3.9.46 \(\int \frac {900 x^3+1250 x^4-125 x^5+25 x^6+e^2 (450 x^2+850 x^3-75 x^4+25 x^5)+(40500 x+51750 x^2-11650 x^3+1900 x^4-200 x^5+e^2 (20250+36000 x-7400 x^2+1650 x^3-200 x^4)) \log (x^2+x^3+e^2 (x+x^2))+(10125 x+5625 x^2-4125 x^3+375 x^4+e^2 (10125+5625 x-4125 x^2+375 x^3)) \log ^2(x^2+x^3+e^2 (x+x^2))}{x+x^2+e^2 (1+x)} \, dx\)

Optimal. Leaf size=27 \[ 5 x \left (x^2+5 (9-x) \log \left (x (1+x) \left (e^2+x\right )\right )\right )^2 \]

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Rubi [C]  time = 4.60, antiderivative size = 1165, normalized size of antiderivative = 43.15, number of steps used = 127, number of rules used = 17, integrand size = 185, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.092, Rules used = {6688, 12, 6742, 1612, 2528, 2523, 893, 2525, 2524, 2418, 2391, 2390, 2301, 2394, 2393, 2392, 2315}

result too large to display

Antiderivative was successfully verified.

[In]

Int[(900*x^3 + 1250*x^4 - 125*x^5 + 25*x^6 + E^2*(450*x^2 + 850*x^3 - 75*x^4 + 25*x^5) + (40500*x + 51750*x^2
- 11650*x^3 + 1900*x^4 - 200*x^5 + E^2*(20250 + 36000*x - 7400*x^2 + 1650*x^3 - 200*x^4))*Log[x^2 + x^3 + E^2*
(x + x^2)] + (10125*x + 5625*x^2 - 4125*x^3 + 375*x^4 + E^2*(10125 + 5625*x - 4125*x^2 + 375*x^3))*Log[x^2 + x
^3 + E^2*(x + x^2)]^2)/(x + x^2 + E^2*(1 + x)),x]

[Out]

182250*x + 20250*(1 + E^2)*x + 125*(1 + E^2)^2*x - 125*(1 + E^2)*(55 + E^2)*x + 300*(1 + E^4)*x - 750*(262 + 1
8*E^2 + E^4)*x - 50*(1 + E^6)*x + 50*(10 + 9*E^4 + E^6)*x - 10125*x^2 + (75*(1 + E^2)*x^2)/2 + (375*(55 + E^2)
*x^2)/2 + 25*(1 + E^4)*x^2 - 25*(10 + 9*E^2 + E^4)*x^2 - 450*x^3 - (50*(1 + E^2)*x^3)/3 + (50*(28 + E^2)*x^3)/
3 + 5*x^5 - 10250*Log[-1 - x]^2 - 20500*Log[-1 - x]*Log[-x] - 40500*E^2*Log[x] - 9000*E^4*Log[x] - 500*E^6*Log
[x] + 500*E^2*(9 + E^2)^2*Log[x] - 67500*Log[1 + x] - 4625*(1 + E^2)*Log[1 + x] + 125*(55 + E^2)*Log[1 + x] -
250*(1 + E^4)*Log[1 + x] + 250*(262 + 18*E^2 + E^4)*Log[1 + x] + 10250*Log[1 + x]^2 - 60750*E^2*Log[E^2 + x] -
6750*E^4*Log[E^2 + x] + 450*E^6*Log[E^2 + x] + 50*E^8*Log[E^2 + x] - 4500*E^2*(1 + E^2)*Log[E^2 + x] - 125*E^
4*(1 + E^2)*Log[E^2 + x] - 50*E^6*(9 + E^2)*Log[E^2 + x] + 125*E^4*(55 + E^2)*Log[E^2 + x] - 250*E^2*(1 + E^4)
*Log[E^2 + x] + 250*E^2*(262 + 18*E^2 + E^4)*Log[E^2 + x] - 20250*E^2*Log[(1 + x)/(1 - E^2)]*Log[E^2 + x] - 45
00*E^4*Log[(1 + x)/(1 - E^2)]*Log[E^2 + x] - 250*E^6*Log[(1 + x)/(1 - E^2)]*Log[E^2 + x] + 250*E^2*(9 + E^2)^2
*Log[(1 + x)/(1 - E^2)]*Log[E^2 + x] - 10125*E^2*Log[E^2 + x]^2 - 2250*E^4*Log[E^2 + x]^2 - 125*E^6*Log[E^2 +
x]^2 + 125*E^2*(9 + E^2)^2*Log[E^2 + x]^2 - 20500*Log[-1 - x]*Log[-((E^2 + x)/(1 - E^2))] + 20500*Log[1 + x]*L
og[-((E^2 + x)/(1 - E^2))] - 60750*x*Log[x*(1 + x)*(E^2 + x)] - 4500*(1 + E^2)*x*Log[x*(1 + x)*(E^2 + x)] - 25
0*(1 + E^4)*x*Log[x*(1 + x)*(E^2 + x)] + 250*(262 + 18*E^2 + E^4)*x*Log[x*(1 + x)*(E^2 + x)] + 6750*x^2*Log[x*
(1 + x)*(E^2 + x)] + 125*(1 + E^2)*x^2*Log[x*(1 + x)*(E^2 + x)] - 125*(55 + E^2)*x^2*Log[x*(1 + x)*(E^2 + x)]
+ 450*x^3*Log[x*(1 + x)*(E^2 + x)] - 50*x^4*Log[x*(1 + x)*(E^2 + x)] + 20500*Log[-1 - x]*Log[x*(1 + x)*(E^2 +
x)] - 20500*Log[1 + x]*Log[x*(1 + x)*(E^2 + x)] + 20250*E^2*Log[E^2 + x]*Log[x*(1 + x)*(E^2 + x)] + 4500*E^4*L
og[E^2 + x]*Log[x*(1 + x)*(E^2 + x)] + 250*E^6*Log[E^2 + x]*Log[x*(1 + x)*(E^2 + x)] - 250*E^2*(9 + E^2)^2*Log
[E^2 + x]*Log[x*(1 + x)*(E^2 + x)] + 10125*x*Log[x*(1 + x)*(E^2 + x)]^2 - 2250*x^2*Log[x*(1 + x)*(E^2 + x)]^2
+ 125*x^3*Log[x*(1 + x)*(E^2 + x)]^2 - 20500*PolyLog[2, -x] + 20250*E^2*PolyLog[2, -(x/E^2)] + 4500*E^4*PolyLo
g[2, -(x/E^2)] + 250*E^6*PolyLog[2, -(x/E^2)] - 250*E^2*(9 + E^2)^2*PolyLog[2, -(x/E^2)] - 20500*PolyLog[2, 1
+ x] - 20250*E^2*PolyLog[2, -((E^2 + x)/(1 - E^2))] - 4500*E^4*PolyLog[2, -((E^2 + x)/(1 - E^2))] - 250*E^6*Po
lyLog[2, -((E^2 + x)/(1 - E^2))] + 250*E^2*(9 + E^2)^2*PolyLog[2, -((E^2 + x)/(1 - E^2))]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 893

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :
> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] &
& NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[p] && ((EqQ[p, 1] && I
ntegersQ[m, n]) || (ILtQ[m, 0] && ILtQ[n, 0]))

Rule 1612

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[E
xpandIntegrand[Px*(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && Poly
Q[Px, x] && IntegersQ[m, n]

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 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 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 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 2523

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

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 2525

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

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]

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 {25 \left (x^2-5 (-9+x) \log \left (x (1+x) \left (e^2+x\right )\right )\right ) \left (x \left (36+50 x-5 x^2+x^3\right )+e^2 \left (18+34 x-3 x^2+x^3\right )-3 \left (e^2+x\right ) \left (-3-2 x+x^2\right ) \log \left (x (1+x) \left (e^2+x\right )\right )\right )}{(1+x) \left (e^2+x\right )} \, dx\\ &=25 \int \frac {\left (x^2-5 (-9+x) \log \left (x (1+x) \left (e^2+x\right )\right )\right ) \left (x \left (36+50 x-5 x^2+x^3\right )+e^2 \left (18+34 x-3 x^2+x^3\right )-3 \left (e^2+x\right ) \left (-3-2 x+x^2\right ) \log \left (x (1+x) \left (e^2+x\right )\right )\right )}{(1+x) \left (e^2+x\right )} \, dx\\ &=25 \int \left (\frac {x^2 \left (18 e^2+2 \left (18+17 e^2\right ) x+\left (50-3 e^2\right ) x^2-\left (5-e^2\right ) x^3+x^4\right )}{(1+x) \left (e^2+x\right )}+\frac {2 \left (405 e^2+810 \left (1+\frac {8 e^2}{9}\right ) x+1035 \left (1-\frac {148 e^2}{1035}\right ) x^2-233 \left (1-\frac {33 e^2}{233}\right ) x^3+38 \left (1-\frac {2 e^2}{19}\right ) x^4-4 x^5\right ) \log \left (x (1+x) \left (e^2+x\right )\right )}{(1+x) \left (e^2+x\right )}+15 (-9+x) (-3+x) \log ^2\left (x (1+x) \left (e^2+x\right )\right )\right ) \, dx\\ &=25 \int \frac {x^2 \left (18 e^2+2 \left (18+17 e^2\right ) x+\left (50-3 e^2\right ) x^2-\left (5-e^2\right ) x^3+x^4\right )}{(1+x) \left (e^2+x\right )} \, dx+50 \int \frac {\left (405 e^2+810 \left (1+\frac {8 e^2}{9}\right ) x+1035 \left (1-\frac {148 e^2}{1035}\right ) x^2-233 \left (1-\frac {33 e^2}{233}\right ) x^3+38 \left (1-\frac {2 e^2}{19}\right ) x^4-4 x^5\right ) \log \left (x (1+x) \left (e^2+x\right )\right )}{(1+x) \left (e^2+x\right )} \, dx+375 \int (-9+x) (-3+x) \log ^2\left (x (1+x) \left (e^2+x\right )\right ) \, dx\\ &=25 \int \left (2 \left (10+9 e^4+e^6\right )-2 \left (10+9 e^2+e^4\right ) x+2 \left (28+e^2\right ) x^2-6 x^3+x^4-\frac {20}{1+x}-\frac {2 e^6 \left (9+e^2\right )}{e^2+x}\right ) \, dx+50 \int \left (5 \left (262+18 e^2+e^4\right ) \log \left (x (1+x) \left (e^2+x\right )\right )-5 \left (55+e^2\right ) x \log \left (x (1+x) \left (e^2+x\right )\right )+42 x^2 \log \left (x (1+x) \left (e^2+x\right )\right )-4 x^3 \log \left (x (1+x) \left (e^2+x\right )\right )-\frac {500 \log \left (x (1+x) \left (e^2+x\right )\right )}{1+x}-\frac {5 e^2 \left (9+e^2\right )^2 \log \left (x (1+x) \left (e^2+x\right )\right )}{e^2+x}\right ) \, dx+375 \int \left (27 \log ^2\left (x (1+x) \left (e^2+x\right )\right )-12 x \log ^2\left (x (1+x) \left (e^2+x\right )\right )+x^2 \log ^2\left (x (1+x) \left (e^2+x\right )\right )\right ) \, dx\\ &=50 \left (10+9 e^4+e^6\right ) x-25 \left (10+9 e^2+e^4\right ) x^2+\frac {50}{3} \left (28+e^2\right ) x^3-\frac {75 x^4}{2}+5 x^5-500 \log (1+x)-50 e^6 \left (9+e^2\right ) \log \left (e^2+x\right )-200 \int x^3 \log \left (x (1+x) \left (e^2+x\right )\right ) \, dx+375 \int x^2 \log ^2\left (x (1+x) \left (e^2+x\right )\right ) \, dx+2100 \int x^2 \log \left (x (1+x) \left (e^2+x\right )\right ) \, dx-4500 \int x \log ^2\left (x (1+x) \left (e^2+x\right )\right ) \, dx+10125 \int \log ^2\left (x (1+x) \left (e^2+x\right )\right ) \, dx-25000 \int \frac {\log \left (x (1+x) \left (e^2+x\right )\right )}{1+x} \, dx-\left (250 e^2 \left (9+e^2\right )^2\right ) \int \frac {\log \left (x (1+x) \left (e^2+x\right )\right )}{e^2+x} \, dx-\left (250 \left (55+e^2\right )\right ) \int x \log \left (x (1+x) \left (e^2+x\right )\right ) \, dx+\left (250 \left (262+18 e^2+e^4\right )\right ) \int \log \left (x (1+x) \left (e^2+x\right )\right ) \, dx\\ &=50 \left (10+9 e^4+e^6\right ) x-25 \left (10+9 e^2+e^4\right ) x^2+\frac {50}{3} \left (28+e^2\right ) x^3-\frac {75 x^4}{2}+5 x^5-500 \log (1+x)-50 e^6 \left (9+e^2\right ) \log \left (e^2+x\right )+250 \left (262+18 e^2+e^4\right ) x \log \left (x (1+x) \left (e^2+x\right )\right )-125 \left (55+e^2\right ) x^2 \log \left (x (1+x) \left (e^2+x\right )\right )+700 x^3 \log \left (x (1+x) \left (e^2+x\right )\right )-50 x^4 \log \left (x (1+x) \left (e^2+x\right )\right )-25000 \log (1+x) \log \left (x (1+x) \left (e^2+x\right )\right )-250 e^2 \left (9+e^2\right )^2 \log \left (e^2+x\right ) \log \left (x (1+x) \left (e^2+x\right )\right )+10125 x \log ^2\left (x (1+x) \left (e^2+x\right )\right )-2250 x^2 \log ^2\left (x (1+x) \left (e^2+x\right )\right )+125 x^3 \log ^2\left (x (1+x) \left (e^2+x\right )\right )+50 \int \frac {x^3 \left (e^2+2 \left (1+e^2\right ) x+3 x^2\right )}{(1+x) \left (e^2+x\right )} \, dx-250 \int \frac {x^2 \left (e^2+2 \left (1+e^2\right ) x+3 x^2\right ) \log \left (x (1+x) \left (e^2+x\right )\right )}{(1+x) \left (e^2+x\right )} \, dx-700 \int \frac {x^2 \left (e^2+2 \left (1+e^2\right ) x+3 x^2\right )}{(1+x) \left (e^2+x\right )} \, dx+4500 \int \frac {x \left (e^2+2 \left (1+e^2\right ) x+3 x^2\right ) \log \left (x (1+x) \left (e^2+x\right )\right )}{(1+x) \left (e^2+x\right )} \, dx-20250 \int \frac {\left (e^2+2 \left (1+e^2\right ) x+3 x^2\right ) \log \left (x (1+x) \left (e^2+x\right )\right )}{(1+x) \left (e^2+x\right )} \, dx+25000 \int \frac {\left (x (1+x)+x \left (e^2+x\right )+(1+x) \left (e^2+x\right )\right ) \log (1+x)}{x (1+x) \left (e^2+x\right )} \, dx+\left (250 e^2 \left (9+e^2\right )^2\right ) \int \frac {\left (x (1+x)+x \left (e^2+x\right )+(1+x) \left (e^2+x\right )\right ) \log \left (e^2+x\right )}{x (1+x) \left (e^2+x\right )} \, dx+\left (125 \left (55+e^2\right )\right ) \int \frac {x \left (e^2+2 \left (1+e^2\right ) x+3 x^2\right )}{(1+x) \left (e^2+x\right )} \, dx-\left (250 \left (262+18 e^2+e^4\right )\right ) \int \frac {e^2+2 \left (1+e^2\right ) x+3 x^2}{(1+x) \left (e^2+x\right )} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.07, size = 25, normalized size = 0.93 \begin {gather*} 5 x \left (x^2-5 (-9+x) \log \left (x (1+x) \left (e^2+x\right )\right )\right )^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(900*x^3 + 1250*x^4 - 125*x^5 + 25*x^6 + E^2*(450*x^2 + 850*x^3 - 75*x^4 + 25*x^5) + (40500*x + 5175
0*x^2 - 11650*x^3 + 1900*x^4 - 200*x^5 + E^2*(20250 + 36000*x - 7400*x^2 + 1650*x^3 - 200*x^4))*Log[x^2 + x^3
+ E^2*(x + x^2)] + (10125*x + 5625*x^2 - 4125*x^3 + 375*x^4 + E^2*(10125 + 5625*x - 4125*x^2 + 375*x^3))*Log[x
^2 + x^3 + E^2*(x + x^2)]^2)/(x + x^2 + E^2*(1 + x)),x]

[Out]

5*x*(x^2 - 5*(-9 + x)*Log[x*(1 + x)*(E^2 + x)])^2

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fricas [B]  time = 0.87, size = 65, normalized size = 2.41 \begin {gather*} 5 \, x^{5} + 125 \, {\left (x^{3} - 18 \, x^{2} + 81 \, x\right )} \log \left (x^{3} + x^{2} + {\left (x^{2} + x\right )} e^{2}\right )^{2} - 50 \, {\left (x^{4} - 9 \, x^{3}\right )} \log \left (x^{3} + x^{2} + {\left (x^{2} + x\right )} e^{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((375*x^3-4125*x^2+5625*x+10125)*exp(2)+375*x^4-4125*x^3+5625*x^2+10125*x)*log((x^2+x)*exp(2)+x^3+x
^2)^2+((-200*x^4+1650*x^3-7400*x^2+36000*x+20250)*exp(2)-200*x^5+1900*x^4-11650*x^3+51750*x^2+40500*x)*log((x^
2+x)*exp(2)+x^3+x^2)+(25*x^5-75*x^4+850*x^3+450*x^2)*exp(2)+25*x^6-125*x^5+1250*x^4+900*x^3)/((x+1)*exp(2)+x^2
+x),x, algorithm="fricas")

[Out]

5*x^5 + 125*(x^3 - 18*x^2 + 81*x)*log(x^3 + x^2 + (x^2 + x)*e^2)^2 - 50*(x^4 - 9*x^3)*log(x^3 + x^2 + (x^2 + x
)*e^2)

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giac [B]  time = 0.91, size = 125, normalized size = 4.63 \begin {gather*} 5 \, x^{5} - 50 \, x^{4} \log \left (x^{3} + x^{2} e^{2} + x^{2} + x e^{2}\right ) + 125 \, x^{3} \log \left (x^{3} + x^{2} e^{2} + x^{2} + x e^{2}\right )^{2} + 450 \, x^{3} \log \left (x^{3} + x^{2} e^{2} + x^{2} + x e^{2}\right ) - 2250 \, x^{2} \log \left (x^{3} + x^{2} e^{2} + x^{2} + x e^{2}\right )^{2} + 10125 \, x \log \left (x^{3} + x^{2} e^{2} + x^{2} + x e^{2}\right )^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((375*x^3-4125*x^2+5625*x+10125)*exp(2)+375*x^4-4125*x^3+5625*x^2+10125*x)*log((x^2+x)*exp(2)+x^3+x
^2)^2+((-200*x^4+1650*x^3-7400*x^2+36000*x+20250)*exp(2)-200*x^5+1900*x^4-11650*x^3+51750*x^2+40500*x)*log((x^
2+x)*exp(2)+x^3+x^2)+(25*x^5-75*x^4+850*x^3+450*x^2)*exp(2)+25*x^6-125*x^5+1250*x^4+900*x^3)/((x+1)*exp(2)+x^2
+x),x, algorithm="giac")

[Out]

5*x^5 - 50*x^4*log(x^3 + x^2*e^2 + x^2 + x*e^2) + 125*x^3*log(x^3 + x^2*e^2 + x^2 + x*e^2)^2 + 450*x^3*log(x^3
+ x^2*e^2 + x^2 + x*e^2) - 2250*x^2*log(x^3 + x^2*e^2 + x^2 + x*e^2)^2 + 10125*x*log(x^3 + x^2*e^2 + x^2 + x*
e^2)^2

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maple [B]  time = 0.31, size = 68, normalized size = 2.52

method	result	size
risch	\(\left (125 x^{3}-2250 x^{2}+10125 x \right ) \ln \left (\left (x^{2}+x \right ) {\mathrm e}^{2}+x^{3}+x^{2}\right )^{2}+\left (-50 x^{4}+450 x^{3}\right ) \ln \left (\left (x^{2}+x \right ) {\mathrm e}^{2}+x^{3}+x^{2}\right )+5 x^{5}\)	\(68\)
norman	\(5 x^{5}+10125 x \ln \left (\left (x^{2}+x \right ) {\mathrm e}^{2}+x^{3}+x^{2}\right )^{2}-2250 x^{2} \ln \left (\left (x^{2}+x \right ) {\mathrm e}^{2}+x^{3}+x^{2}\right )^{2}+450 x^{3} \ln \left (\left (x^{2}+x \right ) {\mathrm e}^{2}+x^{3}+x^{2}\right )+125 x^{3} \ln \left (\left (x^{2}+x \right ) {\mathrm e}^{2}+x^{3}+x^{2}\right )^{2}-50 x^{4} \ln \left (\left (x^{2}+x \right ) {\mathrm e}^{2}+x^{3}+x^{2}\right )\)	\(116\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((375*x^3-4125*x^2+5625*x+10125)*exp(2)+375*x^4-4125*x^3+5625*x^2+10125*x)*ln((x^2+x)*exp(2)+x^3+x^2)^2+(
(-200*x^4+1650*x^3-7400*x^2+36000*x+20250)*exp(2)-200*x^5+1900*x^4-11650*x^3+51750*x^2+40500*x)*ln((x^2+x)*exp
(2)+x^3+x^2)+(25*x^5-75*x^4+850*x^3+450*x^2)*exp(2)+25*x^6-125*x^5+1250*x^4+900*x^3)/((x+1)*exp(2)+x^2+x),x,me
thod=_RETURNVERBOSE)

[Out]

(125*x^3-2250*x^2+10125*x)*ln((x^2+x)*exp(2)+x^3+x^2)^2+(-50*x^4+450*x^3)*ln((x^2+x)*exp(2)+x^3+x^2)+5*x^5

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maxima [B]  time = 0.47, size = 587, normalized size = 21.74 \begin {gather*} 5 \, x^{5} - \frac {25}{4} \, x^{4} {\left (e^{2} + 1\right )} + \frac {25}{4} \, x^{4} + \frac {25}{3} \, x^{3} {\left (e^{4} + e^{2} + 1\right )} - \frac {50}{3} \, x^{3} {\left (e^{2} + 28\right )} + \frac {125}{3} \, x^{3} {\left (e^{2} + 1\right )} + \frac {1250}{3} \, x^{3} - \frac {25}{2} \, x^{2} {\left (e^{6} + e^{4} + e^{2} + 1\right )} + 25 \, x^{2} {\left (e^{4} + 9 \, e^{2} + 10\right )} - \frac {125}{2} \, x^{2} {\left (e^{4} + e^{2} + 1\right )} - 625 \, x^{2} {\left (e^{2} + 1\right )} + 125 \, {\left (x^{3} - 18 \, x^{2} + 81 \, x\right )} \log \left (x + e^{2}\right )^{2} + 125 \, {\left (x^{3} - 18 \, x^{2} + 81 \, x\right )} \log \left (x + 1\right )^{2} + 125 \, {\left (x^{3} - 18 \, x^{2} + 81 \, x\right )} \log \relax (x)^{2} + 450 \, x^{2} + 25 \, x {\left (e^{8} + e^{6} + e^{4} + e^{2} + 1\right )} - 50 \, x {\left (e^{6} + 9 \, e^{4} + 10\right )} + 125 \, x {\left (e^{6} + e^{4} + e^{2} + 1\right )} + 1250 \, x {\left (e^{4} + e^{2} + 1\right )} - 900 \, x {\left (e^{2} + 1\right )} + \frac {25}{12} \, {\left (3 \, x^{4} - 4 \, x^{3} {\left (e^{2} + 1\right )} + 6 \, x^{2} {\left (e^{4} + e^{2} + 1\right )} - 12 \, x {\left (e^{6} + e^{4} + e^{2} + 1\right )} + \frac {12 \, e^{10} \log \left (x + e^{2}\right )}{e^{2} - 1} - \frac {12 \, \log \left (x + 1\right )}{e^{2} - 1}\right )} e^{2} - \frac {25}{2} \, {\left (2 \, x^{3} - 3 \, x^{2} {\left (e^{2} + 1\right )} + 6 \, x {\left (e^{4} + e^{2} + 1\right )} - \frac {6 \, e^{8} \log \left (x + e^{2}\right )}{e^{2} - 1} + \frac {6 \, \log \left (x + 1\right )}{e^{2} - 1}\right )} e^{2} + 425 \, {\left (x^{2} - 2 \, x {\left (e^{2} + 1\right )} + \frac {2 \, e^{6} \log \left (x + e^{2}\right )}{e^{2} - 1} - \frac {2 \, \log \left (x + 1\right )}{e^{2} - 1}\right )} e^{2} + 450 \, {\left (x - \frac {e^{4} \log \left (x + e^{2}\right )}{e^{2} - 1} + \frac {\log \left (x + 1\right )}{e^{2} - 1}\right )} e^{2} - 50 \, {\left (x^{4} - 9 \, x^{3} - 5 \, {\left (x^{3} - 18 \, x^{2} + 81 \, x\right )} \log \left (x + 1\right ) - 5 \, {\left (x^{3} - 18 \, x^{2} + 81 \, x\right )} \log \relax (x) - e^{8} - 9 \, e^{6}\right )} \log \left (x + e^{2}\right ) - 50 \, {\left (x^{4} - 9 \, x^{3} - 5 \, {\left (x^{3} - 18 \, x^{2} + 81 \, x\right )} \log \relax (x) - 10\right )} \log \left (x + 1\right ) - 50 \, {\left (x^{4} - 9 \, x^{3}\right )} \log \relax (x) - \frac {25 \, e^{12} \log \left (x + e^{2}\right )}{e^{2} - 1} - \frac {125 \, e^{10} \log \left (x + e^{2}\right )}{e^{2} - 1} - \frac {1250 \, e^{8} \log \left (x + e^{2}\right )}{e^{2} - 1} + \frac {900 \, e^{6} \log \left (x + e^{2}\right )}{e^{2} - 1} + \frac {500 \, \log \left (x + 1\right )}{e^{2} - 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((375*x^3-4125*x^2+5625*x+10125)*exp(2)+375*x^4-4125*x^3+5625*x^2+10125*x)*log((x^2+x)*exp(2)+x^3+x
^2)^2+((-200*x^4+1650*x^3-7400*x^2+36000*x+20250)*exp(2)-200*x^5+1900*x^4-11650*x^3+51750*x^2+40500*x)*log((x^
2+x)*exp(2)+x^3+x^2)+(25*x^5-75*x^4+850*x^3+450*x^2)*exp(2)+25*x^6-125*x^5+1250*x^4+900*x^3)/((x+1)*exp(2)+x^2
+x),x, algorithm="maxima")

[Out]

5*x^5 - 25/4*x^4*(e^2 + 1) + 25/4*x^4 + 25/3*x^3*(e^4 + e^2 + 1) - 50/3*x^3*(e^2 + 28) + 125/3*x^3*(e^2 + 1) +
1250/3*x^3 - 25/2*x^2*(e^6 + e^4 + e^2 + 1) + 25*x^2*(e^4 + 9*e^2 + 10) - 125/2*x^2*(e^4 + e^2 + 1) - 625*x^2
*(e^2 + 1) + 125*(x^3 - 18*x^2 + 81*x)*log(x + e^2)^2 + 125*(x^3 - 18*x^2 + 81*x)*log(x + 1)^2 + 125*(x^3 - 18
*x^2 + 81*x)*log(x)^2 + 450*x^2 + 25*x*(e^8 + e^6 + e^4 + e^2 + 1) - 50*x*(e^6 + 9*e^4 + 10) + 125*x*(e^6 + e^
4 + e^2 + 1) + 1250*x*(e^4 + e^2 + 1) - 900*x*(e^2 + 1) + 25/12*(3*x^4 - 4*x^3*(e^2 + 1) + 6*x^2*(e^4 + e^2 +
1) - 12*x*(e^6 + e^4 + e^2 + 1) + 12*e^10*log(x + e^2)/(e^2 - 1) - 12*log(x + 1)/(e^2 - 1))*e^2 - 25/2*(2*x^3
- 3*x^2*(e^2 + 1) + 6*x*(e^4 + e^2 + 1) - 6*e^8*log(x + e^2)/(e^2 - 1) + 6*log(x + 1)/(e^2 - 1))*e^2 + 425*(x^
2 - 2*x*(e^2 + 1) + 2*e^6*log(x + e^2)/(e^2 - 1) - 2*log(x + 1)/(e^2 - 1))*e^2 + 450*(x - e^4*log(x + e^2)/(e^
2 - 1) + log(x + 1)/(e^2 - 1))*e^2 - 50*(x^4 - 9*x^3 - 5*(x^3 - 18*x^2 + 81*x)*log(x + 1) - 5*(x^3 - 18*x^2 +
81*x)*log(x) - e^8 - 9*e^6)*log(x + e^2) - 50*(x^4 - 9*x^3 - 5*(x^3 - 18*x^2 + 81*x)*log(x) - 10)*log(x + 1) -
50*(x^4 - 9*x^3)*log(x) - 25*e^12*log(x + e^2)/(e^2 - 1) - 125*e^10*log(x + e^2)/(e^2 - 1) - 1250*e^8*log(x +
e^2)/(e^2 - 1) + 900*e^6*log(x + e^2)/(e^2 - 1) + 500*log(x + 1)/(e^2 - 1)

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mupad [B]  time = 1.43, size = 46, normalized size = 1.70 \begin {gather*} 5\,x\,{\left (45\,\ln \left ({\mathrm {e}}^2\,\left (x^2+x\right )+x^2+x^3\right )-5\,x\,\ln \left ({\mathrm {e}}^2\,\left (x^2+x\right )+x^2+x^3\right )+x^2\right )}^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(exp(2)*(x + x^2) + x^2 + x^3)*(40500*x + exp(2)*(36000*x - 7400*x^2 + 1650*x^3 - 200*x^4 + 20250) + 5
1750*x^2 - 11650*x^3 + 1900*x^4 - 200*x^5) + 900*x^3 + 1250*x^4 - 125*x^5 + 25*x^6 + log(exp(2)*(x + x^2) + x^
2 + x^3)^2*(10125*x + exp(2)*(5625*x - 4125*x^2 + 375*x^3 + 10125) + 5625*x^2 - 4125*x^3 + 375*x^4) + exp(2)*(
450*x^2 + 850*x^3 - 75*x^4 + 25*x^5))/(x + exp(2)*(x + 1) + x^2),x)

[Out]

5*x*(45*log(exp(2)*(x + x^2) + x^2 + x^3) - 5*x*log(exp(2)*(x + x^2) + x^2 + x^3) + x^2)^2

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sympy [B]  time = 0.36, size = 63, normalized size = 2.33 \begin {gather*} 5 x^{5} + \left (- 50 x^{4} + 450 x^{3}\right ) \log {\left (x^{3} + x^{2} + \left (x^{2} + x\right ) e^{2} \right )} + \left (125 x^{3} - 2250 x^{2} + 10125 x\right ) \log {\left (x^{3} + x^{2} + \left (x^{2} + x\right ) e^{2} \right )}^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((375*x**3-4125*x**2+5625*x+10125)*exp(2)+375*x**4-4125*x**3+5625*x**2+10125*x)*ln((x**2+x)*exp(2)+
x**3+x**2)**2+((-200*x**4+1650*x**3-7400*x**2+36000*x+20250)*exp(2)-200*x**5+1900*x**4-11650*x**3+51750*x**2+4
0500*x)*ln((x**2+x)*exp(2)+x**3+x**2)+(25*x**5-75*x**4+850*x**3+450*x**2)*exp(2)+25*x**6-125*x**5+1250*x**4+90
0*x**3)/((x+1)*exp(2)+x**2+x),x)

[Out]

5*x**5 + (-50*x**4 + 450*x**3)*log(x**3 + x**2 + (x**2 + x)*exp(2)) + (125*x**3 - 2250*x**2 + 10125*x)*log(x**
3 + x**2 + (x**2 + x)*exp(2))**2

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