\(\int \frac {55+30 x-18 x^2+102 x^3+110 x^4-56 x^5+6 x^6+100 x^7-40 x^8+4 x^9+e^2 (-50 x^2+20 x^3-2 x^4-100 x^5+40 x^6-4 x^7)+(275+150 x-90 x^2+410 x^3+390 x^4-204 x^5+22 x^6+200 x^7-80 x^8+8 x^9+e^4 (100 x^3-40 x^4+4 x^5)+e^2 (-100 x-210 x^2+96 x^3-10 x^4-300 x^5+120 x^6-12 x^7)) \log (x)+(550+300 x-180 x^2+520 x^3+450 x^4-240 x^5+26 x^6+e^2 (-200 x-270 x^2+132 x^3-14 x^4)) \log ^2(x)+(550+300 x-180 x^2+220 x^3+170 x^4-92 x^5+10 x^6+e^2 (-100 x-110 x^2+56 x^3-6 x^4)) \log ^3(x)+(275+150 x-90 x^2+10 x^3) \log ^4(x)+(55+30 x-18 x^2+2 x^3) \log ^5(x)}{25-10 x+x^2+(125-50 x+5 x^2) \log (x)+(250-100 x+10 x^2) \log ^2(x)+(250-100 x+10 x^2) \log ^3(x)+(125-50 x+5 x^2) \log ^4(x)+(25-10 x+x^2) \log ^5(x)} \, dx\) [4703]

Optimal result

Integrand size = 424, antiderivative size = 36 \[ \int \frac {55+30 x-18 x^2+102 x^3+110 x^4-56 x^5+6 x^6+100 x^7-40 x^8+4 x^9+e^2 \left (-50 x^2+20 x^3-2 x^4-100 x^5+40 x^6-4 x^7\right )+\left (275+150 x-90 x^2+410 x^3+390 x^4-204 x^5+22 x^6+200 x^7-80 x^8+8 x^9+e^4 \left (100 x^3-40 x^4+4 x^5\right )+e^2 \left (-100 x-210 x^2+96 x^3-10 x^4-300 x^5+120 x^6-12 x^7\right )\right ) \log (x)+\left (550+300 x-180 x^2+520 x^3+450 x^4-240 x^5+26 x^6+e^2 \left (-200 x-270 x^2+132 x^3-14 x^4\right )\right ) \log ^2(x)+\left (550+300 x-180 x^2+220 x^3+170 x^4-92 x^5+10 x^6+e^2 \left (-100 x-110 x^2+56 x^3-6 x^4\right )\right ) \log ^3(x)+\left (275+150 x-90 x^2+10 x^3\right ) \log ^4(x)+\left (55+30 x-18 x^2+2 x^3\right ) \log ^5(x)}{25-10 x+x^2+\left (125-50 x+5 x^2\right ) \log (x)+\left (250-100 x+10 x^2\right ) \log ^2(x)+\left (250-100 x+10 x^2\right ) \log ^3(x)+\left (125-50 x+5 x^2\right ) \log ^4(x)+\left (25-10 x+x^2\right ) \log ^5(x)} \, dx=\frac {5}{5-x}+\left (-1-x+\frac {x^2 \left (e^2-x^2\right )}{(1+\log (x))^2}\right )^2 \]

[Out]

5/(5-x)+(-1+(exp(2)-x^2)/(ln(x)+1)^2*x^2-x)^2

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(103\) vs. \(2(36)=72\).

Time = 6.78 (sec) , antiderivative size = 103, normalized size of antiderivative = 2.86, number of steps used = 74, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.019, Rules used = {6873, 6874, 1864, 2395, 2343, 2346, 2209, 2403} \[ \int \frac {55+30 x-18 x^2+102 x^3+110 x^4-56 x^5+6 x^6+100 x^7-40 x^8+4 x^9+e^2 \left (-50 x^2+20 x^3-2 x^4-100 x^5+40 x^6-4 x^7\right )+\left (275+150 x-90 x^2+410 x^3+390 x^4-204 x^5+22 x^6+200 x^7-80 x^8+8 x^9+e^4 \left (100 x^3-40 x^4+4 x^5\right )+e^2 \left (-100 x-210 x^2+96 x^3-10 x^4-300 x^5+120 x^6-12 x^7\right )\right ) \log (x)+\left (550+300 x-180 x^2+520 x^3+450 x^4-240 x^5+26 x^6+e^2 \left (-200 x-270 x^2+132 x^3-14 x^4\right )\right ) \log ^2(x)+\left (550+300 x-180 x^2+220 x^3+170 x^4-92 x^5+10 x^6+e^2 \left (-100 x-110 x^2+56 x^3-6 x^4\right )\right ) \log ^3(x)+\left (275+150 x-90 x^2+10 x^3\right ) \log ^4(x)+\left (55+30 x-18 x^2+2 x^3\right ) \log ^5(x)}{25-10 x+x^2+\left (125-50 x+5 x^2\right ) \log (x)+\left (250-100 x+10 x^2\right ) \log ^2(x)+\left (250-100 x+10 x^2\right ) \log ^3(x)+\left (125-50 x+5 x^2\right ) \log ^4(x)+\left (25-10 x+x^2\right ) \log ^5(x)} \, dx=\frac {x^8}{(\log (x)+1)^4}-\frac {2 e^2 x^6}{(\log (x)+1)^4}+\frac {2 x^5}{(\log (x)+1)^2}+\frac {2 x^4}{(\log (x)+1)^2}+\frac {e^4 x^4}{(\log (x)+1)^4}-\frac {2 e^2 x^3}{(\log (x)+1)^2}+x^2-\frac {2 e^2 x^2}{(\log (x)+1)^2}+2 x+\frac {5}{5-x} \]

[In]

Int[(55 + 30*x - 18*x^2 + 102*x^3 + 110*x^4 - 56*x^5 + 6*x^6 + 100*x^7 - 40*x^8 + 4*x^9 + E^2*(-50*x^2 + 20*x^
3 - 2*x^4 - 100*x^5 + 40*x^6 - 4*x^7) + (275 + 150*x - 90*x^2 + 410*x^3 + 390*x^4 - 204*x^5 + 22*x^6 + 200*x^7
- 80*x^8 + 8*x^9 + E^4*(100*x^3 - 40*x^4 + 4*x^5) + E^2*(-100*x - 210*x^2 + 96*x^3 - 10*x^4 - 300*x^5 + 120*x
^6 - 12*x^7))*Log[x] + (550 + 300*x - 180*x^2 + 520*x^3 + 450*x^4 - 240*x^5 + 26*x^6 + E^2*(-200*x - 270*x^2 +
132*x^3 - 14*x^4))*Log[x]^2 + (550 + 300*x - 180*x^2 + 220*x^3 + 170*x^4 - 92*x^5 + 10*x^6 + E^2*(-100*x - 11
0*x^2 + 56*x^3 - 6*x^4))*Log[x]^3 + (275 + 150*x - 90*x^2 + 10*x^3)*Log[x]^4 + (55 + 30*x - 18*x^2 + 2*x^3)*Lo
g[x]^5)/(25 - 10*x + x^2 + (125 - 50*x + 5*x^2)*Log[x] + (250 - 100*x + 10*x^2)*Log[x]^2 + (250 - 100*x + 10*x
^2)*Log[x]^3 + (125 - 50*x + 5*x^2)*Log[x]^4 + (25 - 10*x + x^2)*Log[x]^5),x]

[Out]

5/(5 - x) + 2*x + x^2 + (E^4*x^4)/(1 + Log[x])^4 - (2*E^2*x^6)/(1 + Log[x])^4 + x^8/(1 + Log[x])^4 - (2*E^2*x^
2)/(1 + Log[x])^2 - (2*E^2*x^3)/(1 + Log[x])^2 + (2*x^4)/(1 + Log[x])^2 + (2*x^5)/(1 + Log[x])^2

Rule 1864

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq*(a + b*x^n)^p, x], x] /; FreeQ[
{a, b, n}, x] && PolyQ[Pq, x] && (IGtQ[p, 0] || EqQ[n, 1])

Rule 2209

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

Rule 2343

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

Rule 2346

Int[((a_.) + Log[(c_.)*(x_)]*(b_.))^(p_)*(x_)^(m_.), x_Symbol] :> Dist[1/c^(m + 1), Subst[Int[E^((m + 1)*x)*(a
+ b*x)^p, x], x, Log[c*x]], x] /; FreeQ[{a, b, c, p}, x] && IntegerQ[m]

Rule 2395

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

Rule 2403

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

Rule 6873

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

Rule 6874

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

Rubi steps \begin{align*} \text {integral}& = \int \frac {55+30 x-18 x^2+102 x^3+110 x^4-56 x^5+6 x^6+100 x^7-40 x^8+4 x^9+e^2 \left (-50 x^2+20 x^3-2 x^4-100 x^5+40 x^6-4 x^7\right )+\left (275+150 x-90 x^2+410 x^3+390 x^4-204 x^5+22 x^6+200 x^7-80 x^8+8 x^9+e^4 \left (100 x^3-40 x^4+4 x^5\right )+e^2 \left (-100 x-210 x^2+96 x^3-10 x^4-300 x^5+120 x^6-12 x^7\right )\right ) \log (x)+\left (550+300 x-180 x^2+520 x^3+450 x^4-240 x^5+26 x^6+e^2 \left (-200 x-270 x^2+132 x^3-14 x^4\right )\right ) \log ^2(x)+\left (550+300 x-180 x^2+220 x^3+170 x^4-92 x^5+10 x^6+e^2 \left (-100 x-110 x^2+56 x^3-6 x^4\right )\right ) \log ^3(x)+\left (275+150 x-90 x^2+10 x^3\right ) \log ^4(x)+\left (55+30 x-18 x^2+2 x^3\right ) \log ^5(x)}{(5-x)^2 (1+\log (x))^5} \, dx \\ & = \int \left (\frac {55+30 x-18 x^2+2 x^3}{(-5+x)^2}-\frac {4 x^3 \left (-e^2+x^2\right )^2}{(1+\log (x))^5}+\frac {4 x^3 \left (e^4-3 e^2 x^2+2 x^4\right )}{(1+\log (x))^4}-\frac {4 x (1+x) \left (-e^2+x^2\right )}{(1+\log (x))^3}+\frac {2 x \left (-2 e^2-3 e^2 x+4 x^2+5 x^3\right )}{(1+\log (x))^2}\right ) \, dx \\ & = 2 \int \frac {x \left (-2 e^2-3 e^2 x+4 x^2+5 x^3\right )}{(1+\log (x))^2} \, dx-4 \int \frac {x^3 \left (-e^2+x^2\right )^2}{(1+\log (x))^5} \, dx+4 \int \frac {x^3 \left (e^4-3 e^2 x^2+2 x^4\right )}{(1+\log (x))^4} \, dx-4 \int \frac {x (1+x) \left (-e^2+x^2\right )}{(1+\log (x))^3} \, dx+\int \frac {55+30 x-18 x^2+2 x^3}{(-5+x)^2} \, dx \\ & = 2 \int \left (-\frac {2 e^2 x}{(1+\log (x))^2}-\frac {3 e^2 x^2}{(1+\log (x))^2}+\frac {4 x^3}{(1+\log (x))^2}+\frac {5 x^4}{(1+\log (x))^2}\right ) \, dx-4 \int \left (\frac {e^4 x^3}{(1+\log (x))^5}-\frac {2 e^2 x^5}{(1+\log (x))^5}+\frac {x^7}{(1+\log (x))^5}\right ) \, dx+4 \int \left (\frac {e^4 x^3}{(1+\log (x))^4}-\frac {3 e^2 x^5}{(1+\log (x))^4}+\frac {2 x^7}{(1+\log (x))^4}\right ) \, dx-4 \int \left (-\frac {e^2 x}{(1+\log (x))^3}-\frac {e^2 x^2}{(1+\log (x))^3}+\frac {x^3}{(1+\log (x))^3}+\frac {x^4}{(1+\log (x))^3}\right ) \, dx+\int \left (2+\frac {5}{(-5+x)^2}+2 x\right ) \, dx \\ & = \frac {5}{5-x}+2 x+x^2-4 \int \frac {x^7}{(1+\log (x))^5} \, dx-4 \int \frac {x^3}{(1+\log (x))^3} \, dx-4 \int \frac {x^4}{(1+\log (x))^3} \, dx+8 \int \frac {x^7}{(1+\log (x))^4} \, dx+8 \int \frac {x^3}{(1+\log (x))^2} \, dx+10 \int \frac {x^4}{(1+\log (x))^2} \, dx+\left (4 e^2\right ) \int \frac {x}{(1+\log (x))^3} \, dx+\left (4 e^2\right ) \int \frac {x^2}{(1+\log (x))^3} \, dx-\left (4 e^2\right ) \int \frac {x}{(1+\log (x))^2} \, dx-\left (6 e^2\right ) \int \frac {x^2}{(1+\log (x))^2} \, dx+\left (8 e^2\right ) \int \frac {x^5}{(1+\log (x))^5} \, dx-\left (12 e^2\right ) \int \frac {x^5}{(1+\log (x))^4} \, dx-\left (4 e^4\right ) \int \frac {x^3}{(1+\log (x))^5} \, dx+\left (4 e^4\right ) \int \frac {x^3}{(1+\log (x))^4} \, dx \\ & = \frac {5}{5-x}+2 x+x^2+\frac {e^4 x^4}{(1+\log (x))^4}-\frac {2 e^2 x^6}{(1+\log (x))^4}+\frac {x^8}{(1+\log (x))^4}-\frac {4 e^4 x^4}{3 (1+\log (x))^3}+\frac {4 e^2 x^6}{(1+\log (x))^3}-\frac {8 x^8}{3 (1+\log (x))^3}-\frac {2 e^2 x^2}{(1+\log (x))^2}-\frac {2 e^2 x^3}{(1+\log (x))^2}+\frac {2 x^4}{(1+\log (x))^2}+\frac {2 x^5}{(1+\log (x))^2}+\frac {4 e^2 x^2}{1+\log (x)}+\frac {6 e^2 x^3}{1+\log (x)}-\frac {8 x^4}{1+\log (x)}-\frac {10 x^5}{1+\log (x)}-8 \int \frac {x^7}{(1+\log (x))^4} \, dx-8 \int \frac {x^3}{(1+\log (x))^2} \, dx-10 \int \frac {x^4}{(1+\log (x))^2} \, dx+\frac {64}{3} \int \frac {x^7}{(1+\log (x))^3} \, dx+32 \int \frac {x^3}{1+\log (x)} \, dx+50 \int \frac {x^4}{1+\log (x)} \, dx+\left (4 e^2\right ) \int \frac {x}{(1+\log (x))^2} \, dx+\left (6 e^2\right ) \int \frac {x^2}{(1+\log (x))^2} \, dx-\left (8 e^2\right ) \int \frac {x}{1+\log (x)} \, dx+\left (12 e^2\right ) \int \frac {x^5}{(1+\log (x))^4} \, dx-\left (18 e^2\right ) \int \frac {x^2}{1+\log (x)} \, dx-\left (24 e^2\right ) \int \frac {x^5}{(1+\log (x))^3} \, dx-\left (4 e^4\right ) \int \frac {x^3}{(1+\log (x))^4} \, dx+\frac {1}{3} \left (16 e^4\right ) \int \frac {x^3}{(1+\log (x))^3} \, dx \\ & = \frac {5}{5-x}+2 x+x^2+\frac {e^4 x^4}{(1+\log (x))^4}-\frac {2 e^2 x^6}{(1+\log (x))^4}+\frac {x^8}{(1+\log (x))^4}-\frac {2 e^2 x^2}{(1+\log (x))^2}-\frac {2 e^2 x^3}{(1+\log (x))^2}+\frac {2 x^4}{(1+\log (x))^2}-\frac {8 e^4 x^4}{3 (1+\log (x))^2}+\frac {2 x^5}{(1+\log (x))^2}+\frac {12 e^2 x^6}{(1+\log (x))^2}-\frac {32 x^8}{3 (1+\log (x))^2}-\frac {64}{3} \int \frac {x^7}{(1+\log (x))^3} \, dx-32 \int \frac {x^3}{1+\log (x)} \, dx+32 \text {Subst}\left (\int \frac {e^{4 x}}{1+x} \, dx,x,\log (x)\right )-50 \int \frac {x^4}{1+\log (x)} \, dx+50 \text {Subst}\left (\int \frac {e^{5 x}}{1+x} \, dx,x,\log (x)\right )+\frac {256}{3} \int \frac {x^7}{(1+\log (x))^2} \, dx+\left (8 e^2\right ) \int \frac {x}{1+\log (x)} \, dx-\left (8 e^2\right ) \text {Subst}\left (\int \frac {e^{2 x}}{1+x} \, dx,x,\log (x)\right )+\left (18 e^2\right ) \int \frac {x^2}{1+\log (x)} \, dx-\left (18 e^2\right ) \text {Subst}\left (\int \frac {e^{3 x}}{1+x} \, dx,x,\log (x)\right )+\left (24 e^2\right ) \int \frac {x^5}{(1+\log (x))^3} \, dx-\left (72 e^2\right ) \int \frac {x^5}{(1+\log (x))^2} \, dx-\frac {1}{3} \left (16 e^4\right ) \int \frac {x^3}{(1+\log (x))^3} \, dx+\frac {1}{3} \left (32 e^4\right ) \int \frac {x^3}{(1+\log (x))^2} \, dx \\ & = \frac {5}{5-x}+2 x+x^2-8 \text {Ei}(2 (1+\log (x)))-\frac {18 \text {Ei}(3 (1+\log (x)))}{e}+\frac {32 \text {Ei}(4 (1+\log (x)))}{e^4}+\frac {50 \text {Ei}(5 (1+\log (x)))}{e^5}+\frac {e^4 x^4}{(1+\log (x))^4}-\frac {2 e^2 x^6}{(1+\log (x))^4}+\frac {x^8}{(1+\log (x))^4}-\frac {2 e^2 x^2}{(1+\log (x))^2}-\frac {2 e^2 x^3}{(1+\log (x))^2}+\frac {2 x^4}{(1+\log (x))^2}+\frac {2 x^5}{(1+\log (x))^2}-\frac {32 e^4 x^4}{3 (1+\log (x))}+\frac {72 e^2 x^6}{1+\log (x)}-\frac {256 x^8}{3 (1+\log (x))}-32 \text {Subst}\left (\int \frac {e^{4 x}}{1+x} \, dx,x,\log (x)\right )-50 \text {Subst}\left (\int \frac {e^{5 x}}{1+x} \, dx,x,\log (x)\right )-\frac {256}{3} \int \frac {x^7}{(1+\log (x))^2} \, dx+\frac {2048}{3} \int \frac {x^7}{1+\log (x)} \, dx+\left (8 e^2\right ) \text {Subst}\left (\int \frac {e^{2 x}}{1+x} \, dx,x,\log (x)\right )+\left (18 e^2\right ) \text {Subst}\left (\int \frac {e^{3 x}}{1+x} \, dx,x,\log (x)\right )+\left (72 e^2\right ) \int \frac {x^5}{(1+\log (x))^2} \, dx-\left (432 e^2\right ) \int \frac {x^5}{1+\log (x)} \, dx-\frac {1}{3} \left (32 e^4\right ) \int \frac {x^3}{(1+\log (x))^2} \, dx+\frac {1}{3} \left (128 e^4\right ) \int \frac {x^3}{1+\log (x)} \, dx \\ & = \frac {5}{5-x}+2 x+x^2+\frac {e^4 x^4}{(1+\log (x))^4}-\frac {2 e^2 x^6}{(1+\log (x))^4}+\frac {x^8}{(1+\log (x))^4}-\frac {2 e^2 x^2}{(1+\log (x))^2}-\frac {2 e^2 x^3}{(1+\log (x))^2}+\frac {2 x^4}{(1+\log (x))^2}+\frac {2 x^5}{(1+\log (x))^2}-\frac {2048}{3} \int \frac {x^7}{1+\log (x)} \, dx+\frac {2048}{3} \text {Subst}\left (\int \frac {e^{8 x}}{1+x} \, dx,x,\log (x)\right )+\left (432 e^2\right ) \int \frac {x^5}{1+\log (x)} \, dx-\left (432 e^2\right ) \text {Subst}\left (\int \frac {e^{6 x}}{1+x} \, dx,x,\log (x)\right )-\frac {1}{3} \left (128 e^4\right ) \int \frac {x^3}{1+\log (x)} \, dx+\frac {1}{3} \left (128 e^4\right ) \text {Subst}\left (\int \frac {e^{4 x}}{1+x} \, dx,x,\log (x)\right ) \\ & = \frac {5}{5-x}+2 x+x^2+\frac {128}{3} \text {Ei}(4 (1+\log (x)))-\frac {432 \text {Ei}(6 (1+\log (x)))}{e^4}+\frac {2048 \text {Ei}(8 (1+\log (x)))}{3 e^8}+\frac {e^4 x^4}{(1+\log (x))^4}-\frac {2 e^2 x^6}{(1+\log (x))^4}+\frac {x^8}{(1+\log (x))^4}-\frac {2 e^2 x^2}{(1+\log (x))^2}-\frac {2 e^2 x^3}{(1+\log (x))^2}+\frac {2 x^4}{(1+\log (x))^2}+\frac {2 x^5}{(1+\log (x))^2}-\frac {2048}{3} \text {Subst}\left (\int \frac {e^{8 x}}{1+x} \, dx,x,\log (x)\right )+\left (432 e^2\right ) \text {Subst}\left (\int \frac {e^{6 x}}{1+x} \, dx,x,\log (x)\right )-\frac {1}{3} \left (128 e^4\right ) \text {Subst}\left (\int \frac {e^{4 x}}{1+x} \, dx,x,\log (x)\right ) \\ & = \frac {5}{5-x}+2 x+x^2+\frac {e^4 x^4}{(1+\log (x))^4}-\frac {2 e^2 x^6}{(1+\log (x))^4}+\frac {x^8}{(1+\log (x))^4}-\frac {2 e^2 x^2}{(1+\log (x))^2}-\frac {2 e^2 x^3}{(1+\log (x))^2}+\frac {2 x^4}{(1+\log (x))^2}+\frac {2 x^5}{(1+\log (x))^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.20 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.61 \[ \int \frac {55+30 x-18 x^2+102 x^3+110 x^4-56 x^5+6 x^6+100 x^7-40 x^8+4 x^9+e^2 \left (-50 x^2+20 x^3-2 x^4-100 x^5+40 x^6-4 x^7\right )+\left (275+150 x-90 x^2+410 x^3+390 x^4-204 x^5+22 x^6+200 x^7-80 x^8+8 x^9+e^4 \left (100 x^3-40 x^4+4 x^5\right )+e^2 \left (-100 x-210 x^2+96 x^3-10 x^4-300 x^5+120 x^6-12 x^7\right )\right ) \log (x)+\left (550+300 x-180 x^2+520 x^3+450 x^4-240 x^5+26 x^6+e^2 \left (-200 x-270 x^2+132 x^3-14 x^4\right )\right ) \log ^2(x)+\left (550+300 x-180 x^2+220 x^3+170 x^4-92 x^5+10 x^6+e^2 \left (-100 x-110 x^2+56 x^3-6 x^4\right )\right ) \log ^3(x)+\left (275+150 x-90 x^2+10 x^3\right ) \log ^4(x)+\left (55+30 x-18 x^2+2 x^3\right ) \log ^5(x)}{25-10 x+x^2+\left (125-50 x+5 x^2\right ) \log (x)+\left (250-100 x+10 x^2\right ) \log ^2(x)+\left (250-100 x+10 x^2\right ) \log ^3(x)+\left (125-50 x+5 x^2\right ) \log ^4(x)+\left (25-10 x+x^2\right ) \log ^5(x)} \, dx=-\frac {5}{-5+x}+2 x+x^2+\frac {x^4 \left (e^2-x^2\right )^2}{(1+\log (x))^4}+\frac {2 x^2 (1+x) \left (-e^2+x^2\right )}{(1+\log (x))^2} \]

[In]

Integrate[(55 + 30*x - 18*x^2 + 102*x^3 + 110*x^4 - 56*x^5 + 6*x^6 + 100*x^7 - 40*x^8 + 4*x^9 + E^2*(-50*x^2 +
20*x^3 - 2*x^4 - 100*x^5 + 40*x^6 - 4*x^7) + (275 + 150*x - 90*x^2 + 410*x^3 + 390*x^4 - 204*x^5 + 22*x^6 + 2
00*x^7 - 80*x^8 + 8*x^9 + E^4*(100*x^3 - 40*x^4 + 4*x^5) + E^2*(-100*x - 210*x^2 + 96*x^3 - 10*x^4 - 300*x^5 +
120*x^6 - 12*x^7))*Log[x] + (550 + 300*x - 180*x^2 + 520*x^3 + 450*x^4 - 240*x^5 + 26*x^6 + E^2*(-200*x - 270
*x^2 + 132*x^3 - 14*x^4))*Log[x]^2 + (550 + 300*x - 180*x^2 + 220*x^3 + 170*x^4 - 92*x^5 + 10*x^6 + E^2*(-100*
x - 110*x^2 + 56*x^3 - 6*x^4))*Log[x]^3 + (275 + 150*x - 90*x^2 + 10*x^3)*Log[x]^4 + (55 + 30*x - 18*x^2 + 2*x
^3)*Log[x]^5)/(25 - 10*x + x^2 + (125 - 50*x + 5*x^2)*Log[x] + (250 - 100*x + 10*x^2)*Log[x]^2 + (250 - 100*x
+ 10*x^2)*Log[x]^3 + (125 - 50*x + 5*x^2)*Log[x]^4 + (25 - 10*x + x^2)*Log[x]^5),x]

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(111\) vs. \(2(35)=70\).

Time = 0.61 (sec) , antiderivative size = 112, normalized size of antiderivative = 3.11

method	result	size
default	\(\frac {x^{8}}{\left (\ln \left (x \right )+1\right )^{4}}-\frac {2 \,{\mathrm e}^{2} x^{6}}{\left (\ln \left (x \right )+1\right )^{4}}+\frac {2 x^{5}}{\left (\ln \left (x \right )+1\right )^{2}}+\frac {\left ({\mathrm e}^{4}+2 \ln \left (x \right )^{2}+4 \ln \left (x \right )+2\right ) x^{4}}{\left (\ln \left (x \right )+1\right )^{4}}-\frac {2 \,{\mathrm e}^{2} x^{3}}{\left (\ln \left (x \right )+1\right )^{2}}-\frac {\left (-\ln \left (x \right )^{2}+2 \,{\mathrm e}^{2}-2 \ln \left (x \right )-1\right ) x^{2}}{\left (\ln \left (x \right )+1\right )^{2}}+2 x -\frac {5}{-5+x}\)	\(112\)
risch	\(\frac {x^{3}-3 x^{2}-10 x -5}{-5+x}+\frac {x^{2} \left (x^{6}-2 x^{4} {\mathrm e}^{2}+2 x^{3} \ln \left (x \right )^{2}+x^{2} {\mathrm e}^{4}-2 x \,{\mathrm e}^{2} \ln \left (x \right )^{2}+4 x^{3} \ln \left (x \right )+2 x^{2} \ln \left (x \right )^{2}-4 x \,{\mathrm e}^{2} \ln \left (x \right )-2 \,{\mathrm e}^{2} \ln \left (x \right )^{2}+2 x^{3}+4 x^{2} \ln \left (x \right )-2 \,{\mathrm e}^{2} x -4 \,{\mathrm e}^{2} \ln \left (x \right )+2 x^{2}-2 \,{\mathrm e}^{2}\right )}{\left (\ln \left (x \right )+1\right )^{4}}\)	\(129\)
parallelrisch	\(\frac {-55+16 x^{3} {\mathrm e}^{2} \ln \left (x \right )+10 x^{6} {\mathrm e}^{2}+2 x^{6} \ln \left (x \right )^{2}+4 x^{6} \ln \left (x \right )+4 x^{3} \ln \left (x \right )^{3}+8 \,{\mathrm e}^{2} \ln \left (x \right )^{2} x^{3}-4 \,{\mathrm e}^{2} \ln \left (x \right ) x^{4}+10 \,{\mathrm e}^{2} \ln \left (x \right )^{2} x^{2}-2 \,{\mathrm e}^{2} \ln \left (x \right )^{2} x^{4}-16 x^{5} \ln \left (x \right )+x^{3} \ln \left (x \right )^{4}-10 x^{4} \ln \left (x \right )^{2}+8 x^{3} {\mathrm e}^{2}+10 x^{2} {\mathrm e}^{2}-20 x^{4} \ln \left (x \right )-2 x^{4} {\mathrm e}^{2}-2 \,{\mathrm e}^{2} x^{7}-8 x^{5} \ln \left (x \right )^{2}-5 x^{4} {\mathrm e}^{4}+x^{5} {\mathrm e}^{4}-12 x^{2} \ln \left (x \right )^{3}+6 x^{3} \ln \left (x \right )^{2}-18 x^{2} \ln \left (x \right )^{2}-3 x^{2} \ln \left (x \right )^{4}+20 x^{2} {\mathrm e}^{2} \ln \left (x \right )-5 x^{8}+x^{9}-55 \ln \left (x \right )^{4}-220 \ln \left (x \right )^{3}+2 x^{6}-8 x^{5}-220 \ln \left (x \right )-330 \ln \left (x \right )^{2}-10 x^{4}+x^{3}-3 x^{2}+4 x^{3} \ln \left (x \right )-12 x^{2} \ln \left (x \right )}{\left (\ln \left (x \right )^{4}+4 \ln \left (x \right )^{3}+6 \ln \left (x \right )^{2}+4 \ln \left (x \right )+1\right ) \left (-5+x \right )}\)	\(313\)

[In]

int(((2*x^3-18*x^2+30*x+55)*ln(x)^5+(10*x^3-90*x^2+150*x+275)*ln(x)^4+((-6*x^4+56*x^3-110*x^2-100*x)*exp(2)+10
*x^6-92*x^5+170*x^4+220*x^3-180*x^2+300*x+550)*ln(x)^3+((-14*x^4+132*x^3-270*x^2-200*x)*exp(2)+26*x^6-240*x^5+
450*x^4+520*x^3-180*x^2+300*x+550)*ln(x)^2+((4*x^5-40*x^4+100*x^3)*exp(2)^2+(-12*x^7+120*x^6-300*x^5-10*x^4+96
*x^3-210*x^2-100*x)*exp(2)+8*x^9-80*x^8+200*x^7+22*x^6-204*x^5+390*x^4+410*x^3-90*x^2+150*x+275)*ln(x)+(-4*x^7
+40*x^6-100*x^5-2*x^4+20*x^3-50*x^2)*exp(2)+4*x^9-40*x^8+100*x^7+6*x^6-56*x^5+110*x^4+102*x^3-18*x^2+30*x+55)/
((x^2-10*x+25)*ln(x)^5+(5*x^2-50*x+125)*ln(x)^4+(10*x^2-100*x+250)*ln(x)^3+(10*x^2-100*x+250)*ln(x)^2+(5*x^2-5
0*x+125)*ln(x)+x^2-10*x+25),x,method=_RETURNVERBOSE)

[Out]

1/(ln(x)+1)^4*x^8-2*exp(2)/(ln(x)+1)^4*x^6+2/(ln(x)+1)^2*x^5+(exp(2)^2+2*ln(x)^2+4*ln(x)+2)/(ln(x)+1)^4*x^4-2*
exp(2)/(ln(x)+1)^2*x^3-(-ln(x)^2+2*exp(2)-2*ln(x)-1)/(ln(x)+1)^2*x^2+2*x-5/(-5+x)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 250 vs. \(2 (31) = 62\).

Time = 0.25 (sec) , antiderivative size = 250, normalized size of antiderivative = 6.94 \[ \int \frac {55+30 x-18 x^2+102 x^3+110 x^4-56 x^5+6 x^6+100 x^7-40 x^8+4 x^9+e^2 \left (-50 x^2+20 x^3-2 x^4-100 x^5+40 x^6-4 x^7\right )+\left (275+150 x-90 x^2+410 x^3+390 x^4-204 x^5+22 x^6+200 x^7-80 x^8+8 x^9+e^4 \left (100 x^3-40 x^4+4 x^5\right )+e^2 \left (-100 x-210 x^2+96 x^3-10 x^4-300 x^5+120 x^6-12 x^7\right )\right ) \log (x)+\left (550+300 x-180 x^2+520 x^3+450 x^4-240 x^5+26 x^6+e^2 \left (-200 x-270 x^2+132 x^3-14 x^4\right )\right ) \log ^2(x)+\left (550+300 x-180 x^2+220 x^3+170 x^4-92 x^5+10 x^6+e^2 \left (-100 x-110 x^2+56 x^3-6 x^4\right )\right ) \log ^3(x)+\left (275+150 x-90 x^2+10 x^3\right ) \log ^4(x)+\left (55+30 x-18 x^2+2 x^3\right ) \log ^5(x)}{25-10 x+x^2+\left (125-50 x+5 x^2\right ) \log (x)+\left (250-100 x+10 x^2\right ) \log ^2(x)+\left (250-100 x+10 x^2\right ) \log ^3(x)+\left (125-50 x+5 x^2\right ) \log ^4(x)+\left (25-10 x+x^2\right ) \log ^5(x)} \, dx=\frac {x^{9} - 5 \, x^{8} + 2 \, x^{6} - 8 \, x^{5} + {\left (x^{3} - 3 \, x^{2} - 10 \, x - 5\right )} \log \left (x\right )^{4} - 10 \, x^{4} + 4 \, {\left (x^{3} - 3 \, x^{2} - 10 \, x - 5\right )} \log \left (x\right )^{3} + x^{3} + 2 \, {\left (x^{6} - 4 \, x^{5} - 5 \, x^{4} + 3 \, x^{3} - 9 \, x^{2} - {\left (x^{4} - 4 \, x^{3} - 5 \, x^{2}\right )} e^{2} - 30 \, x - 15\right )} \log \left (x\right )^{2} - 3 \, x^{2} + {\left (x^{5} - 5 \, x^{4}\right )} e^{4} - 2 \, {\left (x^{7} - 5 \, x^{6} + x^{4} - 4 \, x^{3} - 5 \, x^{2}\right )} e^{2} + 4 \, {\left (x^{6} - 4 \, x^{5} - 5 \, x^{4} + x^{3} - 3 \, x^{2} - {\left (x^{4} - 4 \, x^{3} - 5 \, x^{2}\right )} e^{2} - 10 \, x - 5\right )} \log \left (x\right ) - 10 \, x - 5}{{\left (x - 5\right )} \log \left (x\right )^{4} + 4 \, {\left (x - 5\right )} \log \left (x\right )^{3} + 6 \, {\left (x - 5\right )} \log \left (x\right )^{2} + 4 \, {\left (x - 5\right )} \log \left (x\right ) + x - 5} \]

[In]

integrate(((2*x^3-18*x^2+30*x+55)*log(x)^5+(10*x^3-90*x^2+150*x+275)*log(x)^4+((-6*x^4+56*x^3-110*x^2-100*x)*e
xp(2)+10*x^6-92*x^5+170*x^4+220*x^3-180*x^2+300*x+550)*log(x)^3+((-14*x^4+132*x^3-270*x^2-200*x)*exp(2)+26*x^6
-240*x^5+450*x^4+520*x^3-180*x^2+300*x+550)*log(x)^2+((4*x^5-40*x^4+100*x^3)*exp(2)^2+(-12*x^7+120*x^6-300*x^5
-10*x^4+96*x^3-210*x^2-100*x)*exp(2)+8*x^9-80*x^8+200*x^7+22*x^6-204*x^5+390*x^4+410*x^3-90*x^2+150*x+275)*log
(x)+(-4*x^7+40*x^6-100*x^5-2*x^4+20*x^3-50*x^2)*exp(2)+4*x^9-40*x^8+100*x^7+6*x^6-56*x^5+110*x^4+102*x^3-18*x^
2+30*x+55)/((x^2-10*x+25)*log(x)^5+(5*x^2-50*x+125)*log(x)^4+(10*x^2-100*x+250)*log(x)^3+(10*x^2-100*x+250)*lo
g(x)^2+(5*x^2-50*x+125)*log(x)+x^2-10*x+25),x, algorithm="fricas")

[Out]

(x^9 - 5*x^8 + 2*x^6 - 8*x^5 + (x^3 - 3*x^2 - 10*x - 5)*log(x)^4 - 10*x^4 + 4*(x^3 - 3*x^2 - 10*x - 5)*log(x)^
3 + x^3 + 2*(x^6 - 4*x^5 - 5*x^4 + 3*x^3 - 9*x^2 - (x^4 - 4*x^3 - 5*x^2)*e^2 - 30*x - 15)*log(x)^2 - 3*x^2 + (
x^5 - 5*x^4)*e^4 - 2*(x^7 - 5*x^6 + x^4 - 4*x^3 - 5*x^2)*e^2 + 4*(x^6 - 4*x^5 - 5*x^4 + x^3 - 3*x^2 - (x^4 - 4
*x^3 - 5*x^2)*e^2 - 10*x - 5)*log(x) - 10*x - 5)/((x - 5)*log(x)^4 + 4*(x - 5)*log(x)^3 + 6*(x - 5)*log(x)^2 +
4*(x - 5)*log(x) + x - 5)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 144 vs. \(2 (26) = 52\).

Time = 0.27 (sec) , antiderivative size = 144, normalized size of antiderivative = 4.00 \[ \int \frac {55+30 x-18 x^2+102 x^3+110 x^4-56 x^5+6 x^6+100 x^7-40 x^8+4 x^9+e^2 \left (-50 x^2+20 x^3-2 x^4-100 x^5+40 x^6-4 x^7\right )+\left (275+150 x-90 x^2+410 x^3+390 x^4-204 x^5+22 x^6+200 x^7-80 x^8+8 x^9+e^4 \left (100 x^3-40 x^4+4 x^5\right )+e^2 \left (-100 x-210 x^2+96 x^3-10 x^4-300 x^5+120 x^6-12 x^7\right )\right ) \log (x)+\left (550+300 x-180 x^2+520 x^3+450 x^4-240 x^5+26 x^6+e^2 \left (-200 x-270 x^2+132 x^3-14 x^4\right )\right ) \log ^2(x)+\left (550+300 x-180 x^2+220 x^3+170 x^4-92 x^5+10 x^6+e^2 \left (-100 x-110 x^2+56 x^3-6 x^4\right )\right ) \log ^3(x)+\left (275+150 x-90 x^2+10 x^3\right ) \log ^4(x)+\left (55+30 x-18 x^2+2 x^3\right ) \log ^5(x)}{25-10 x+x^2+\left (125-50 x+5 x^2\right ) \log (x)+\left (250-100 x+10 x^2\right ) \log ^2(x)+\left (250-100 x+10 x^2\right ) \log ^3(x)+\left (125-50 x+5 x^2\right ) \log ^4(x)+\left (25-10 x+x^2\right ) \log ^5(x)} \, dx=x^{2} + 2 x + \frac {x^{8} - 2 x^{6} e^{2} + 2 x^{5} + 2 x^{4} + x^{4} e^{4} - 2 x^{3} e^{2} - 2 x^{2} e^{2} + \left (2 x^{5} + 2 x^{4} - 2 x^{3} e^{2} - 2 x^{2} e^{2}\right ) \log {\left (x \right )}^{2} + \left (4 x^{5} + 4 x^{4} - 4 x^{3} e^{2} - 4 x^{2} e^{2}\right ) \log {\left (x \right )}}{\log {\left (x \right )}^{4} + 4 \log {\left (x \right )}^{3} + 6 \log {\left (x \right )}^{2} + 4 \log {\left (x \right )} + 1} - \frac {5}{x - 5} \]

[In]

integrate(((2*x**3-18*x**2+30*x+55)*ln(x)**5+(10*x**3-90*x**2+150*x+275)*ln(x)**4+((-6*x**4+56*x**3-110*x**2-1
00*x)*exp(2)+10*x**6-92*x**5+170*x**4+220*x**3-180*x**2+300*x+550)*ln(x)**3+((-14*x**4+132*x**3-270*x**2-200*x
)*exp(2)+26*x**6-240*x**5+450*x**4+520*x**3-180*x**2+300*x+550)*ln(x)**2+((4*x**5-40*x**4+100*x**3)*exp(2)**2+
(-12*x**7+120*x**6-300*x**5-10*x**4+96*x**3-210*x**2-100*x)*exp(2)+8*x**9-80*x**8+200*x**7+22*x**6-204*x**5+39
0*x**4+410*x**3-90*x**2+150*x+275)*ln(x)+(-4*x**7+40*x**6-100*x**5-2*x**4+20*x**3-50*x**2)*exp(2)+4*x**9-40*x*
*8+100*x**7+6*x**6-56*x**5+110*x**4+102*x**3-18*x**2+30*x+55)/((x**2-10*x+25)*ln(x)**5+(5*x**2-50*x+125)*ln(x)
**4+(10*x**2-100*x+250)*ln(x)**3+(10*x**2-100*x+250)*ln(x)**2+(5*x**2-50*x+125)*ln(x)+x**2-10*x+25),x)

[Out]

x**2 + 2*x + (x**8 - 2*x**6*exp(2) + 2*x**5 + 2*x**4 + x**4*exp(4) - 2*x**3*exp(2) - 2*x**2*exp(2) + (2*x**5 +
2*x**4 - 2*x**3*exp(2) - 2*x**2*exp(2))*log(x)**2 + (4*x**5 + 4*x**4 - 4*x**3*exp(2) - 4*x**2*exp(2))*log(x))
/(log(x)**4 + 4*log(x)**3 + 6*log(x)**2 + 4*log(x) + 1) - 5/(x - 5)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 244 vs. \(2 (31) = 62\).

Time = 0.30 (sec) , antiderivative size = 244, normalized size of antiderivative = 6.78 \[ \int \frac {55+30 x-18 x^2+102 x^3+110 x^4-56 x^5+6 x^6+100 x^7-40 x^8+4 x^9+e^2 \left (-50 x^2+20 x^3-2 x^4-100 x^5+40 x^6-4 x^7\right )+\left (275+150 x-90 x^2+410 x^3+390 x^4-204 x^5+22 x^6+200 x^7-80 x^8+8 x^9+e^4 \left (100 x^3-40 x^4+4 x^5\right )+e^2 \left (-100 x-210 x^2+96 x^3-10 x^4-300 x^5+120 x^6-12 x^7\right )\right ) \log (x)+\left (550+300 x-180 x^2+520 x^3+450 x^4-240 x^5+26 x^6+e^2 \left (-200 x-270 x^2+132 x^3-14 x^4\right )\right ) \log ^2(x)+\left (550+300 x-180 x^2+220 x^3+170 x^4-92 x^5+10 x^6+e^2 \left (-100 x-110 x^2+56 x^3-6 x^4\right )\right ) \log ^3(x)+\left (275+150 x-90 x^2+10 x^3\right ) \log ^4(x)+\left (55+30 x-18 x^2+2 x^3\right ) \log ^5(x)}{25-10 x+x^2+\left (125-50 x+5 x^2\right ) \log (x)+\left (250-100 x+10 x^2\right ) \log ^2(x)+\left (250-100 x+10 x^2\right ) \log ^3(x)+\left (125-50 x+5 x^2\right ) \log ^4(x)+\left (25-10 x+x^2\right ) \log ^5(x)} \, dx=\frac {x^{9} - 5 \, x^{8} - 2 \, x^{7} e^{2} + 2 \, x^{6} {\left (5 \, e^{2} + 1\right )} + x^{5} {\left (e^{4} - 8\right )} - x^{4} {\left (5 \, e^{4} + 2 \, e^{2} + 10\right )} + {\left (x^{3} - 3 \, x^{2} - 10 \, x - 5\right )} \log \left (x\right )^{4} + x^{3} {\left (8 \, e^{2} + 1\right )} + 4 \, {\left (x^{3} - 3 \, x^{2} - 10 \, x - 5\right )} \log \left (x\right )^{3} + x^{2} {\left (10 \, e^{2} - 3\right )} + 2 \, {\left (x^{6} - 4 \, x^{5} - x^{4} {\left (e^{2} + 5\right )} + x^{3} {\left (4 \, e^{2} + 3\right )} + x^{2} {\left (5 \, e^{2} - 9\right )} - 30 \, x - 15\right )} \log \left (x\right )^{2} + 4 \, {\left (x^{6} - 4 \, x^{5} - x^{4} {\left (e^{2} + 5\right )} + x^{3} {\left (4 \, e^{2} + 1\right )} + x^{2} {\left (5 \, e^{2} - 3\right )} - 10 \, x - 5\right )} \log \left (x\right ) - 10 \, x - 5}{{\left (x - 5\right )} \log \left (x\right )^{4} + 4 \, {\left (x - 5\right )} \log \left (x\right )^{3} + 6 \, {\left (x - 5\right )} \log \left (x\right )^{2} + 4 \, {\left (x - 5\right )} \log \left (x\right ) + x - 5} \]

[In]

integrate(((2*x^3-18*x^2+30*x+55)*log(x)^5+(10*x^3-90*x^2+150*x+275)*log(x)^4+((-6*x^4+56*x^3-110*x^2-100*x)*e
xp(2)+10*x^6-92*x^5+170*x^4+220*x^3-180*x^2+300*x+550)*log(x)^3+((-14*x^4+132*x^3-270*x^2-200*x)*exp(2)+26*x^6
-240*x^5+450*x^4+520*x^3-180*x^2+300*x+550)*log(x)^2+((4*x^5-40*x^4+100*x^3)*exp(2)^2+(-12*x^7+120*x^6-300*x^5
-10*x^4+96*x^3-210*x^2-100*x)*exp(2)+8*x^9-80*x^8+200*x^7+22*x^6-204*x^5+390*x^4+410*x^3-90*x^2+150*x+275)*log
(x)+(-4*x^7+40*x^6-100*x^5-2*x^4+20*x^3-50*x^2)*exp(2)+4*x^9-40*x^8+100*x^7+6*x^6-56*x^5+110*x^4+102*x^3-18*x^
2+30*x+55)/((x^2-10*x+25)*log(x)^5+(5*x^2-50*x+125)*log(x)^4+(10*x^2-100*x+250)*log(x)^3+(10*x^2-100*x+250)*lo
g(x)^2+(5*x^2-50*x+125)*log(x)+x^2-10*x+25),x, algorithm="maxima")

[Out]

(x^9 - 5*x^8 - 2*x^7*e^2 + 2*x^6*(5*e^2 + 1) + x^5*(e^4 - 8) - x^4*(5*e^4 + 2*e^2 + 10) + (x^3 - 3*x^2 - 10*x
- 5)*log(x)^4 + x^3*(8*e^2 + 1) + 4*(x^3 - 3*x^2 - 10*x - 5)*log(x)^3 + x^2*(10*e^2 - 3) + 2*(x^6 - 4*x^5 - x^
4*(e^2 + 5) + x^3*(4*e^2 + 3) + x^2*(5*e^2 - 9) - 30*x - 15)*log(x)^2 + 4*(x^6 - 4*x^5 - x^4*(e^2 + 5) + x^3*(
4*e^2 + 1) + x^2*(5*e^2 - 3) - 10*x - 5)*log(x) - 10*x - 5)/((x - 5)*log(x)^4 + 4*(x - 5)*log(x)^3 + 6*(x - 5)
*log(x)^2 + 4*(x - 5)*log(x) + x - 5)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 360 vs. \(2 (31) = 62\).

Time = 0.34 (sec) , antiderivative size = 360, normalized size of antiderivative = 10.00 \[ \int \frac {55+30 x-18 x^2+102 x^3+110 x^4-56 x^5+6 x^6+100 x^7-40 x^8+4 x^9+e^2 \left (-50 x^2+20 x^3-2 x^4-100 x^5+40 x^6-4 x^7\right )+\left (275+150 x-90 x^2+410 x^3+390 x^4-204 x^5+22 x^6+200 x^7-80 x^8+8 x^9+e^4 \left (100 x^3-40 x^4+4 x^5\right )+e^2 \left (-100 x-210 x^2+96 x^3-10 x^4-300 x^5+120 x^6-12 x^7\right )\right ) \log (x)+\left (550+300 x-180 x^2+520 x^3+450 x^4-240 x^5+26 x^6+e^2 \left (-200 x-270 x^2+132 x^3-14 x^4\right )\right ) \log ^2(x)+\left (550+300 x-180 x^2+220 x^3+170 x^4-92 x^5+10 x^6+e^2 \left (-100 x-110 x^2+56 x^3-6 x^4\right )\right ) \log ^3(x)+\left (275+150 x-90 x^2+10 x^3\right ) \log ^4(x)+\left (55+30 x-18 x^2+2 x^3\right ) \log ^5(x)}{25-10 x+x^2+\left (125-50 x+5 x^2\right ) \log (x)+\left (250-100 x+10 x^2\right ) \log ^2(x)+\left (250-100 x+10 x^2\right ) \log ^3(x)+\left (125-50 x+5 x^2\right ) \log ^4(x)+\left (25-10 x+x^2\right ) \log ^5(x)} \, dx=\frac {x^{9} - 5 \, x^{8} - 2 \, x^{7} e^{2} + 2 \, x^{6} \log \left (x\right )^{2} + 10 \, x^{6} e^{2} + 4 \, x^{6} \log \left (x\right ) - 8 \, x^{5} \log \left (x\right )^{2} - 2 \, x^{4} e^{2} \log \left (x\right )^{2} + x^{3} \log \left (x\right )^{4} + 2 \, x^{6} + x^{5} e^{4} - 16 \, x^{5} \log \left (x\right ) - 4 \, x^{4} e^{2} \log \left (x\right ) - 10 \, x^{4} \log \left (x\right )^{2} + 8 \, x^{3} e^{2} \log \left (x\right )^{2} + 4 \, x^{3} \log \left (x\right )^{3} - 3 \, x^{2} \log \left (x\right )^{4} - 8 \, x^{5} - 5 \, x^{4} e^{4} - 2 \, x^{4} e^{2} - 20 \, x^{4} \log \left (x\right ) + 16 \, x^{3} e^{2} \log \left (x\right ) + 6 \, x^{3} \log \left (x\right )^{2} + 10 \, x^{2} e^{2} \log \left (x\right )^{2} - 12 \, x^{2} \log \left (x\right )^{3} - 10 \, x \log \left (x\right )^{4} - 10 \, x^{4} + 8 \, x^{3} e^{2} + 4 \, x^{3} \log \left (x\right ) + 20 \, x^{2} e^{2} \log \left (x\right ) - 18 \, x^{2} \log \left (x\right )^{2} - 40 \, x \log \left (x\right )^{3} - 5 \, \log \left (x\right )^{4} + x^{3} + 10 \, x^{2} e^{2} - 12 \, x^{2} \log \left (x\right ) - 60 \, x \log \left (x\right )^{2} - 20 \, \log \left (x\right )^{3} - 3 \, x^{2} - 40 \, x \log \left (x\right ) - 30 \, \log \left (x\right )^{2} - 10 \, x - 20 \, \log \left (x\right ) - 5}{x \log \left (x\right )^{4} + 4 \, x \log \left (x\right )^{3} - 5 \, \log \left (x\right )^{4} + 6 \, x \log \left (x\right )^{2} - 20 \, \log \left (x\right )^{3} + 4 \, x \log \left (x\right ) - 30 \, \log \left (x\right )^{2} + x - 20 \, \log \left (x\right ) - 5} \]

[In]

integrate(((2*x^3-18*x^2+30*x+55)*log(x)^5+(10*x^3-90*x^2+150*x+275)*log(x)^4+((-6*x^4+56*x^3-110*x^2-100*x)*e
xp(2)+10*x^6-92*x^5+170*x^4+220*x^3-180*x^2+300*x+550)*log(x)^3+((-14*x^4+132*x^3-270*x^2-200*x)*exp(2)+26*x^6
-240*x^5+450*x^4+520*x^3-180*x^2+300*x+550)*log(x)^2+((4*x^5-40*x^4+100*x^3)*exp(2)^2+(-12*x^7+120*x^6-300*x^5
-10*x^4+96*x^3-210*x^2-100*x)*exp(2)+8*x^9-80*x^8+200*x^7+22*x^6-204*x^5+390*x^4+410*x^3-90*x^2+150*x+275)*log
(x)+(-4*x^7+40*x^6-100*x^5-2*x^4+20*x^3-50*x^2)*exp(2)+4*x^9-40*x^8+100*x^7+6*x^6-56*x^5+110*x^4+102*x^3-18*x^
2+30*x+55)/((x^2-10*x+25)*log(x)^5+(5*x^2-50*x+125)*log(x)^4+(10*x^2-100*x+250)*log(x)^3+(10*x^2-100*x+250)*lo
g(x)^2+(5*x^2-50*x+125)*log(x)+x^2-10*x+25),x, algorithm="giac")

[Out]

(x^9 - 5*x^8 - 2*x^7*e^2 + 2*x^6*log(x)^2 + 10*x^6*e^2 + 4*x^6*log(x) - 8*x^5*log(x)^2 - 2*x^4*e^2*log(x)^2 +
x^3*log(x)^4 + 2*x^6 + x^5*e^4 - 16*x^5*log(x) - 4*x^4*e^2*log(x) - 10*x^4*log(x)^2 + 8*x^3*e^2*log(x)^2 + 4*x
^3*log(x)^3 - 3*x^2*log(x)^4 - 8*x^5 - 5*x^4*e^4 - 2*x^4*e^2 - 20*x^4*log(x) + 16*x^3*e^2*log(x) + 6*x^3*log(x
)^2 + 10*x^2*e^2*log(x)^2 - 12*x^2*log(x)^3 - 10*x*log(x)^4 - 10*x^4 + 8*x^3*e^2 + 4*x^3*log(x) + 20*x^2*e^2*l
og(x) - 18*x^2*log(x)^2 - 40*x*log(x)^3 - 5*log(x)^4 + x^3 + 10*x^2*e^2 - 12*x^2*log(x) - 60*x*log(x)^2 - 20*l
og(x)^3 - 3*x^2 - 40*x*log(x) - 30*log(x)^2 - 10*x - 20*log(x) - 5)/(x*log(x)^4 + 4*x*log(x)^3 - 5*log(x)^4 +
6*x*log(x)^2 - 20*log(x)^3 + 4*x*log(x) - 30*log(x)^2 + x - 20*log(x) - 5)

Mupad [B] (verification not implemented)

Time = 12.43 (sec) , antiderivative size = 783, normalized size of antiderivative = 21.75 \[ \int \frac {55+30 x-18 x^2+102 x^3+110 x^4-56 x^5+6 x^6+100 x^7-40 x^8+4 x^9+e^2 \left (-50 x^2+20 x^3-2 x^4-100 x^5+40 x^6-4 x^7\right )+\left (275+150 x-90 x^2+410 x^3+390 x^4-204 x^5+22 x^6+200 x^7-80 x^8+8 x^9+e^4 \left (100 x^3-40 x^4+4 x^5\right )+e^2 \left (-100 x-210 x^2+96 x^3-10 x^4-300 x^5+120 x^6-12 x^7\right )\right ) \log (x)+\left (550+300 x-180 x^2+520 x^3+450 x^4-240 x^5+26 x^6+e^2 \left (-200 x-270 x^2+132 x^3-14 x^4\right )\right ) \log ^2(x)+\left (550+300 x-180 x^2+220 x^3+170 x^4-92 x^5+10 x^6+e^2 \left (-100 x-110 x^2+56 x^3-6 x^4\right )\right ) \log ^3(x)+\left (275+150 x-90 x^2+10 x^3\right ) \log ^4(x)+\left (55+30 x-18 x^2+2 x^3\right ) \log ^5(x)}{25-10 x+x^2+\left (125-50 x+5 x^2\right ) \log (x)+\left (250-100 x+10 x^2\right ) \log ^2(x)+\left (250-100 x+10 x^2\right ) \log ^3(x)+\left (125-50 x+5 x^2\right ) \log ^4(x)+\left (25-10 x+x^2\right ) \log ^5(x)} \, dx=2\,x-\frac {5}{x-5}-\frac {-\frac {x\,\left (-125\,x^4-64\,x^3+27\,{\mathrm {e}}^2\,x^2+8\,{\mathrm {e}}^2\,x\right )\,{\ln \left (x\right )}^3}{12}-\frac {x\,\left (-475\,x^4-256\,x^3+117\,{\mathrm {e}}^2\,x^2+40\,{\mathrm {e}}^2\,x\right )\,{\ln \left (x\right )}^2}{12}+\frac {x\,\left (32\,x^3\,{\mathrm {e}}^4-147\,x^2\,{\mathrm {e}}^2-52\,x\,{\mathrm {e}}^2-216\,x^5\,{\mathrm {e}}^2+312\,x^3+565\,x^4+256\,x^7\right )\,\ln \left (x\right )}{12}+\frac {x\,\left (16\,x^3\,{\mathrm {e}}^4-53\,x^2\,{\mathrm {e}}^2-16\,x\,{\mathrm {e}}^2-144\,x^5\,{\mathrm {e}}^2+116\,x^3+211\,x^4+192\,x^7\right )}{12}}{{\ln \left (x\right )}^2+2\,\ln \left (x\right )+1}-\ln \left (x\right )\,\left (-\frac {375\,x^5}{2}-\frac {256\,x^4}{3}+\frac {63\,{\mathrm {e}}^2\,x^3}{2}+8\,{\mathrm {e}}^2\,x^2\right )-x^2\,\left (\frac {32\,{\mathrm {e}}^2}{3}-1\right )+x^4\,\left (\frac {32\,{\mathrm {e}}^4}{3}+80\right )-\frac {135\,x^3\,{\mathrm {e}}^2}{4}-108\,x^6\,{\mathrm {e}}^2-\frac {-\frac {x\,\left (-625\,x^4-256\,x^3+81\,{\mathrm {e}}^2\,x^2+16\,{\mathrm {e}}^2\,x\right )\,{\ln \left (x\right )}^3}{12}-\frac {x\,\left (-1375\,x^4-608\,x^3+216\,{\mathrm {e}}^2\,x^2+52\,{\mathrm {e}}^2\,x\right )\,{\ln \left (x\right )}^2}{6}+\frac {x\,\left (128\,x^3\,{\mathrm {e}}^4-675\,x^2\,{\mathrm {e}}^2-184\,x\,{\mathrm {e}}^2-1296\,x^5\,{\mathrm {e}}^2+1760\,x^3+3775\,x^4+2048\,x^7\right )\,\ln \left (x\right )}{12}+\frac {x\,\left (48\,x^3\,{\mathrm {e}}^4-153\,x^2\,{\mathrm {e}}^2-42\,x\,{\mathrm {e}}^2-540\,x^5\,{\mathrm {e}}^2+388\,x^3+810\,x^4+896\,x^7\right )}{6}}{\ln \left (x\right )+1}-{\ln \left (x\right )}^2\,\left (-\frac {625\,x^5}{12}-\frac {64\,x^4}{3}+\frac {27\,{\mathrm {e}}^2\,x^3}{4}+\frac {4\,{\mathrm {e}}^2\,x^2}{3}\right )-\frac {-\frac {x\,\left (-5\,x^4-4\,x^3+3\,{\mathrm {e}}^2\,x^2+2\,{\mathrm {e}}^2\,x\right )\,{\ln \left (x\right )}^3}{2}-\frac {x\,\left (-13\,x^4-10\,x^3+7\,{\mathrm {e}}^2\,x^2+4\,{\mathrm {e}}^2\,x\right )\,{\ln \left (x\right )}^2}{2}+\frac {x\,\left (2\,x^3\,{\mathrm {e}}^4-5\,x^2\,{\mathrm {e}}^2-2\,x\,{\mathrm {e}}^2-6\,x^5\,{\mathrm {e}}^2+8\,x^3+11\,x^4+4\,x^7\right )\,\ln \left (x\right )}{2}+\frac {x\,\left (2\,x^7-2\,{\mathrm {e}}^2\,x^5+3\,x^4+2\,x^3-{\mathrm {e}}^2\,x^2\right )}{2}}{{\ln \left (x\right )}^4+4\,{\ln \left (x\right )}^3+6\,{\ln \left (x\right )}^2+4\,\ln \left (x\right )+1}+\frac {1925\,x^5}{12}+\frac {512\,x^8}{3}-\frac {-\frac {x\,\left (-25\,x^4-16\,x^3+9\,{\mathrm {e}}^2\,x^2+4\,{\mathrm {e}}^2\,x\right )\,{\ln \left (x\right )}^3}{6}-\frac {x\,\left (-40\,x^4-26\,x^3+15\,{\mathrm {e}}^2\,x^2+7\,{\mathrm {e}}^2\,x\right )\,{\ln \left (x\right )}^2}{3}+\frac {x\,\left (8\,x^3\,{\mathrm {e}}^4-29\,x^2\,{\mathrm {e}}^2-12\,x\,{\mathrm {e}}^2-36\,x^5\,{\mathrm {e}}^2+52\,x^3+81\,x^4+32\,x^7\right )\,\ln \left (x\right )}{6}+\frac {x\,\left (x^3\,{\mathrm {e}}^4-4\,x^2\,{\mathrm {e}}^2-x\,{\mathrm {e}}^2-9\,x^5\,{\mathrm {e}}^2+8\,x^3+13\,x^4+10\,x^7\right )}{3}}{{\ln \left (x\right )}^3+3\,{\ln \left (x\right )}^2+3\,\ln \left (x\right )+1} \]

[In]

int((30*x + log(x)*(150*x - exp(2)*(100*x + 210*x^2 - 96*x^3 + 10*x^4 + 300*x^5 - 120*x^6 + 12*x^7) + exp(4)*(
100*x^3 - 40*x^4 + 4*x^5) - 90*x^2 + 410*x^3 + 390*x^4 - 204*x^5 + 22*x^6 + 200*x^7 - 80*x^8 + 8*x^9 + 275) +
log(x)^5*(30*x - 18*x^2 + 2*x^3 + 55) + log(x)^4*(150*x - 90*x^2 + 10*x^3 + 275) - exp(2)*(50*x^2 - 20*x^3 + 2
*x^4 + 100*x^5 - 40*x^6 + 4*x^7) + log(x)^3*(300*x - exp(2)*(100*x + 110*x^2 - 56*x^3 + 6*x^4) - 180*x^2 + 220
*x^3 + 170*x^4 - 92*x^5 + 10*x^6 + 550) + log(x)^2*(300*x - exp(2)*(200*x + 270*x^2 - 132*x^3 + 14*x^4) - 180*
x^2 + 520*x^3 + 450*x^4 - 240*x^5 + 26*x^6 + 550) - 18*x^2 + 102*x^3 + 110*x^4 - 56*x^5 + 6*x^6 + 100*x^7 - 40
*x^8 + 4*x^9 + 55)/(log(x)^4*(5*x^2 - 50*x + 125) - 10*x + log(x)^2*(10*x^2 - 100*x + 250) + log(x)^3*(10*x^2
- 100*x + 250) + log(x)*(5*x^2 - 50*x + 125) + log(x)^5*(x^2 - 10*x + 25) + x^2 + 25),x)

[Out]

2*x - 5/(x - 5) - ((x*(16*x^3*exp(4) - 53*x^2*exp(2) - 16*x*exp(2) - 144*x^5*exp(2) + 116*x^3 + 211*x^4 + 192*
x^7))/12 - (x*log(x)^3*(8*x*exp(2) + 27*x^2*exp(2) - 64*x^3 - 125*x^4))/12 - (x*log(x)^2*(40*x*exp(2) + 117*x^
2*exp(2) - 256*x^3 - 475*x^4))/12 + (x*log(x)*(32*x^3*exp(4) - 147*x^2*exp(2) - 52*x*exp(2) - 216*x^5*exp(2) +
312*x^3 + 565*x^4 + 256*x^7))/12)/(2*log(x) + log(x)^2 + 1) - log(x)*(8*x^2*exp(2) + (63*x^3*exp(2))/2 - (256
*x^4)/3 - (375*x^5)/2) - x^2*((32*exp(2))/3 - 1) + x^4*((32*exp(4))/3 + 80) - (135*x^3*exp(2))/4 - 108*x^6*exp
(2) - ((x*(48*x^3*exp(4) - 153*x^2*exp(2) - 42*x*exp(2) - 540*x^5*exp(2) + 388*x^3 + 810*x^4 + 896*x^7))/6 - (
x*log(x)^3*(16*x*exp(2) + 81*x^2*exp(2) - 256*x^3 - 625*x^4))/12 - (x*log(x)^2*(52*x*exp(2) + 216*x^2*exp(2) -
608*x^3 - 1375*x^4))/6 + (x*log(x)*(128*x^3*exp(4) - 675*x^2*exp(2) - 184*x*exp(2) - 1296*x^5*exp(2) + 1760*x
^3 + 3775*x^4 + 2048*x^7))/12)/(log(x) + 1) - log(x)^2*((4*x^2*exp(2))/3 + (27*x^3*exp(2))/4 - (64*x^4)/3 - (6
25*x^5)/12) - ((x*(2*x^3 - 2*x^5*exp(2) - x^2*exp(2) + 3*x^4 + 2*x^7))/2 - (x*log(x)^3*(2*x*exp(2) + 3*x^2*exp
(2) - 4*x^3 - 5*x^4))/2 - (x*log(x)^2*(4*x*exp(2) + 7*x^2*exp(2) - 10*x^3 - 13*x^4))/2 + (x*log(x)*(2*x^3*exp(
4) - 5*x^2*exp(2) - 2*x*exp(2) - 6*x^5*exp(2) + 8*x^3 + 11*x^4 + 4*x^7))/2)/(4*log(x) + 6*log(x)^2 + 4*log(x)^
3 + log(x)^4 + 1) + (1925*x^5)/12 + (512*x^8)/3 - ((x*(x^3*exp(4) - 4*x^2*exp(2) - x*exp(2) - 9*x^5*exp(2) + 8
*x^3 + 13*x^4 + 10*x^7))/3 - (x*log(x)^3*(4*x*exp(2) + 9*x^2*exp(2) - 16*x^3 - 25*x^4))/6 - (x*log(x)^2*(7*x*e
xp(2) + 15*x^2*exp(2) - 26*x^3 - 40*x^4))/3 + (x*log(x)*(8*x^3*exp(4) - 29*x^2*exp(2) - 12*x*exp(2) - 36*x^5*e
xp(2) + 52*x^3 + 81*x^4 + 32*x^7))/6)/(3*log(x) + 3*log(x)^2 + log(x)^3 + 1)