3.78.76 \(\int \frac {-375 x+15 x^2+(10125+10845 x+4925 x^2+1035 x^3+75 x^4-5 x^5) \log (-25+x)+(-405 x-450 x^2-215 x^3-50 x^4-5 x^5+(11250 x+10300 x^2+3320 x^3+350 x^4-20 x^5) \log (-25+x)) \log (x)}{-25 x+x^2} \, dx\)

Optimal. Leaf size=26 \[ 5 \left (-6+3 x-\left (x-(3+x)^2\right )^2 \log (-25+x) \log (x)\right ) \]

________________________________________________________________________________________

Rubi [C]  time = 2.84, antiderivative size = 106, normalized size of antiderivative = 4.08, number of steps used = 134, number of rules used = 30, integrand size = 105, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {1593, 6688, 12, 6742, 14, 2418, 2389, 2295, 2394, 2315, 2395, 43, 698, 2417, 2304, 2370, 2411, 2351, 2317, 2391, 2376, 2392, 2316, 2433, 2375, 2374, 6589, 2346, 2301, 2334} \begin {gather*} 2868750 \text {Li}_2\left (1-\frac {x}{25}\right )+2868750 \text {Li}_2\left (\frac {x}{25}\right )-5 x^4 \log (x-25) \log (x)-50 x^3 \log (x-25) \log (x)-215 x^2 \log (x-25) \log (x)+15 x-2880405 \log (25) \log (x-25)-11655 \log (x-25) \log \left (\frac {x}{25}\right )+2868750 \log (25-x) \log (x)+450 (25-x) \log (x-25) \log (x)-2868750 \log (25) \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-375*x + 15*x^2 + (10125 + 10845*x + 4925*x^2 + 1035*x^3 + 75*x^4 - 5*x^5)*Log[-25 + x] + (-405*x - 450*x
^2 - 215*x^3 - 50*x^4 - 5*x^5 + (11250*x + 10300*x^2 + 3320*x^3 + 350*x^4 - 20*x^5)*Log[-25 + x])*Log[x])/(-25
*x + x^2),x]

[Out]

15*x - 2880405*Log[25]*Log[-25 + x] - 11655*Log[-25 + x]*Log[x/25] - 2868750*Log[25]*Log[x] + 2868750*Log[25 -
x]*Log[x] + 450*(25 - x)*Log[-25 + x]*Log[x] - 215*x^2*Log[-25 + x]*Log[x] - 50*x^3*Log[-25 + x]*Log[x] - 5*x
^4*Log[-25 + x]*Log[x] + 2868750*PolyLog[2, 1 - x/25] + 2868750*PolyLog[2, x/25]

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 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 698

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*
e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

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 2295

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

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 2304

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

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 2317

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

Rule 2334

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(x_)^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> With[{u = I
ntHide[x^m*(d + e*x^r)^q, x]}, Simp[u*(a + b*Log[c*x^n]), x] - Dist[b*n, Int[SimplifyIntegrand[u/x, x], x], x]
] /; FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[q, 0] && IntegerQ[m] &&  !(EqQ[q, 1] && EqQ[m, -1])

Rule 2346

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_.))/(x_), x_Symbol] :> Dist[d, Int[((d
+ e*x)^(q - 1)*(a + b*Log[c*x^n])^p)/x, x], x] + Dist[e, Int[(d + e*x)^(q - 1)*(a + b*Log[c*x^n])^p, x], x] /
; FreeQ[{a, b, c, d, e, n}, x] && IGtQ[p, 0] && GtQ[q, 0] && IntegerQ[2*q]

Rule 2351

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> Wit
h[{u = ExpandIntegrand[a + b*Log[c*x^n], (f*x)^m*(d + e*x^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c,
d, e, f, m, n, q, r}, x] && IntegerQ[q] && (GtQ[q, 0] || (IntegerQ[m] && IntegerQ[r]))

Rule 2370

Int[Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))^(r_.)]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> With[
{u = IntHide[Log[d*(e + f*x^m)^r], x]}, Dist[(a + b*Log[c*x^n])^p, u, x] - Dist[b*n*p, Int[Dist[(a + b*Log[c*x
^n])^(p - 1)/x, u, x], x], x]] /; FreeQ[{a, b, c, d, e, f, r, m, n}, x] && IGtQ[p, 0] && RationalQ[m] && (EqQ[
p, 1] || (FractionQ[m] && IntegerQ[1/m]) || (EqQ[r, 1] && EqQ[m, 1] && EqQ[d*e, 1]))

Rule 2374

Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.))/(x_), x_Symbol] :> -Sim
p[(PolyLog[2, -(d*f*x^m)]*(a + b*Log[c*x^n])^p)/m, x] + Dist[(b*n*p)/m, Int[(PolyLog[2, -(d*f*x^m)]*(a + b*Log
[c*x^n])^(p - 1))/x, x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[p, 0] && EqQ[d*e, 1]

Rule 2375

Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))^(r_.)]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.))/(x_), x_Symbol] :
> Simp[(Log[d*(e + f*x^m)^r]*(a + b*Log[c*x^n])^(p + 1))/(b*n*(p + 1)), x] - Dist[(f*m*r)/(b*n*(p + 1)), Int[(
x^(m - 1)*(a + b*Log[c*x^n])^(p + 1))/(e + f*x^m), x], x] /; FreeQ[{a, b, c, d, e, f, r, m, n}, x] && IGtQ[p,
0] && NeQ[d*e, 1]

Rule 2376

Int[Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))^(r_.)]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((g_.)*(x_))^(q_.), x_Sym
bol] :> With[{u = IntHide[(g*x)^q*Log[d*(e + f*x^m)^r], x]}, Dist[a + b*Log[c*x^n], u, x] - Dist[b*n, Int[Dist
[1/x, u, x], x], x]] /; FreeQ[{a, b, c, d, e, f, g, r, m, n, q}, x] && (IntegerQ[(q + 1)/m] || (RationalQ[m] &
& RationalQ[q])) && NeQ[q, -1]

Rule 2389

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.), x_Symbol] :> Dist[1/e, Subst[Int[(a + b*Log[c*
x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, n, p}, x]

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 2392

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*d])*Log[x], x] + Dist[
b, Int[Log[1 + (e*x)/d]/x, x], x] /; FreeQ[{a, b, c, d, e}, x] && GtQ[c*d, 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 2395

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

Rule 2411

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + (g_.)*(x_))^(q_.)*((h_.) + (i_.)*(x_))
^(r_.), x_Symbol] :> Dist[1/e, Subst[Int[((g*x)/e)^q*((e*h - d*i)/e + (i*x)/e)^r*(a + b*Log[c*x^n])^p, x], x,
d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, n, p, q, r}, x] && EqQ[e*f - d*g, 0] && (IGtQ[p, 0] || IGtQ[
r, 0]) && IntegerQ[2*r]

Rule 2417

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*(Polyx_), x_Symbol] :> Int[ExpandIntegrand[Poly
x*(a + b*Log[c*(d + e*x)^n])^p, x], x] /; FreeQ[{a, b, c, d, e, n, p}, x] && PolynomialQ[Polyx, x]

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 2433

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + Log[(h_.)*((i_.) + (j_.)*(x_))^(m_.)]*
(g_.))*((k_.) + (l_.)*(x_))^(r_.), x_Symbol] :> Dist[1/e, Subst[Int[((k*x)/d)^r*(a + b*Log[c*x^n])^p*(f + g*Lo
g[h*((e*i - d*j)/e + (j*x)/e)^m]), x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, j, k, l, n, p, r},
x] && EqQ[e*k - d*l, 0]

Rule 6589

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

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 {-375 x+15 x^2+\left (10125+10845 x+4925 x^2+1035 x^3+75 x^4-5 x^5\right ) \log (-25+x)+\left (-405 x-450 x^2-215 x^3-50 x^4-5 x^5+\left (11250 x+10300 x^2+3320 x^3+350 x^4-20 x^5\right ) \log (-25+x)\right ) \log (x)}{(-25+x) x} \, dx\\ &=\int \frac {5 \left (\left (-225-116 x-20 x^2+x^3\right ) \log (-25+x) \left (9+5 x+x^2+2 x (5+2 x) \log (x)\right )+x \left (75-3 x+\left (9+5 x+x^2\right )^2 \log (x)\right )\right )}{(25-x) x} \, dx\\ &=5 \int \frac {\left (-225-116 x-20 x^2+x^3\right ) \log (-25+x) \left (9+5 x+x^2+2 x (5+2 x) \log (x)\right )+x \left (75-3 x+\left (9+5 x+x^2\right )^2 \log (x)\right )}{(25-x) x} \, dx\\ &=5 \int \left (\frac {3 x-81 \log (-25+x)-90 x \log (-25+x)-43 x^2 \log (-25+x)-10 x^3 \log (-25+x)-x^4 \log (-25+x)}{x}-\frac {\left (9+5 x+x^2\right ) \left (9+5 x+x^2-250 \log (-25+x)-90 x \log (-25+x)+4 x^2 \log (-25+x)\right ) \log (x)}{-25+x}\right ) \, dx\\ &=5 \int \frac {3 x-81 \log (-25+x)-90 x \log (-25+x)-43 x^2 \log (-25+x)-10 x^3 \log (-25+x)-x^4 \log (-25+x)}{x} \, dx-5 \int \frac {\left (9+5 x+x^2\right ) \left (9+5 x+x^2-250 \log (-25+x)-90 x \log (-25+x)+4 x^2 \log (-25+x)\right ) \log (x)}{-25+x} \, dx\\ &=5 \int \left (3-\frac {\left (9+5 x+x^2\right )^2 \log (-25+x)}{x}\right ) \, dx-5 \int \left (30 \left (9+5 x+x^2-250 \log (-25+x)-90 x \log (-25+x)+4 x^2 \log (-25+x)\right ) \log (x)+\frac {759 \left (9+5 x+x^2-250 \log (-25+x)-90 x \log (-25+x)+4 x^2 \log (-25+x)\right ) \log (x)}{-25+x}+x \left (9+5 x+x^2-250 \log (-25+x)-90 x \log (-25+x)+4 x^2 \log (-25+x)\right ) \log (x)\right ) \, dx\\ &=15 x-5 \int \frac {\left (9+5 x+x^2\right )^2 \log (-25+x)}{x} \, dx-5 \int x \left (9+5 x+x^2-250 \log (-25+x)-90 x \log (-25+x)+4 x^2 \log (-25+x)\right ) \log (x) \, dx-150 \int \left (9+5 x+x^2-250 \log (-25+x)-90 x \log (-25+x)+4 x^2 \log (-25+x)\right ) \log (x) \, dx-3795 \int \frac {\left (9+5 x+x^2-250 \log (-25+x)-90 x \log (-25+x)+4 x^2 \log (-25+x)\right ) \log (x)}{-25+x} \, dx\\ &=15 x-5 \int \left (90 \log (-25+x)+\frac {81 \log (-25+x)}{x}+43 x \log (-25+x)+10 x^2 \log (-25+x)+x^3 \log (-25+x)\right ) \, dx-5 \int \left (9 x \log (x)+5 x^2 \log (x)+x^3 \log (x)-250 x \log (-25+x) \log (x)-90 x^2 \log (-25+x) \log (x)+4 x^3 \log (-25+x) \log (x)\right ) \, dx-150 \int \left (9 \log (x)+5 x \log (x)+x^2 \log (x)-250 \log (-25+x) \log (x)-90 x \log (-25+x) \log (x)+4 x^2 \log (-25+x) \log (x)\right ) \, dx-3795 \int \left (\frac {9 \log (x)}{-25+x}+\frac {5 x \log (x)}{-25+x}+\frac {x^2 \log (x)}{-25+x}-\frac {250 \log (-25+x) \log (x)}{-25+x}-\frac {90 x \log (-25+x) \log (x)}{-25+x}+\frac {4 x^2 \log (-25+x) \log (x)}{-25+x}\right ) \, dx\\ &=15 x-5 \int x^3 \log (-25+x) \, dx-5 \int x^3 \log (x) \, dx-20 \int x^3 \log (-25+x) \log (x) \, dx-25 \int x^2 \log (x) \, dx-45 \int x \log (x) \, dx-50 \int x^2 \log (-25+x) \, dx-150 \int x^2 \log (x) \, dx-215 \int x \log (-25+x) \, dx-405 \int \frac {\log (-25+x)}{x} \, dx-450 \int \log (-25+x) \, dx+450 \int x^2 \log (-25+x) \log (x) \, dx-600 \int x^2 \log (-25+x) \log (x) \, dx-750 \int x \log (x) \, dx+1250 \int x \log (-25+x) \log (x) \, dx-1350 \int \log (x) \, dx-3795 \int \frac {x^2 \log (x)}{-25+x} \, dx+13500 \int x \log (-25+x) \log (x) \, dx-15180 \int \frac {x^2 \log (-25+x) \log (x)}{-25+x} \, dx-18975 \int \frac {x \log (x)}{-25+x} \, dx-34155 \int \frac {\log (x)}{-25+x} \, dx+37500 \int \log (-25+x) \log (x) \, dx+341550 \int \frac {x \log (-25+x) \log (x)}{-25+x} \, dx+948750 \int \frac {\log (-25+x) \log (x)}{-25+x} \, dx\\ &=1365 x+\frac {795 x^2}{4}+\frac {175 x^3}{9}+\frac {5 x^4}{16}-\frac {215}{2} x^2 \log (-25+x)-\frac {50}{3} x^3 \log (-25+x)-\frac {5}{4} x^4 \log (-25+x)-34155 \log (25) \log (-25+x)-405 \log (-25+x) \log \left (\frac {x}{25}\right )-113850 x \log (x)-\frac {3795}{2} x^2 \log (x)-1875000 \log (25-x) \log (x)-37500 (25-x) \log (-25+x) \log (x)+7375 x^2 \log (-25+x) \log (x)-50 x^3 \log (-25+x) \log (x)-5 x^4 \log (-25+x) \log (x)+\frac {5}{4} \int \frac {x^4}{-25+x} \, dx+\frac {50}{3} \int \frac {x^3}{-25+x} \, dx+20 \int \left (-\frac {15625}{4}-\frac {625 x}{8}-\frac {25 x^2}{12}-\frac {x^3}{16}-\frac {390625 \log (25-x)}{4 x}+\frac {1}{4} x^3 \log (-25+x)\right ) \, dx+\frac {215}{2} \int \frac {x^2}{-25+x} \, dx+405 \int \frac {\log \left (\frac {x}{25}\right )}{-25+x} \, dx-450 \int \left (-\frac {625}{3}-\frac {25 x}{6}-\frac {x^2}{9}-\frac {15625 \log (25-x)}{3 x}+\frac {1}{3} x^2 \log (-25+x)\right ) \, dx-450 \operatorname {Subst}(\int \log (x) \, dx,x,-25+x)+600 \int \left (-\frac {625}{3}-\frac {25 x}{6}-\frac {x^2}{9}-\frac {15625 \log (25-x)}{3 x}+\frac {1}{3} x^2 \log (-25+x)\right ) \, dx-1250 \int \left (-\frac {25}{2}-\frac {x}{4}-\frac {625 \log (25-x)}{2 x}+\frac {1}{2} x \log (-25+x)\right ) \, dx-3795 \int \left (25 \log (x)+\frac {625 \log (x)}{-25+x}+x \log (x)\right ) \, dx-13500 \int \left (-\frac {25}{2}-\frac {x}{4}-\frac {625 \log (25-x)}{2 x}+\frac {1}{2} x \log (-25+x)\right ) \, dx-15180 \int \left (25 \log (-25+x) \log (x)+\frac {625 \log (-25+x) \log (x)}{-25+x}+x \log (-25+x) \log (x)\right ) \, dx-18975 \int \left (\log (x)+\frac {25 \log (x)}{-25+x}\right ) \, dx-34155 \int \frac {\log \left (\frac {x}{25}\right )}{-25+x} \, dx-37500 \int \left (-1-\frac {(25-x) \log (-25+x)}{x}\right ) \, dx+341550 \int \left (\log (-25+x) \log (x)+\frac {25 \log (-25+x) \log (x)}{-25+x}\right ) \, dx+948750 \operatorname {Subst}\left (\int \frac {\log (x) \log (25+x)}{x} \, dx,x,-25+x\right )\\ &=114315 x+\frac {3795 x^2}{4}+450 (25-x) \log (-25+x)-\frac {215}{2} x^2 \log (-25+x)-\frac {50}{3} x^3 \log (-25+x)-\frac {5}{4} x^4 \log (-25+x)-34155 \log (25) \log (-25+x)-405 \log (-25+x) \log \left (\frac {x}{25}\right )-113850 x \log (x)-\frac {3795}{2} x^2 \log (x)-1875000 \log (25-x) \log (x)-37500 (25-x) \log (-25+x) \log (x)+7375 x^2 \log (-25+x) \log (x)-50 x^3 \log (-25+x) \log (x)-5 x^4 \log (-25+x) \log (x)+474375 \log ^2(-25+x) \log (x)+33750 \text {Li}_2\left (1-\frac {x}{25}\right )+\frac {5}{4} \int \left (15625+\frac {390625}{-25+x}+625 x+25 x^2+x^3\right ) \, dx+5 \int x^3 \log (-25+x) \, dx+\frac {50}{3} \int \left (625+\frac {15625}{-25+x}+25 x+x^2\right ) \, dx+\frac {215}{2} \int \left (25+\frac {625}{-25+x}+x\right ) \, dx-150 \int x^2 \log (-25+x) \, dx+200 \int x^2 \log (-25+x) \, dx-625 \int x \log (-25+x) \, dx-3795 \int x \log (x) \, dx-6750 \int x \log (-25+x) \, dx-15180 \int x \log (-25+x) \log (x) \, dx-18975 \int \log (x) \, dx+37500 \int \frac {(25-x) \log (-25+x)}{x} \, dx-94875 \int \log (x) \, dx+341550 \int \log (-25+x) \log (x) \, dx-379500 \int \log (-25+x) \log (x) \, dx+390625 \int \frac {\log (25-x)}{x} \, dx-474375 \int \frac {\log (x)}{-25+x} \, dx-474375 \operatorname {Subst}\left (\int \frac {\log ^2(x)}{25+x} \, dx,x,-25+x\right )-1953125 \int \frac {\log (25-x)}{x} \, dx+2343750 \int \frac {\log (25-x)}{x} \, dx-2371875 \int \frac {\log (x)}{-25+x} \, dx-3125000 \int \frac {\log (25-x)}{x} \, dx+4218750 \int \frac {\log (25-x)}{x} \, dx+8538750 \int \frac {\log (-25+x) \log (x)}{-25+x} \, dx-9487500 \int \frac {\log (-25+x) \log (x)}{-25+x} \, dx\\ &=\frac {3129605 x}{12}+\frac {61205 x^2}{24}+\frac {575 x^3}{36}+\frac {5 x^4}{16}+\frac {9790625}{12} \log (25-x)+450 (25-x) \log (-25+x)-3795 x^2 \log (-25+x)-2880405 \log (25) \log (-25+x)-405 \log (-25+x) \log \left (\frac {x}{25}\right )-474375 \log ^2(-25+x) \log \left (\frac {x}{25}\right )+1875000 \log (25) \log (x)+2868750 \log (25-x) \log (x)+450 (25-x) \log (-25+x) \log (x)-215 x^2 \log (-25+x) \log (x)-50 x^3 \log (-25+x) \log (x)-5 x^4 \log (-25+x) \log (x)+474375 \log ^2(-25+x) \log (x)+33750 \text {Li}_2\left (1-\frac {x}{25}\right )-\frac {5}{4} \int \frac {x^4}{-25+x} \, dx+50 \int \frac {x^3}{-25+x} \, dx-\frac {200}{3} \int \frac {x^3}{-25+x} \, dx+\frac {625}{2} \int \frac {x^2}{-25+x} \, dx+3375 \int \frac {x^2}{-25+x} \, dx+15180 \int \left (-\frac {25}{2}-\frac {x}{4}-\frac {625 \log (25-x)}{2 x}+\frac {1}{2} x \log (-25+x)\right ) \, dx-37500 \operatorname {Subst}\left (\int \frac {x \log (x)}{25+x} \, dx,x,-25+x\right )-341550 \int \left (-1-\frac {(25-x) \log (-25+x)}{x}\right ) \, dx+379500 \int \left (-1-\frac {(25-x) \log (-25+x)}{x}\right ) \, dx+390625 \int \frac {\log \left (1-\frac {x}{25}\right )}{x} \, dx-474375 \int \frac {\log \left (\frac {x}{25}\right )}{-25+x} \, dx+948750 \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {x}{25}\right ) \log (x)}{x} \, dx,x,-25+x\right )-1953125 \int \frac {\log \left (1-\frac {x}{25}\right )}{x} \, dx+2343750 \int \frac {\log \left (1-\frac {x}{25}\right )}{x} \, dx-2371875 \int \frac {\log \left (\frac {x}{25}\right )}{-25+x} \, dx-3125000 \int \frac {\log \left (1-\frac {x}{25}\right )}{x} \, dx+4218750 \int \frac {\log \left (1-\frac {x}{25}\right )}{x} \, dx+8538750 \operatorname {Subst}\left (\int \frac {\log (x) \log (25+x)}{x} \, dx,x,-25+x\right )-9487500 \operatorname {Subst}\left (\int \frac {\log (x) \log (25+x)}{x} \, dx,x,-25+x\right )\\ &=\frac {397205 x}{12}+\frac {15665 x^2}{24}+\frac {575 x^3}{36}+\frac {5 x^4}{16}+\frac {9790625}{12} \log (25-x)+450 (25-x) \log (-25+x)-3795 x^2 \log (-25+x)-2880405 \log (25) \log (-25+x)-405 \log (-25+x) \log \left (\frac {x}{25}\right )-474375 \log ^2(-25+x) \log \left (\frac {x}{25}\right )+1875000 \log (25) \log (x)+2868750 \log (25-x) \log (x)+450 (25-x) \log (-25+x) \log (x)-215 x^2 \log (-25+x) \log (x)-50 x^3 \log (-25+x) \log (x)-5 x^4 \log (-25+x) \log (x)+2880000 \text {Li}_2\left (1-\frac {x}{25}\right )-948750 \log (-25+x) \text {Li}_2\left (1-\frac {x}{25}\right )-1875000 \text {Li}_2\left (\frac {x}{25}\right )-\frac {5}{4} \int \left (15625+\frac {390625}{-25+x}+625 x+25 x^2+x^3\right ) \, dx+50 \int \left (625+\frac {15625}{-25+x}+25 x+x^2\right ) \, dx-\frac {200}{3} \int \left (625+\frac {15625}{-25+x}+25 x+x^2\right ) \, dx+\frac {625}{2} \int \left (25+\frac {625}{-25+x}+x\right ) \, dx+3375 \int \left (25+\frac {625}{-25+x}+x\right ) \, dx+7590 \int x \log (-25+x) \, dx-37500 \operatorname {Subst}\left (\int \left (\log (x)-\frac {25 \log (x)}{25+x}\right ) \, dx,x,-25+x\right )+341550 \int \frac {(25-x) \log (-25+x)}{x} \, dx-379500 \int \frac {(25-x) \log (-25+x)}{x} \, dx+948750 \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {x}{25}\right )}{x} \, dx,x,-25+x\right )-4269375 \operatorname {Subst}\left (\int \frac {\log ^2(x)}{25+x} \, dx,x,-25+x\right )-4743750 \int \frac {\log (25-x)}{x} \, dx+4743750 \operatorname {Subst}\left (\int \frac {\log ^2(x)}{25+x} \, dx,x,-25+x\right )\\ &=95340 x+\frac {3795 x^2}{2}+2371875 \log (25-x)+450 (25-x) \log (-25+x)-2880405 \log (25) \log (-25+x)-405 \log (-25+x) \log \left (\frac {x}{25}\right )-2868750 \log (25) \log (x)+2868750 \log (25-x) \log (x)+450 (25-x) \log (-25+x) \log (x)-215 x^2 \log (-25+x) \log (x)-50 x^3 \log (-25+x) \log (x)-5 x^4 \log (-25+x) \log (x)+2880000 \text {Li}_2\left (1-\frac {x}{25}\right )-948750 \log (-25+x) \text {Li}_2\left (1-\frac {x}{25}\right )-1875000 \text {Li}_2\left (\frac {x}{25}\right )+948750 \text {Li}_3\left (1-\frac {x}{25}\right )-3795 \int \frac {x^2}{-25+x} \, dx-37500 \operatorname {Subst}(\int \log (x) \, dx,x,-25+x)-341550 \operatorname {Subst}\left (\int \frac {x \log (x)}{25+x} \, dx,x,-25+x\right )+379500 \operatorname {Subst}\left (\int \frac {x \log (x)}{25+x} \, dx,x,-25+x\right )+937500 \operatorname {Subst}\left (\int \frac {\log (x)}{25+x} \, dx,x,-25+x\right )-4743750 \int \frac {\log \left (1-\frac {x}{25}\right )}{x} \, dx+8538750 \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {x}{25}\right ) \log (x)}{x} \, dx,x,-25+x\right )-9487500 \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {x}{25}\right ) \log (x)}{x} \, dx,x,-25+x\right )\\ &=132840 x+\frac {3795 x^2}{2}+2371875 \log (25-x)+37950 (25-x) \log (-25+x)-2880405 \log (25) \log (-25+x)+937095 \log (-25+x) \log \left (\frac {x}{25}\right )-2868750 \log (25) \log (x)+2868750 \log (25-x) \log (x)+450 (25-x) \log (-25+x) \log (x)-215 x^2 \log (-25+x) \log (x)-50 x^3 \log (-25+x) \log (x)-5 x^4 \log (-25+x) \log (x)+2880000 \text {Li}_2\left (1-\frac {x}{25}\right )+2868750 \text {Li}_2\left (\frac {x}{25}\right )+948750 \text {Li}_3\left (1-\frac {x}{25}\right )-3795 \int \left (25+\frac {625}{-25+x}+x\right ) \, dx-341550 \operatorname {Subst}\left (\int \left (\log (x)-\frac {25 \log (x)}{25+x}\right ) \, dx,x,-25+x\right )+379500 \operatorname {Subst}\left (\int \left (\log (x)-\frac {25 \log (x)}{25+x}\right ) \, dx,x,-25+x\right )-937500 \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {x}{25}\right )}{x} \, dx,x,-25+x\right )+8538750 \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {x}{25}\right )}{x} \, dx,x,-25+x\right )-9487500 \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {x}{25}\right )}{x} \, dx,x,-25+x\right )\\ &=37965 x+37950 (25-x) \log (-25+x)-2880405 \log (25) \log (-25+x)+937095 \log (-25+x) \log \left (\frac {x}{25}\right )-2868750 \log (25) \log (x)+2868750 \log (25-x) \log (x)+450 (25-x) \log (-25+x) \log (x)-215 x^2 \log (-25+x) \log (x)-50 x^3 \log (-25+x) \log (x)-5 x^4 \log (-25+x) \log (x)+3817500 \text {Li}_2\left (1-\frac {x}{25}\right )+2868750 \text {Li}_2\left (\frac {x}{25}\right )-341550 \operatorname {Subst}(\int \log (x) \, dx,x,-25+x)+379500 \operatorname {Subst}(\int \log (x) \, dx,x,-25+x)+8538750 \operatorname {Subst}\left (\int \frac {\log (x)}{25+x} \, dx,x,-25+x\right )-9487500 \operatorname {Subst}\left (\int \frac {\log (x)}{25+x} \, dx,x,-25+x\right )\\ &=15 x-2880405 \log (25) \log (-25+x)-11655 \log (-25+x) \log \left (\frac {x}{25}\right )-2868750 \log (25) \log (x)+2868750 \log (25-x) \log (x)+450 (25-x) \log (-25+x) \log (x)-215 x^2 \log (-25+x) \log (x)-50 x^3 \log (-25+x) \log (x)-5 x^4 \log (-25+x) \log (x)+3817500 \text {Li}_2\left (1-\frac {x}{25}\right )+2868750 \text {Li}_2\left (\frac {x}{25}\right )-8538750 \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {x}{25}\right )}{x} \, dx,x,-25+x\right )+9487500 \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {x}{25}\right )}{x} \, dx,x,-25+x\right )\\ &=15 x-2880405 \log (25) \log (-25+x)-11655 \log (-25+x) \log \left (\frac {x}{25}\right )-2868750 \log (25) \log (x)+2868750 \log (25-x) \log (x)+450 (25-x) \log (-25+x) \log (x)-215 x^2 \log (-25+x) \log (x)-50 x^3 \log (-25+x) \log (x)-5 x^4 \log (-25+x) \log (x)+2868750 \text {Li}_2\left (1-\frac {x}{25}\right )+2868750 \text {Li}_2\left (\frac {x}{25}\right )\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.23, size = 23, normalized size = 0.88 \begin {gather*} -5 \left (-3 x+\left (9+5 x+x^2\right )^2 \log (-25+x) \log (x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-375*x + 15*x^2 + (10125 + 10845*x + 4925*x^2 + 1035*x^3 + 75*x^4 - 5*x^5)*Log[-25 + x] + (-405*x -
450*x^2 - 215*x^3 - 50*x^4 - 5*x^5 + (11250*x + 10300*x^2 + 3320*x^3 + 350*x^4 - 20*x^5)*Log[-25 + x])*Log[x]
)/(-25*x + x^2),x]

[Out]

-5*(-3*x + (9 + 5*x + x^2)^2*Log[-25 + x]*Log[x])

________________________________________________________________________________________

fricas [A]  time = 0.87, size = 30, normalized size = 1.15 \begin {gather*} -5 \, {\left (x^{4} + 10 \, x^{3} + 43 \, x^{2} + 90 \, x + 81\right )} \log \left (x - 25\right ) \log \relax (x) + 15 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-20*x^5+350*x^4+3320*x^3+10300*x^2+11250*x)*log(x-25)-5*x^5-50*x^4-215*x^3-450*x^2-405*x)*log(x)+
(-5*x^5+75*x^4+1035*x^3+4925*x^2+10845*x+10125)*log(x-25)+15*x^2-375*x)/(x^2-25*x),x, algorithm="fricas")

[Out]

-5*(x^4 + 10*x^3 + 43*x^2 + 90*x + 81)*log(x - 25)*log(x) + 15*x

________________________________________________________________________________________

giac [A]  time = 0.15, size = 35, normalized size = 1.35 \begin {gather*} -5 \, {\left ({\left (x^{4} + 10 \, x^{3} + 43 \, x^{2} + 90 \, x\right )} \log \relax (x) + 81 \, \log \relax (x)\right )} \log \left (x - 25\right ) + 15 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-20*x^5+350*x^4+3320*x^3+10300*x^2+11250*x)*log(x-25)-5*x^5-50*x^4-215*x^3-450*x^2-405*x)*log(x)+
(-5*x^5+75*x^4+1035*x^3+4925*x^2+10845*x+10125)*log(x-25)+15*x^2-375*x)/(x^2-25*x),x, algorithm="giac")

[Out]

-5*((x^4 + 10*x^3 + 43*x^2 + 90*x)*log(x) + 81*log(x))*log(x - 25) + 15*x

________________________________________________________________________________________

maple [A]  time = 0.26, size = 41, normalized size = 1.58

method	result	size
risch	\(\left (-5 x^{4} \ln \relax (x )-50 x^{3} \ln \relax (x )-215 x^{2} \ln \relax (x )-450 x \ln \relax (x )-405 \ln \relax (x )\right ) \ln \left (x -25\right )+15 x\)	\(41\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-20*x^5+350*x^4+3320*x^3+10300*x^2+11250*x)*ln(x-25)-5*x^5-50*x^4-215*x^3-450*x^2-405*x)*ln(x)+(-5*x^5+
75*x^4+1035*x^3+4925*x^2+10845*x+10125)*ln(x-25)+15*x^2-375*x)/(x^2-25*x),x,method=_RETURNVERBOSE)

[Out]

(-5*x^4*ln(x)-50*x^3*ln(x)-215*x^2*ln(x)-450*x*ln(x)-405*ln(x))*ln(x-25)+15*x

________________________________________________________________________________________

maxima [A]  time = 0.41, size = 30, normalized size = 1.15 \begin {gather*} -5 \, {\left (x^{4} + 10 \, x^{3} + 43 \, x^{2} + 90 \, x + 81\right )} \log \left (x - 25\right ) \log \relax (x) + 15 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-20*x^5+350*x^4+3320*x^3+10300*x^2+11250*x)*log(x-25)-5*x^5-50*x^4-215*x^3-450*x^2-405*x)*log(x)+
(-5*x^5+75*x^4+1035*x^3+4925*x^2+10845*x+10125)*log(x-25)+15*x^2-375*x)/(x^2-25*x),x, algorithm="maxima")

[Out]

-5*(x^4 + 10*x^3 + 43*x^2 + 90*x + 81)*log(x - 25)*log(x) + 15*x

________________________________________________________________________________________

mupad [B]  time = 5.40, size = 32, normalized size = 1.23 \begin {gather*} 15\,x-\ln \left (x-25\right )\,\ln \relax (x)\,\left (5\,x^4+50\,x^3+215\,x^2+450\,x+405\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((375*x + log(x)*(405*x - log(x - 25)*(11250*x + 10300*x^2 + 3320*x^3 + 350*x^4 - 20*x^5) + 450*x^2 + 215*x
^3 + 50*x^4 + 5*x^5) - log(x - 25)*(10845*x + 4925*x^2 + 1035*x^3 + 75*x^4 - 5*x^5 + 10125) - 15*x^2)/(25*x -
x^2),x)

[Out]

15*x - log(x - 25)*log(x)*(450*x + 215*x^2 + 50*x^3 + 5*x^4 + 405)

________________________________________________________________________________________

sympy [B]  time = 0.48, size = 46, normalized size = 1.77 \begin {gather*} 15 x + \left (- 5 x^{4} \log {\relax (x )} - 50 x^{3} \log {\relax (x )} - 215 x^{2} \log {\relax (x )} - 450 x \log {\relax (x )} - 405 \log {\relax (x )}\right ) \log {\left (x - 25 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-20*x**5+350*x**4+3320*x**3+10300*x**2+11250*x)*ln(x-25)-5*x**5-50*x**4-215*x**3-450*x**2-405*x)*
ln(x)+(-5*x**5+75*x**4+1035*x**3+4925*x**2+10845*x+10125)*ln(x-25)+15*x**2-375*x)/(x**2-25*x),x)

[Out]

15*x + (-5*x**4*log(x) - 50*x**3*log(x) - 215*x**2*log(x) - 450*x*log(x) - 405*log(x))*log(x - 25)

________________________________________________________________________________________