )}{5 (5+x)}\right ) \, dx-\frac {134 \int \frac {\text {Ei}\left (\frac {4}{5}+\frac {4}{x}\right )}{x} \, dx}{e^{4/5}}-\frac {586 \int \frac {\text {Ei}\left (\frac {4}{5}+\frac {4}{x}\right )}{x} \, dx}{e^{4/5}}-\frac {720 \operatorname {Subst}\left (\int \frac {\text {Ei}\left (\frac {4}{5}+4 x\right )}{x} \, dx,x,\frac {1}{x}\right )}{e^{4/5}}\\ &=-30 e^{4/x} x+\frac {450 \text {Ei}\left (\frac {4}{5}+\frac {4}{x}\right )}{e^{4/5}}-90 \text {Ei}\left (\frac {4}{x}\right )+\frac {2880 \, _3F_3\left (1,1,1;2,2,2;\frac {4}{x}\right )}{x}+360 \log ^2\left (-\frac {4}{x}\right )+720 \left (E_1\left (-\frac {4}{x}\right )+\text {Ei}\left (\frac {4}{x}\right )\right ) \log \left (\frac {1}{x}\right )-720 \gamma \log (x)+84 e^{4/x} x \log \left (1+\frac {5}{x}\right ) \log (x)-12 e^{4/x} x^2 \log \left (1+\frac {5}{x}\right ) \log (x)-3 \int e^{4/x} \, dx+33 \int e^{4/x} \, dx+\frac {688}{15} \int \frac {\text {Ei}\left (\frac {4}{x}\right )}{x} \, dx-\frac {688}{15} \int \frac {\text {Ei}\left (\frac {4}{x}\right )}{5+x} \, dx-56 \int \frac {e^{4/x}}{x} \, dx-\frac {544}{5} \int \frac {\text {Ei}\left (\frac {4}{x}\right )}{5+x} \, dx-190 \operatorname {Subst}\left (\int \frac {\text {Ei}(4 x)}{x} \, dx,x,\frac {1}{x}\right )+\frac {2912}{15} \int \frac {\text {Ei}\left (\frac {4}{x}\right )}{x} \, dx-\frac {2912}{15} \int \frac {\text {Ei}\left (\frac {4}{x}\right )}{5+x} \, dx-\frac {667}{3} \int \frac {e^{4/x}}{5+x} \, dx-\frac {683}{3} \int \frac {e^{4/x}}{5+x} \, dx-290 \operatorname {Subst}\left (\int \frac {\text {Ei}(4 x)}{x} \, dx,x,\frac {1}{x}\right )+296 \int \frac {e^{4/x}}{x} \, dx+\frac {1744}{5} \int \frac {\text {Ei}\left (\frac {4}{x}\right )}{5+x} \, dx+\frac {134 \operatorname {Subst}\left (\int \frac {\text {Ei}\left (\frac {4}{5}+4 x\right )}{x} \, dx,x,\frac {1}{x}\right )}{e^{4/5}}+\frac {586 \operatorname {Subst}\left (\int \frac {\text {Ei}\left (\frac {4}{5}+4 x\right )}{x} \, dx,x,\frac {1}{x}\right )}{e^{4/5}}-\frac {720 \operatorname {Subst}\left (\int \frac {\text {Ei}\left (\frac {4}{5}+4 x\right )}{x} \, dx,x,\frac {1}{x}\right )}{e^{4/5}}\\ &=\frac {450 \text {Ei}\left (\frac {4}{5}+\frac {4}{x}\right )}{e^{4/5}}-330 \text {Ei}\left (\frac {4}{x}\right )+\frac {2880 \, _3F_3\left (1,1,1;2,2,2;\frac {4}{x}\right )}{x}+360 \log ^2\left (-\frac {4}{x}\right )+240 \left (E_1\left (-\frac {4}{x}\right )+\text {Ei}\left (\frac {4}{x}\right )\right ) \log \left (\frac {1}{x}\right )-720 \gamma \log (x)+84 e^{4/x} x \log \left (1+\frac {5}{x}\right ) \log (x)-12 e^{4/x} x^2 \log \left (1+\frac {5}{x}\right ) \log (x)-12 \int \frac {e^{4/x}}{x} \, dx-\frac {688}{15} \int \frac {\text {Ei}\left (\frac {4}{x}\right )}{5+x} \, dx-\frac {688}{15} \operatorname {Subst}\left (\int \frac {\text {Ei}(4 x)}{x} \, dx,x,\frac {1}{x}\right )-\frac {544}{5} \int \frac {\text {Ei}\left (\frac {4}{x}\right )}{5+x} \, dx+132 \int \frac {e^{4/x}}{x} \, dx+190 \operatorname {Subst}\left (\int \frac {E_1(-4 x)}{x} \, dx,x,\frac {1}{x}\right )-\frac {2912}{15} \int \frac {\text {Ei}\left (\frac {4}{x}\right )}{5+x} \, dx-\frac {2912}{15} \operatorname {Subst}\left (\int \frac {\text {Ei}(4 x)}{x} \, dx,x,\frac {1}{x}\right )-\frac {667}{3} \int \frac {e^{4/x}}{x} \, dx-\frac {683}{3} \int \frac {e^{4/x}}{x} \, dx+290 \operatorname {Subst}\left (\int \frac {E_1(-4 x)}{x} \, dx,x,\frac {1}{x}\right )+\frac {1744}{5} \int \frac {\text {Ei}\left (\frac {4}{x}\right )}{5+x} \, dx+\frac {3335}{3} \int \frac {e^{4/x}}{x (5+x)} \, dx+\frac {3415}{3} \int \frac {e^{4/x}}{x (5+x)} \, dx+\frac {134 \operatorname {Subst}\left (\int \frac {\text {Ei}\left (\frac {4}{5}+4 x\right )}{x} \, dx,x,\frac {1}{x}\right )}{e^{4/5}}+\frac {586 \operatorname {Subst}\left (\int \frac {\text {Ei}\left (\frac {4}{5}+4 x\right )}{x} \, dx,x,\frac {1}{x}\right )}{e^{4/5}}-\frac {720 \operatorname {Subst}\left (\int \frac {\text {Ei}\left (\frac {4}{5}+4 x\right )}{x} \, dx,x,\frac {1}{x}\right )}{e^{4/5}}\\ &=\frac {450 \text {Ei}\left (\frac {4}{5}+\frac {4}{x}\right )}{e^{4/5}}+\frac {960 \, _3F_3\left (1,1,1;2,2,2;\frac {4}{x}\right )}{x}+120 \log ^2\left (-\frac {4}{x}\right )-240 \gamma \log (x)+84 e^{4/x} x \log \left (1+\frac {5}{x}\right ) \log (x)-12 e^{4/x} x^2 \log \left (1+\frac {5}{x}\right ) \log (x)-\frac {688}{15} \int \frac {\text {Ei}\left (\frac {4}{x}\right )}{5+x} \, dx+\frac {688}{15} \operatorname {Subst}\left (\int \frac {E_1(-4 x)}{x} \, dx,x,\frac {1}{x}\right )-\frac {544}{5} \int \frac {\text {Ei}\left (\frac {4}{x}\right )}{5+x} \, dx-\frac {2912}{15} \int \frac {\text {Ei}\left (\frac {4}{x}\right )}{5+x} \, dx+\frac {2912}{15} \operatorname {Subst}\left (\int \frac {E_1(-4 x)}{x} \, dx,x,\frac {1}{x}\right )-\frac {667}{3} \operatorname {Subst}\left (\int \frac {e^{-\frac {4}{5}+\frac {4 x}{5}}}{x} \, dx,x,\frac {5+x}{x}\right )-\frac {683}{3} \operatorname {Subst}\left (\int \frac {e^{-\frac {4}{5}+\frac {4 x}{5}}}{x} \, dx,x,\frac {5+x}{x}\right )+\frac {1744}{5} \int \frac {\text {Ei}\left (\frac {4}{x}\right )}{5+x} \, dx+\frac {134 \operatorname {Subst}\left (\int \frac {\text {Ei}\left (\frac {4}{5}+4 x\right )}{x} \, dx,x,\frac {1}{x}\right )}{e^{4/5}}+\frac {586 \operatorname {Subst}\left (\int \frac {\text {Ei}\left (\frac {4}{5}+4 x\right )}{x} \, dx,x,\frac {1}{x}\right )}{e^{4/5}}-\frac {720 \operatorname {Subst}\left (\int \frac {\text {Ei}\left (\frac {4}{5}+4 x\right )}{x} \, dx,x,\frac {1}{x}\right )}{e^{4/5}}\\ &=84 e^{4/x} x \log \left (1+\frac {5}{x}\right ) \log (x)-12 e^{4/x} x^2 \log \left (1+\frac {5}{x}\right ) \log (x)-\frac {688}{15} \int \frac {\text {Ei}\left (\frac {4}{x}\right )}{5+x} \, dx-\frac {544}{5} \int \frac {\text {Ei}\left (\frac {4}{x}\right )}{5+x} \, dx-\frac {2912}{15} \int \frac {\text {Ei}\left (\frac {4}{x}\right )}{5+x} \, dx+\frac {1744}{5} \int \frac {\text {Ei}\left (\frac {4}{x}\right )}{5+x} \, dx+\frac {134 \operatorname {Subst}\left (\int \frac {\text {Ei}\left (\frac {4}{5}+4 x\right )}{x} \, dx,x,\frac {1}{x}\right )}{e^{4/5}}+\frac {586 \operatorname {Subst}\left (\int \frac {\text {Ei}\left (\frac {4}{5}+4 x\right )}{x} \, dx,x,\frac {1}{x}\right )}{e^{4/5}}-\frac {720 \operatorname {Subst}\left (\int \frac {\text {Ei}\left (\frac {4}{5}+4 x\right )}{x} \, dx,x,\frac {1}{x}\right )}{e^{4/5}}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 5.06, size = 23, normalized size = 0.92 \begin {gather*} -12 e^{4/x} (-7+x) x \log (x) \log \left (\frac {5+x}{x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(4/x)*(420*x + 24*x^2 - 12*x^3)*Log[(5 + x)/x] + Log[x]*(E^(4/x)*(-420*x + 60*x^2) + E^(4/x)*(-16
80 + 324*x + 12*x^2 - 24*x^3)*Log[(5 + x)/x]))/(5*x + x^2),x]

[Out]

-12*E^(4/x)*(-7 + x)*x*Log[x]*Log[(5 + x)/x]

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fricas [A]  time = 0.87, size = 25, normalized size = 1.00 \begin {gather*} -12 \, {\left (x^{2} - 7 \, x\right )} e^{\frac {4}{x}} \log \relax (x) \log \left (\frac {x + 5}{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-24*x^3+12*x^2+324*x-1680)*exp(4/x)*log(1/x*(5+x))+(60*x^2-420*x)*exp(4/x))*log(x)+(-12*x^3+24*x^
2+420*x)*exp(4/x)*log(1/x*(5+x)))/(x^2+5*x),x, algorithm="fricas")

[Out]

-12*(x^2 - 7*x)*e^(4/x)*log(x)*log((x + 5)/x)

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-24*x^3+12*x^2+324*x-1680)*exp(4/x)*log(1/x*(5+x))+(60*x^2-420*x)*exp(4/x))*log(x)+(-12*x^3+24*x^
2+420*x)*exp(4/x)*log(1/x*(5+x)))/(x^2+5*x),x, algorithm="giac")

[Out]

Timed out

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maple [C]  time = 0.34, size = 308, normalized size = 12.32

method	result	size
risch	\(-12 x \left (x -7\right ) {\mathrm e}^{\frac {4}{x}} \ln \relax (x ) \ln \left (5+x \right )-6 i \pi \,x^{2} \mathrm {csgn}\left (i \left (5+x \right )\right ) \mathrm {csgn}\left (\frac {i \left (5+x \right )}{x}\right )^{2} {\mathrm e}^{\frac {4}{x}} \ln \relax (x )+6 i \pi \,x^{2} \mathrm {csgn}\left (i \left (5+x \right )\right ) \mathrm {csgn}\left (\frac {i \left (5+x \right )}{x}\right ) \mathrm {csgn}\left (\frac {i}{x}\right ) {\mathrm e}^{\frac {4}{x}} \ln \relax (x )+6 i \pi \,x^{2} \mathrm {csgn}\left (\frac {i \left (5+x \right )}{x}\right )^{3} {\mathrm e}^{\frac {4}{x}} \ln \relax (x )-6 i \pi \,x^{2} \mathrm {csgn}\left (\frac {i \left (5+x \right )}{x}\right )^{2} \mathrm {csgn}\left (\frac {i}{x}\right ) {\mathrm e}^{\frac {4}{x}} \ln \relax (x )+42 i \pi x \,\mathrm {csgn}\left (i \left (5+x \right )\right ) \mathrm {csgn}\left (\frac {i \left (5+x \right )}{x}\right )^{2} {\mathrm e}^{\frac {4}{x}} \ln \relax (x )-42 i \pi x \,\mathrm {csgn}\left (i \left (5+x \right )\right ) \mathrm {csgn}\left (\frac {i \left (5+x \right )}{x}\right ) \mathrm {csgn}\left (\frac {i}{x}\right ) {\mathrm e}^{\frac {4}{x}} \ln \relax (x )-42 i \pi x \mathrm {csgn}\left (\frac {i \left (5+x \right )}{x}\right )^{3} {\mathrm e}^{\frac {4}{x}} \ln \relax (x )+42 i \pi x \mathrm {csgn}\left (\frac {i \left (5+x \right )}{x}\right )^{2} \mathrm {csgn}\left (\frac {i}{x}\right ) {\mathrm e}^{\frac {4}{x}} \ln \relax (x )+12 x^{2} {\mathrm e}^{\frac {4}{x}} \ln \relax (x )^{2}-84 x \,{\mathrm e}^{\frac {4}{x}} \ln \relax (x )^{2}\)	\(308\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-24*x^3+12*x^2+324*x-1680)*exp(4/x)*ln(1/x*(5+x))+(60*x^2-420*x)*exp(4/x))*ln(x)+(-12*x^3+24*x^2+420*x)
*exp(4/x)*ln(1/x*(5+x)))/(x^2+5*x),x,method=_RETURNVERBOSE)

[Out]

-12*x*(x-7)*exp(4/x)*ln(x)*ln(5+x)-6*I*Pi*x^2*csgn(I*(5+x))*csgn(I/x*(5+x))^2*exp(4/x)*ln(x)+6*I*Pi*x^2*csgn(I
*(5+x))*csgn(I/x*(5+x))*csgn(I/x)*exp(4/x)*ln(x)+6*I*Pi*x^2*csgn(I/x*(5+x))^3*exp(4/x)*ln(x)-6*I*Pi*x^2*csgn(I
/x*(5+x))^2*csgn(I/x)*exp(4/x)*ln(x)+42*I*Pi*x*csgn(I*(5+x))*csgn(I/x*(5+x))^2*exp(4/x)*ln(x)-42*I*Pi*x*csgn(I
*(5+x))*csgn(I/x*(5+x))*csgn(I/x)*exp(4/x)*ln(x)-42*I*Pi*x*csgn(I/x*(5+x))^3*exp(4/x)*ln(x)+42*I*Pi*x*csgn(I/x
*(5+x))^2*csgn(I/x)*exp(4/x)*ln(x)+12*x^2*exp(4/x)*ln(x)^2-84*x*exp(4/x)*ln(x)^2

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maxima [A]  time = 0.40, size = 36, normalized size = 1.44 \begin {gather*} -12 \, {\left ({\left (x^{2} - 7 \, x\right )} \log \left (x + 5\right ) \log \relax (x) - {\left (x^{2} - 7 \, x\right )} \log \relax (x)^{2}\right )} e^{\frac {4}{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-24*x^3+12*x^2+324*x-1680)*exp(4/x)*log(1/x*(5+x))+(60*x^2-420*x)*exp(4/x))*log(x)+(-12*x^3+24*x^
2+420*x)*exp(4/x)*log(1/x*(5+x)))/(x^2+5*x),x, algorithm="maxima")

[Out]

-12*((x^2 - 7*x)*log(x + 5)*log(x) - (x^2 - 7*x)*log(x)^2)*e^(4/x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} -\int \frac {\ln \relax (x)\,\left ({\mathrm {e}}^{4/x}\,\left (420\,x-60\,x^2\right )-{\mathrm {e}}^{4/x}\,\ln \left (\frac {x+5}{x}\right )\,\left (-24\,x^3+12\,x^2+324\,x-1680\right )\right )-{\mathrm {e}}^{4/x}\,\ln \left (\frac {x+5}{x}\right )\,\left (-12\,x^3+24\,x^2+420\,x\right )}{x^2+5\,x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log(x)*(exp(4/x)*(420*x - 60*x^2) - exp(4/x)*log((x + 5)/x)*(324*x + 12*x^2 - 24*x^3 - 1680)) - exp(4/x)
*log((x + 5)/x)*(420*x + 24*x^2 - 12*x^3))/(5*x + x^2),x)

[Out]

-int((log(x)*(exp(4/x)*(420*x - 60*x^2) - exp(4/x)*log((x + 5)/x)*(324*x + 12*x^2 - 24*x^3 - 1680)) - exp(4/x)
*log((x + 5)/x)*(420*x + 24*x^2 - 12*x^3))/(5*x + x^2), x)

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sympy [A]  time = 6.85, size = 32, normalized size = 1.28 \begin {gather*} \left (- 12 x^{2} \log {\relax (x )} \log {\left (\frac {x + 5}{x} \right )} + 84 x \log {\relax (x )} \log {\left (\frac {x + 5}{x} \right )}\right ) e^{\frac {4}{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-24*x**3+12*x**2+324*x-1680)*exp(4/x)*ln(1/x*(5+x))+(60*x**2-420*x)*exp(4/x))*ln(x)+(-12*x**3+24*
x**2+420*x)*exp(4/x)*ln(1/x*(5+x)))/(x**2+5*x),x)

[Out]

(-12*x**2*log(x)*log((x + 5)/x) + 84*x*log(x)*log((x + 5)/x))*exp(4/x)

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