3.9.16 \(\int \frac {(-780+320 x-32 x^2) \log (\frac {6400-2560 x+256 x^2}{e^3 (114075 x^2-46800 x^3+4800 x^4)})}{195 x-79 x^2+8 x^3} \, dx\)

Optimal. Leaf size=28 \[ \log ^2\left (\frac {4}{3 e^3 \left (5-\frac {5}{8 (5-x)}\right )^2 x^2}\right ) \]

________________________________________________________________________________________

Rubi [B]  time = 1.58, antiderivative size = 203, normalized size of antiderivative = 7.25, number of steps used = 48, number of rules used = 14, integrand size = 60, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.233, Rules used = {1594, 2528, 2524, 12, 27, 2418, 2390, 2301, 2394, 2315, 2393, 2391, 2357, 2316} \begin {gather*} 4 \log \left (\frac {256 \left (x^2-10 x+25\right )}{75 e^3 \left (64 x^4-624 x^3+1521 x^2\right )}\right ) \log (x-5)-4 \log (x) \log \left (\frac {256 \left (x^2-10 x+25\right )}{75 e^3 \left (64 x^4-624 x^3+1521 x^2\right )}\right )-4 \log (8 x-39) \log \left (\frac {256 \left (x^2-10 x+25\right )}{75 e^3 \left (64 x^4-624 x^3+1521 x^2\right )}\right )-4 \log ^2(x-5)-4 \log ^2(x)-4 \log ^2(8 x-39)+8 \log \left (\frac {x}{5}\right ) \log (x-5)+8 \log (8 x-39) \log (x-5)+8 \log (5) \log (x-5)-8 \log \left (\frac {8 x}{39}\right ) \log (8 x-39)-8 \log \left (\frac {39}{8}\right ) \log (8 x-39) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((-780 + 320*x - 32*x^2)*Log[(6400 - 2560*x + 256*x^2)/(E^3*(114075*x^2 - 46800*x^3 + 4800*x^4))])/(195*x
- 79*x^2 + 8*x^3),x]

[Out]

8*Log[5]*Log[-5 + x] - 4*Log[-5 + x]^2 + 8*Log[-5 + x]*Log[x/5] - 4*Log[x]^2 - 8*Log[39/8]*Log[-39 + 8*x] + 8*
Log[-5 + x]*Log[-39 + 8*x] - 8*Log[(8*x)/39]*Log[-39 + 8*x] - 4*Log[-39 + 8*x]^2 + 4*Log[-5 + x]*Log[(256*(25
- 10*x + x^2))/(75*E^3*(1521*x^2 - 624*x^3 + 64*x^4))] - 4*Log[x]*Log[(256*(25 - 10*x + x^2))/(75*E^3*(1521*x^
2 - 624*x^3 + 64*x^4))] - 4*Log[-39 + 8*x]*Log[(256*(25 - 10*x + x^2))/(75*E^3*(1521*x^2 - 624*x^3 + 64*x^4))]

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 1594

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 2301

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]

Rule 2315

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[PolyLog[2, 1 - c*x]/e, x] /; FreeQ[{c, d, e}, x] &
& EqQ[e + c*d, 0]

Rule 2316

Int[((a_.) + Log[(c_.)*(x_)]*(b_.))/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[((a + b*Log[-((c*d)/e)])*Log[d + e*
x])/e, x] + Dist[b, Int[Log[-((e*x)/d)]/(d + e*x), x], x] /; FreeQ[{a, b, c, d, e}, x] && GtQ[-((c*d)/e), 0]

Rule 2357

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(RFx_), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*Log[c*x^
n])^p, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, n}, x] && RationalFunctionQ[RFx, x] && IGtQ[p, 0]

Rule 2390

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_) + (g_.)*(x_))^(q_.), x_Symbol] :> Dist[1/
e, Subst[Int[((f*x)/d)^q*(a + b*Log[c*x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p, q}, x]
&& EqQ[e*f - d*g, 0]

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 2393

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Dist[1/g, Subst[Int[(a +
b*Log[1 + (c*e*x)/g])/x, x], x, f + g*x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && EqQ[g
+ c*(e*f - d*g), 0]

Rule 2394

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

Rule 2418

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*(RFx_), x_Symbol] :> With[{u = ExpandIntegrand[
(a + b*Log[c*(d + e*x)^n])^p, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, n}, x] && RationalFunct
ionQ[RFx, x] && IntegerQ[p]

Rule 2524

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[(Log[d + e*x]*(a + b
*Log[c*RFx^p])^n)/e, x] - Dist[(b*n*p)/e, Int[(Log[d + e*x]*(a + b*Log[c*RFx^p])^(n - 1)*D[RFx, x])/RFx, x], x
] /; FreeQ[{a, b, c, d, e, p}, x] && RationalFunctionQ[RFx, x] && IGtQ[n, 0]

Rule 2528

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)*(RGx_), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*Log[c*
RFx^p])^n, RGx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, p}, x] && RationalFunctionQ[RFx, x] && RationalF
unctionQ[RGx, x] && IGtQ[n, 0]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\left (-780+320 x-32 x^2\right ) \log \left (\frac {6400-2560 x+256 x^2}{e^3 \left (114075 x^2-46800 x^3+4800 x^4\right )}\right )}{x \left (195-79 x+8 x^2\right )} \, dx\\ &=\int \left (\frac {4 \log \left (\frac {6400-2560 x+256 x^2}{e^3 \left (114075 x^2-46800 x^3+4800 x^4\right )}\right )}{-5+x}-\frac {4 \log \left (\frac {6400-2560 x+256 x^2}{e^3 \left (114075 x^2-46800 x^3+4800 x^4\right )}\right )}{x}-\frac {32 \log \left (\frac {6400-2560 x+256 x^2}{e^3 \left (114075 x^2-46800 x^3+4800 x^4\right )}\right )}{-39+8 x}\right ) \, dx\\ &=4 \int \frac {\log \left (\frac {6400-2560 x+256 x^2}{e^3 \left (114075 x^2-46800 x^3+4800 x^4\right )}\right )}{-5+x} \, dx-4 \int \frac {\log \left (\frac {6400-2560 x+256 x^2}{e^3 \left (114075 x^2-46800 x^3+4800 x^4\right )}\right )}{x} \, dx-32 \int \frac {\log \left (\frac {6400-2560 x+256 x^2}{e^3 \left (114075 x^2-46800 x^3+4800 x^4\right )}\right )}{-39+8 x} \, dx\\ &=4 \log (-5+x) \log \left (\frac {256 \left (25-10 x+x^2\right )}{75 e^3 \left (1521 x^2-624 x^3+64 x^4\right )}\right )-4 \log (x) \log \left (\frac {256 \left (25-10 x+x^2\right )}{75 e^3 \left (1521 x^2-624 x^3+64 x^4\right )}\right )-4 \log (-39+8 x) \log \left (\frac {256 \left (25-10 x+x^2\right )}{75 e^3 \left (1521 x^2-624 x^3+64 x^4\right )}\right )-4 \int \frac {e^3 \left (114075 x^2-46800 x^3+4800 x^4\right ) \left (-\frac {\left (6400-2560 x+256 x^2\right ) \left (228150 x-140400 x^2+19200 x^3\right )}{e^3 \left (114075 x^2-46800 x^3+4800 x^4\right )^2}+\frac {-2560+512 x}{e^3 \left (114075 x^2-46800 x^3+4800 x^4\right )}\right ) \log (-5+x)}{6400-2560 x+256 x^2} \, dx+4 \int \frac {e^3 \left (114075 x^2-46800 x^3+4800 x^4\right ) \left (-\frac {\left (6400-2560 x+256 x^2\right ) \left (228150 x-140400 x^2+19200 x^3\right )}{e^3 \left (114075 x^2-46800 x^3+4800 x^4\right )^2}+\frac {-2560+512 x}{e^3 \left (114075 x^2-46800 x^3+4800 x^4\right )}\right ) \log (x)}{6400-2560 x+256 x^2} \, dx+4 \int \frac {e^3 \left (114075 x^2-46800 x^3+4800 x^4\right ) \left (-\frac {\left (6400-2560 x+256 x^2\right ) \left (228150 x-140400 x^2+19200 x^3\right )}{e^3 \left (114075 x^2-46800 x^3+4800 x^4\right )^2}+\frac {-2560+512 x}{e^3 \left (114075 x^2-46800 x^3+4800 x^4\right )}\right ) \log (-39+8 x)}{6400-2560 x+256 x^2} \, dx\\ &=4 \log (-5+x) \log \left (\frac {256 \left (25-10 x+x^2\right )}{75 e^3 \left (1521 x^2-624 x^3+64 x^4\right )}\right )-4 \log (x) \log \left (\frac {256 \left (25-10 x+x^2\right )}{75 e^3 \left (1521 x^2-624 x^3+64 x^4\right )}\right )-4 \log (-39+8 x) \log \left (\frac {256 \left (25-10 x+x^2\right )}{75 e^3 \left (1521 x^2-624 x^3+64 x^4\right )}\right )-\left (4 e^3\right ) \int \frac {\left (114075 x^2-46800 x^3+4800 x^4\right ) \left (-\frac {\left (6400-2560 x+256 x^2\right ) \left (228150 x-140400 x^2+19200 x^3\right )}{e^3 \left (114075 x^2-46800 x^3+4800 x^4\right )^2}+\frac {-2560+512 x}{e^3 \left (114075 x^2-46800 x^3+4800 x^4\right )}\right ) \log (-5+x)}{6400-2560 x+256 x^2} \, dx+\left (4 e^3\right ) \int \frac {\left (114075 x^2-46800 x^3+4800 x^4\right ) \left (-\frac {\left (6400-2560 x+256 x^2\right ) \left (228150 x-140400 x^2+19200 x^3\right )}{e^3 \left (114075 x^2-46800 x^3+4800 x^4\right )^2}+\frac {-2560+512 x}{e^3 \left (114075 x^2-46800 x^3+4800 x^4\right )}\right ) \log (x)}{6400-2560 x+256 x^2} \, dx+\left (4 e^3\right ) \int \frac {\left (114075 x^2-46800 x^3+4800 x^4\right ) \left (-\frac {\left (6400-2560 x+256 x^2\right ) \left (228150 x-140400 x^2+19200 x^3\right )}{e^3 \left (114075 x^2-46800 x^3+4800 x^4\right )^2}+\frac {-2560+512 x}{e^3 \left (114075 x^2-46800 x^3+4800 x^4\right )}\right ) \log (-39+8 x)}{6400-2560 x+256 x^2} \, dx\\ &=4 \log (-5+x) \log \left (\frac {256 \left (25-10 x+x^2\right )}{75 e^3 \left (1521 x^2-624 x^3+64 x^4\right )}\right )-4 \log (x) \log \left (\frac {256 \left (25-10 x+x^2\right )}{75 e^3 \left (1521 x^2-624 x^3+64 x^4\right )}\right )-4 \log (-39+8 x) \log \left (\frac {256 \left (25-10 x+x^2\right )}{75 e^3 \left (1521 x^2-624 x^3+64 x^4\right )}\right )-\left (4 e^3\right ) \int \frac {\left (114075 x^2-46800 x^3+4800 x^4\right ) \left (-\frac {\left (6400-2560 x+256 x^2\right ) \left (228150 x-140400 x^2+19200 x^3\right )}{e^3 \left (114075 x^2-46800 x^3+4800 x^4\right )^2}+\frac {-2560+512 x}{e^3 \left (114075 x^2-46800 x^3+4800 x^4\right )}\right ) \log (-5+x)}{256 (-5+x)^2} \, dx+\left (4 e^3\right ) \int \frac {\left (114075 x^2-46800 x^3+4800 x^4\right ) \left (-\frac {\left (6400-2560 x+256 x^2\right ) \left (228150 x-140400 x^2+19200 x^3\right )}{e^3 \left (114075 x^2-46800 x^3+4800 x^4\right )^2}+\frac {-2560+512 x}{e^3 \left (114075 x^2-46800 x^3+4800 x^4\right )}\right ) \log (x)}{256 (-5+x)^2} \, dx+\left (4 e^3\right ) \int \frac {\left (114075 x^2-46800 x^3+4800 x^4\right ) \left (-\frac {\left (6400-2560 x+256 x^2\right ) \left (228150 x-140400 x^2+19200 x^3\right )}{e^3 \left (114075 x^2-46800 x^3+4800 x^4\right )^2}+\frac {-2560+512 x}{e^3 \left (114075 x^2-46800 x^3+4800 x^4\right )}\right ) \log (-39+8 x)}{256 (-5+x)^2} \, dx\\ &=4 \log (-5+x) \log \left (\frac {256 \left (25-10 x+x^2\right )}{75 e^3 \left (1521 x^2-624 x^3+64 x^4\right )}\right )-4 \log (x) \log \left (\frac {256 \left (25-10 x+x^2\right )}{75 e^3 \left (1521 x^2-624 x^3+64 x^4\right )}\right )-4 \log (-39+8 x) \log \left (\frac {256 \left (25-10 x+x^2\right )}{75 e^3 \left (1521 x^2-624 x^3+64 x^4\right )}\right )-\frac {1}{64} e^3 \int \frac {\left (114075 x^2-46800 x^3+4800 x^4\right ) \left (-\frac {\left (6400-2560 x+256 x^2\right ) \left (228150 x-140400 x^2+19200 x^3\right )}{e^3 \left (114075 x^2-46800 x^3+4800 x^4\right )^2}+\frac {-2560+512 x}{e^3 \left (114075 x^2-46800 x^3+4800 x^4\right )}\right ) \log (-5+x)}{(-5+x)^2} \, dx+\frac {1}{64} e^3 \int \frac {\left (114075 x^2-46800 x^3+4800 x^4\right ) \left (-\frac {\left (6400-2560 x+256 x^2\right ) \left (228150 x-140400 x^2+19200 x^3\right )}{e^3 \left (114075 x^2-46800 x^3+4800 x^4\right )^2}+\frac {-2560+512 x}{e^3 \left (114075 x^2-46800 x^3+4800 x^4\right )}\right ) \log (x)}{(-5+x)^2} \, dx+\frac {1}{64} e^3 \int \frac {\left (114075 x^2-46800 x^3+4800 x^4\right ) \left (-\frac {\left (6400-2560 x+256 x^2\right ) \left (228150 x-140400 x^2+19200 x^3\right )}{e^3 \left (114075 x^2-46800 x^3+4800 x^4\right )^2}+\frac {-2560+512 x}{e^3 \left (114075 x^2-46800 x^3+4800 x^4\right )}\right ) \log (-39+8 x)}{(-5+x)^2} \, dx\\ &=4 \log (-5+x) \log \left (\frac {256 \left (25-10 x+x^2\right )}{75 e^3 \left (1521 x^2-624 x^3+64 x^4\right )}\right )-4 \log (x) \log \left (\frac {256 \left (25-10 x+x^2\right )}{75 e^3 \left (1521 x^2-624 x^3+64 x^4\right )}\right )-4 \log (-39+8 x) \log \left (\frac {256 \left (25-10 x+x^2\right )}{75 e^3 \left (1521 x^2-624 x^3+64 x^4\right )}\right )-\frac {1}{64} e^3 \int \frac {x^2 \left (114075-46800 x+4800 x^2\right ) \left (-\frac {\left (6400-2560 x+256 x^2\right ) \left (228150 x-140400 x^2+19200 x^3\right )}{e^3 \left (114075 x^2-46800 x^3+4800 x^4\right )^2}+\frac {-2560+512 x}{e^3 \left (114075 x^2-46800 x^3+4800 x^4\right )}\right ) \log (-5+x)}{(-5+x)^2} \, dx+\frac {1}{64} e^3 \int \frac {x^2 \left (114075-46800 x+4800 x^2\right ) \left (-\frac {\left (6400-2560 x+256 x^2\right ) \left (228150 x-140400 x^2+19200 x^3\right )}{e^3 \left (114075 x^2-46800 x^3+4800 x^4\right )^2}+\frac {-2560+512 x}{e^3 \left (114075 x^2-46800 x^3+4800 x^4\right )}\right ) \log (x)}{(-5+x)^2} \, dx+\frac {1}{64} e^3 \int \frac {x^2 \left (114075-46800 x+4800 x^2\right ) \left (-\frac {\left (6400-2560 x+256 x^2\right ) \left (228150 x-140400 x^2+19200 x^3\right )}{e^3 \left (114075 x^2-46800 x^3+4800 x^4\right )^2}+\frac {-2560+512 x}{e^3 \left (114075 x^2-46800 x^3+4800 x^4\right )}\right ) \log (-39+8 x)}{(-5+x)^2} \, dx\\ &=4 \log (-5+x) \log \left (\frac {256 \left (25-10 x+x^2\right )}{75 e^3 \left (1521 x^2-624 x^3+64 x^4\right )}\right )-4 \log (x) \log \left (\frac {256 \left (25-10 x+x^2\right )}{75 e^3 \left (1521 x^2-624 x^3+64 x^4\right )}\right )-4 \log (-39+8 x) \log \left (\frac {256 \left (25-10 x+x^2\right )}{75 e^3 \left (1521 x^2-624 x^3+64 x^4\right )}\right )-\frac {1}{64} e^3 \int \frac {75 x^2 (-39+8 x)^2 \left (-\frac {\left (6400-2560 x+256 x^2\right ) \left (228150 x-140400 x^2+19200 x^3\right )}{e^3 \left (114075 x^2-46800 x^3+4800 x^4\right )^2}+\frac {-2560+512 x}{e^3 \left (114075 x^2-46800 x^3+4800 x^4\right )}\right ) \log (-5+x)}{(-5+x)^2} \, dx+\frac {1}{64} e^3 \int \frac {75 x^2 (-39+8 x)^2 \left (-\frac {\left (6400-2560 x+256 x^2\right ) \left (228150 x-140400 x^2+19200 x^3\right )}{e^3 \left (114075 x^2-46800 x^3+4800 x^4\right )^2}+\frac {-2560+512 x}{e^3 \left (114075 x^2-46800 x^3+4800 x^4\right )}\right ) \log (x)}{(-5+x)^2} \, dx+\frac {1}{64} e^3 \int \frac {75 x^2 (-39+8 x)^2 \left (-\frac {\left (6400-2560 x+256 x^2\right ) \left (228150 x-140400 x^2+19200 x^3\right )}{e^3 \left (114075 x^2-46800 x^3+4800 x^4\right )^2}+\frac {-2560+512 x}{e^3 \left (114075 x^2-46800 x^3+4800 x^4\right )}\right ) \log (-39+8 x)}{(-5+x)^2} \, dx\\ &=4 \log (-5+x) \log \left (\frac {256 \left (25-10 x+x^2\right )}{75 e^3 \left (1521 x^2-624 x^3+64 x^4\right )}\right )-4 \log (x) \log \left (\frac {256 \left (25-10 x+x^2\right )}{75 e^3 \left (1521 x^2-624 x^3+64 x^4\right )}\right )-4 \log (-39+8 x) \log \left (\frac {256 \left (25-10 x+x^2\right )}{75 e^3 \left (1521 x^2-624 x^3+64 x^4\right )}\right )-\frac {1}{64} \left (75 e^3\right ) \int \frac {x^2 (-39+8 x)^2 \left (-\frac {\left (6400-2560 x+256 x^2\right ) \left (228150 x-140400 x^2+19200 x^3\right )}{e^3 \left (114075 x^2-46800 x^3+4800 x^4\right )^2}+\frac {-2560+512 x}{e^3 \left (114075 x^2-46800 x^3+4800 x^4\right )}\right ) \log (-5+x)}{(-5+x)^2} \, dx+\frac {1}{64} \left (75 e^3\right ) \int \frac {x^2 (-39+8 x)^2 \left (-\frac {\left (6400-2560 x+256 x^2\right ) \left (228150 x-140400 x^2+19200 x^3\right )}{e^3 \left (114075 x^2-46800 x^3+4800 x^4\right )^2}+\frac {-2560+512 x}{e^3 \left (114075 x^2-46800 x^3+4800 x^4\right )}\right ) \log (x)}{(-5+x)^2} \, dx+\frac {1}{64} \left (75 e^3\right ) \int \frac {x^2 (-39+8 x)^2 \left (-\frac {\left (6400-2560 x+256 x^2\right ) \left (228150 x-140400 x^2+19200 x^3\right )}{e^3 \left (114075 x^2-46800 x^3+4800 x^4\right )^2}+\frac {-2560+512 x}{e^3 \left (114075 x^2-46800 x^3+4800 x^4\right )}\right ) \log (-39+8 x)}{(-5+x)^2} \, dx\\ &=4 \log (-5+x) \log \left (\frac {256 \left (25-10 x+x^2\right )}{75 e^3 \left (1521 x^2-624 x^3+64 x^4\right )}\right )-4 \log (x) \log \left (\frac {256 \left (25-10 x+x^2\right )}{75 e^3 \left (1521 x^2-624 x^3+64 x^4\right )}\right )-4 \log (-39+8 x) \log \left (\frac {256 \left (25-10 x+x^2\right )}{75 e^3 \left (1521 x^2-624 x^3+64 x^4\right )}\right )-\frac {1}{64} \left (75 e^3\right ) \int \left (\frac {512 \log (-5+x)}{75 e^3 (-5+x)}-\frac {512 \log (-5+x)}{75 e^3 x}-\frac {4096 \log (-5+x)}{75 e^3 (-39+8 x)}\right ) \, dx+\frac {1}{64} \left (75 e^3\right ) \int \left (\frac {512 \log (x)}{75 e^3 (-5+x)}-\frac {512 \log (x)}{75 e^3 x}-\frac {4096 \log (x)}{75 e^3 (-39+8 x)}\right ) \, dx+\frac {1}{64} \left (75 e^3\right ) \int \left (\frac {512 \log (-39+8 x)}{75 e^3 (-5+x)}-\frac {512 \log (-39+8 x)}{75 e^3 x}-\frac {4096 \log (-39+8 x)}{75 e^3 (-39+8 x)}\right ) \, dx\\ &=4 \log (-5+x) \log \left (\frac {256 \left (25-10 x+x^2\right )}{75 e^3 \left (1521 x^2-624 x^3+64 x^4\right )}\right )-4 \log (x) \log \left (\frac {256 \left (25-10 x+x^2\right )}{75 e^3 \left (1521 x^2-624 x^3+64 x^4\right )}\right )-4 \log (-39+8 x) \log \left (\frac {256 \left (25-10 x+x^2\right )}{75 e^3 \left (1521 x^2-624 x^3+64 x^4\right )}\right )-8 \int \frac {\log (-5+x)}{-5+x} \, dx+8 \int \frac {\log (-5+x)}{x} \, dx+8 \int \frac {\log (x)}{-5+x} \, dx-8 \int \frac {\log (x)}{x} \, dx+8 \int \frac {\log (-39+8 x)}{-5+x} \, dx-8 \int \frac {\log (-39+8 x)}{x} \, dx+64 \int \frac {\log (-5+x)}{-39+8 x} \, dx-64 \int \frac {\log (x)}{-39+8 x} \, dx-64 \int \frac {\log (-39+8 x)}{-39+8 x} \, dx\\ &=8 \log (5) \log (-5+x)+8 \log (-5+x) \log \left (\frac {x}{5}\right )-4 \log ^2(x)-8 \log \left (\frac {39}{8}\right ) \log (-39+8 x)+8 \log (-5+x) \log (-39+8 x)-8 \log \left (\frac {8 x}{39}\right ) \log (-39+8 x)+4 \log (-5+x) \log \left (\frac {256 \left (25-10 x+x^2\right )}{75 e^3 \left (1521 x^2-624 x^3+64 x^4\right )}\right )-4 \log (x) \log \left (\frac {256 \left (25-10 x+x^2\right )}{75 e^3 \left (1521 x^2-624 x^3+64 x^4\right )}\right )-4 \log (-39+8 x) \log \left (\frac {256 \left (25-10 x+x^2\right )}{75 e^3 \left (1521 x^2-624 x^3+64 x^4\right )}\right )-8 \int \frac {\log (-39+8 x)}{-5+x} \, dx-8 \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,-5+x\right )-8 \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,-39+8 x\right )+8 \operatorname {Subst}\left (\int \frac {\log (1+8 x)}{x} \, dx,x,-5+x\right )\\ &=8 \log (5) \log (-5+x)-4 \log ^2(-5+x)+8 \log (-5+x) \log \left (\frac {x}{5}\right )-4 \log ^2(x)-8 \log \left (\frac {39}{8}\right ) \log (-39+8 x)+8 \log (-5+x) \log (-39+8 x)-8 \log \left (\frac {8 x}{39}\right ) \log (-39+8 x)-4 \log ^2(-39+8 x)+4 \log (-5+x) \log \left (\frac {256 \left (25-10 x+x^2\right )}{75 e^3 \left (1521 x^2-624 x^3+64 x^4\right )}\right )-4 \log (x) \log \left (\frac {256 \left (25-10 x+x^2\right )}{75 e^3 \left (1521 x^2-624 x^3+64 x^4\right )}\right )-4 \log (-39+8 x) \log \left (\frac {256 \left (25-10 x+x^2\right )}{75 e^3 \left (1521 x^2-624 x^3+64 x^4\right )}\right )-8 \text {Li}_2(8 (5-x))-8 \operatorname {Subst}\left (\int \frac {\log (1+8 x)}{x} \, dx,x,-5+x\right )\\ &=8 \log (5) \log (-5+x)-4 \log ^2(-5+x)+8 \log (-5+x) \log \left (\frac {x}{5}\right )-4 \log ^2(x)-8 \log \left (\frac {39}{8}\right ) \log (-39+8 x)+8 \log (-5+x) \log (-39+8 x)-8 \log \left (\frac {8 x}{39}\right ) \log (-39+8 x)-4 \log ^2(-39+8 x)+4 \log (-5+x) \log \left (\frac {256 \left (25-10 x+x^2\right )}{75 e^3 \left (1521 x^2-624 x^3+64 x^4\right )}\right )-4 \log (x) \log \left (\frac {256 \left (25-10 x+x^2\right )}{75 e^3 \left (1521 x^2-624 x^3+64 x^4\right )}\right )-4 \log (-39+8 x) \log \left (\frac {256 \left (25-10 x+x^2\right )}{75 e^3 \left (1521 x^2-624 x^3+64 x^4\right )}\right )\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [C]  time = 0.09, size = 197, normalized size = 7.04 \begin {gather*} -4 \left (\log ^2(5-x)-2 \log (5) \log (-5+x)-2 \log (5) \log (x)+\log ^2(x)-\log (5-x) \left (-3+\log \left (\frac {256}{75}\right )+\log \left (\frac {(-5+x)^2}{x^2 (-39+8 x)^2}\right )\right )+\log (x) \left (-3+\log \left (\frac {256}{75}\right )+\log \left (\frac {(-5+x)^2}{x^2 (-39+8 x)^2}\right )\right )+2 \log \left (\frac {39}{8}\right ) \log (-39+8 x)+\log (64) \log (-39+8 x)+2 \log \left (\frac {8 x}{39}\right ) \log (-39+8 x)+\left (-3+\log \left (\frac {256}{75}\right )+\log \left (\frac {(-5+x)^2}{x^2 (-39+8 x)^2}\right )\right ) \log (-39+8 x)+\log ^2(-39+8 x)+2 \text {Li}_2(40-8 x)+2 \text {Li}_2\left (1-\frac {x}{5}\right )+2 \text {Li}_2\left (\frac {x}{5}\right )+2 \text {Li}_2(-39+8 x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((-780 + 320*x - 32*x^2)*Log[(6400 - 2560*x + 256*x^2)/(E^3*(114075*x^2 - 46800*x^3 + 4800*x^4))])/(
195*x - 79*x^2 + 8*x^3),x]

[Out]

-4*(Log[5 - x]^2 - 2*Log[5]*Log[-5 + x] - 2*Log[5]*Log[x] + Log[x]^2 - Log[5 - x]*(-3 + Log[256/75] + Log[(-5
+ x)^2/(x^2*(-39 + 8*x)^2)]) + Log[x]*(-3 + Log[256/75] + Log[(-5 + x)^2/(x^2*(-39 + 8*x)^2)]) + 2*Log[39/8]*L
og[-39 + 8*x] + Log[64]*Log[-39 + 8*x] + 2*Log[(8*x)/39]*Log[-39 + 8*x] + (-3 + Log[256/75] + Log[(-5 + x)^2/(
x^2*(-39 + 8*x)^2)])*Log[-39 + 8*x] + Log[-39 + 8*x]^2 + 2*PolyLog[2, 40 - 8*x] + 2*PolyLog[2, 1 - x/5] + 2*Po
lyLog[2, x/5] + 2*PolyLog[2, -39 + 8*x])

________________________________________________________________________________________

fricas [A]  time = 0.64, size = 33, normalized size = 1.18 \begin {gather*} \log \left (\frac {256 \, {\left (x^{2} - 10 \, x + 25\right )} e^{\left (-3\right )}}{75 \, {\left (64 \, x^{4} - 624 \, x^{3} + 1521 \, x^{2}\right )}}\right )^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-32*x^2+320*x-780)*log((256*x^2-2560*x+6400)/(4800*x^4-46800*x^3+114075*x^2)/exp(3))/(8*x^3-79*x^2+
195*x),x, algorithm="fricas")

[Out]

log(256/75*(x^2 - 10*x + 25)*e^(-3)/(64*x^4 - 624*x^3 + 1521*x^2))^2

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -\frac {4 \, {\left (8 \, x^{2} - 80 \, x + 195\right )} \log \left (\frac {256 \, {\left (x^{2} - 10 \, x + 25\right )} e^{\left (-3\right )}}{75 \, {\left (64 \, x^{4} - 624 \, x^{3} + 1521 \, x^{2}\right )}}\right )}{8 \, x^{3} - 79 \, x^{2} + 195 \, x}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-32*x^2+320*x-780)*log((256*x^2-2560*x+6400)/(4800*x^4-46800*x^3+114075*x^2)/exp(3))/(8*x^3-79*x^2+
195*x),x, algorithm="giac")

[Out]

integrate(-4*(8*x^2 - 80*x + 195)*log(256/75*(x^2 - 10*x + 25)*e^(-3)/(64*x^4 - 624*x^3 + 1521*x^2))/(8*x^3 -
79*x^2 + 195*x), x)

________________________________________________________________________________________

maple [A]  time = 0.22, size = 37, normalized size = 1.32

method	result	size
norman	\(\ln \left (\frac {\left (256 x^{2}-2560 x +6400\right ) {\mathrm e}^{-3}}{4800 x^{4}-46800 x^{3}+114075 x^{2}}\right )^{2}\)	\(37\)
risch	\(-4 \ln \relax (x ) \ln \left (\frac {\left (x^{2}-10 x +25\right ) {\mathrm e}^{-3}}{x^{2} \left (64 x^{2}-624 x +1521\right )}\right )+8 \ln \left (-\frac {8 x}{39}+1\right ) \ln \left (\frac {8 x}{39}\right )-8 \ln \left (-\frac {8 x}{39}+1\right ) \ln \relax (x )-8 \ln \left (1-\frac {x}{5}\right ) \ln \left (\frac {x}{5}\right )+8 \ln \left (1-\frac {x}{5}\right ) \ln \relax (x )-4 \ln \relax (x )^{2}-4 \ln \left (8 x -39\right ) \ln \left (\frac {\left (x^{2}-10 x +25\right ) {\mathrm e}^{-3}}{x^{2} \left (64 x^{2}-624 x +1521\right )}\right )-4 \ln \left (8 x -39\right )^{2}-8 \ln \left (8 x -39\right ) \ln \left (\frac {8 x}{39}\right )+4 \ln \left (x -5\right ) \ln \left (\frac {\left (x^{2}-10 x +25\right ) {\mathrm e}^{-3}}{x^{2} \left (64 x^{2}-624 x +1521\right )}\right )+8 \ln \left (x -5\right ) \ln \left (\frac {x}{5}\right )-4 \ln \left (x -5\right )^{2}+8 \ln \left (x -5\right ) \ln \left (8 x -39\right )-32 \ln \relax (2) \ln \left (8 x -39\right )+32 \ln \relax (2) \ln \left (x -5\right )-32 \ln \relax (2) \ln \relax (x )+4 \ln \left (75\right ) \ln \left (8 x -39\right )-4 \ln \left (75\right ) \ln \left (x -5\right )+4 \ln \left (75\right ) \ln \relax (x )\)	\(251\)
default	\(-4 \ln \left (75\right ) \ln \left (x -5\right )+4 \ln \left (75\right ) \ln \relax (x )+4 \ln \left (75\right ) \ln \left (8 x -39\right )+12 \ln \relax (x )+12 \ln \left (8 x -39\right )-12 \ln \left (x -5\right )+32 \ln \relax (2) \ln \left (x -5\right )-32 \ln \relax (2) \ln \relax (x )-32 \ln \relax (2) \ln \left (8 x -39\right )-4 \ln \relax (x ) \ln \left (\frac {x^{2}-10 x +25}{x^{2} \left (64 x^{2}-624 x +1521\right )}\right )-4 \ln \relax (x )^{2}-8 \left (\ln \relax (x )-\ln \left (\frac {8 x}{39}\right )\right ) \ln \left (-\frac {8 x}{39}+1\right )+8 \left (\ln \relax (x )-\ln \left (\frac {x}{5}\right )\right ) \ln \left (1-\frac {x}{5}\right )-4 \ln \left (8 x -39\right ) \ln \left (\frac {x^{2}-10 x +25}{x^{2} \left (64 x^{2}-624 x +1521\right )}\right )-4 \ln \left (8 x -39\right )^{2}-8 \ln \left (8 x -39\right ) \ln \left (\frac {8 x}{39}\right )+4 \ln \left (x -5\right ) \ln \left (\frac {x^{2}-10 x +25}{x^{2} \left (64 x^{2}-624 x +1521\right )}\right )+8 \ln \left (x -5\right ) \ln \left (\frac {x}{5}\right )-4 \ln \left (x -5\right )^{2}+8 \ln \left (x -5\right ) \ln \left (8 x -39\right )\)	\(253\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-32*x^2+320*x-780)*ln((256*x^2-2560*x+6400)/(4800*x^4-46800*x^3+114075*x^2)/exp(3))/(8*x^3-79*x^2+195*x),
x,method=_RETURNVERBOSE)

[Out]

ln((256*x^2-2560*x+6400)/(4800*x^4-46800*x^3+114075*x^2)/exp(3))^2

________________________________________________________________________________________

maxima [B]  time = 0.36, size = 98, normalized size = 3.50 \begin {gather*} 8 \, {\left (\log \left (x - 5\right ) - \log \relax (x)\right )} \log \left (8 \, x - 39\right ) - 4 \, \log \left (8 \, x - 39\right )^{2} - 4 \, \log \left (x - 5\right )^{2} + 8 \, \log \left (x - 5\right ) \log \relax (x) - 4 \, \log \relax (x)^{2} - 4 \, {\left (\log \left (8 \, x - 39\right ) - \log \left (x - 5\right ) + \log \relax (x)\right )} \log \left (\frac {256 \, {\left (x^{2} - 10 \, x + 25\right )} e^{\left (-3\right )}}{75 \, {\left (64 \, x^{4} - 624 \, x^{3} + 1521 \, x^{2}\right )}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-32*x^2+320*x-780)*log((256*x^2-2560*x+6400)/(4800*x^4-46800*x^3+114075*x^2)/exp(3))/(8*x^3-79*x^2+
195*x),x, algorithm="maxima")

[Out]

8*(log(x - 5) - log(x))*log(8*x - 39) - 4*log(8*x - 39)^2 - 4*log(x - 5)^2 + 8*log(x - 5)*log(x) - 4*log(x)^2
- 4*(log(8*x - 39) - log(x - 5) + log(x))*log(256/75*(x^2 - 10*x + 25)*e^(-3)/(64*x^4 - 624*x^3 + 1521*x^2))

________________________________________________________________________________________

mupad [B]  time = 2.07, size = 34, normalized size = 1.21 \begin {gather*} {\left (\ln \left (\frac {256\,x^2-2560\,x+6400}{4800\,x^4-46800\,x^3+114075\,x^2}\right )-3\right )}^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log((exp(-3)*(256*x^2 - 2560*x + 6400))/(114075*x^2 - 46800*x^3 + 4800*x^4))*(32*x^2 - 320*x + 780))/(19
5*x - 79*x^2 + 8*x^3),x)

[Out]

(log((256*x^2 - 2560*x + 6400)/(114075*x^2 - 46800*x^3 + 4800*x^4)) - 3)^2

________________________________________________________________________________________

sympy [A]  time = 0.21, size = 31, normalized size = 1.11 \begin {gather*} \log {\left (\frac {256 x^{2} - 2560 x + 6400}{\left (4800 x^{4} - 46800 x^{3} + 114075 x^{2}\right ) e^{3}} \right )}^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-32*x**2+320*x-780)*ln((256*x**2-2560*x+6400)/(4800*x**4-46800*x**3+114075*x**2)/exp(3))/(8*x**3-79
*x**2+195*x),x)

[Out]

log((256*x**2 - 2560*x + 6400)*exp(-3)/(4800*x**4 - 46800*x**3 + 114075*x**2))**2

________________________________________________________________________________________