3.81.2 \(\int \frac {e^x (-40 x+192 x^2-156 x^3-20 x^4+18 x^5)+e^{2 x} (-40 x^2+36 x^3+56 x^4-42 x^5-16 x^6+6 x^7)+(-40+192 x-156 x^2-20 x^3+18 x^4+e^x (-40 x+36 x^2+56 x^3-42 x^4-16 x^5+6 x^6)) \log (\frac {10 x-24 x^2+17 x^3-3 x^4}{2+x})}{-20 x+38 x^2-10 x^3-11 x^4+3 x^5} \, dx\) [8002]

Optimal. Leaf size=30 \[ \left (e^x x+\log \left (\left (x-x^2\right ) \left (5+3 x-\frac {25 x}{2+x}\right )\right )\right )^2 \]

[Out]

(ln((-x^2+x)*(3*x+5-x/(2/25+1/25*x)))+exp(x)*x)^2

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Rubi [C] Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
time = 22.84, antiderivative size = 5064, normalized size of antiderivative = 168.80, number of steps used = 458, number of rules used = 23, integrand size = 173, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {6820, 12, 6874, 2227, 2207, 2225, 2608, 2604, 2465, 2441, 2440, 2438, 2437, 2338, 2352, 2439, 2404, 2354, 2353, 6860, 2209, 2302, 2634} \begin {gather*} \text {Too large to display} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^x*(-40*x + 192*x^2 - 156*x^3 - 20*x^4 + 18*x^5) + E^(2*x)*(-40*x^2 + 36*x^3 + 56*x^4 - 42*x^5 - 16*x^6
+ 6*x^7) + (-40 + 192*x - 156*x^2 - 20*x^3 + 18*x^4 + E^x*(-40*x + 36*x^2 + 56*x^3 - 42*x^4 - 16*x^5 + 6*x^6))
*Log[(10*x - 24*x^2 + 17*x^3 - 3*x^4)/(2 + x)])/(-20*x + 38*x^2 - 10*x^3 - 11*x^4 + 3*x^5),x]

[Out]

E^(2*x)*x^2 - (2*(323 - 71*Sqrt[19])*E^((7 + Sqrt[19])/3)*ExpIntegralEi[(-7 - Sqrt[19] + 3*x)/3])/95 + (96*(15
2 - 29*Sqrt[19])*E^((7 + Sqrt[19])/3)*ExpIntegralEi[(-7 - Sqrt[19] + 3*x)/3])/475 - (78*(171 - 17*Sqrt[19])*E^
((7 + Sqrt[19])/3)*ExpIntegralEi[(-7 - Sqrt[19] + 3*x)/3])/475 - (2*(7 + Sqrt[19])*E^((7 + Sqrt[19])/3)*ExpInt
egralEi[(-7 - Sqrt[19] + 3*x)/3])/3 - (4*(437 + 26*Sqrt[19])*E^((7 + Sqrt[19])/3)*ExpIntegralEi[(-7 - Sqrt[19]
+ 3*x)/3])/285 + (2*(3553 + 619*Sqrt[19])*E^((7 + Sqrt[19])/3)*ExpIntegralEi[(-7 - Sqrt[19] + 3*x)/3])/475 +
(2*(3553 - 619*Sqrt[19])*E^(7/3 - Sqrt[19]/3)*ExpIntegralEi[(-7 + Sqrt[19] + 3*x)/3])/475 - (4*(437 - 26*Sqrt[
19])*E^(7/3 - Sqrt[19]/3)*ExpIntegralEi[(-7 + Sqrt[19] + 3*x)/3])/285 - (2*(7 - Sqrt[19])*E^(7/3 - Sqrt[19]/3)
*ExpIntegralEi[(-7 + Sqrt[19] + 3*x)/3])/3 - (78*(171 + 17*Sqrt[19])*E^(7/3 - Sqrt[19]/3)*ExpIntegralEi[(-7 +
Sqrt[19] + 3*x)/3])/475 + (96*(152 + 29*Sqrt[19])*E^(7/3 - Sqrt[19]/3)*ExpIntegralEi[(-7 + Sqrt[19] + 3*x)/3])
/475 - (2*(323 + 71*Sqrt[19])*E^(7/3 - Sqrt[19]/3)*ExpIntegralEi[(-7 + Sqrt[19] + 3*x)/3])/95 - (3*(437 - 26*S
qrt[19])*Log[-2*(7 - Sqrt[19] - 3*x)]^2)/475 + ((171 + 17*Sqrt[19])*Log[-2*(7 - Sqrt[19] - 3*x)]^2)/95 + (39*(
152 + 29*Sqrt[19])*Log[-2*(7 - Sqrt[19] - 3*x)]^2)/475 - (24*(323 + 71*Sqrt[19])*Log[-2*(7 - Sqrt[19] - 3*x)]^
2)/475 + ((361 + 82*Sqrt[19])*Log[-2*(7 - Sqrt[19] - 3*x)]^2)/95 + ((361 - 82*Sqrt[19])*Log[-2*(7 + Sqrt[19] -
3*x)]^2)/95 - (24*(323 - 71*Sqrt[19])*Log[-2*(7 + Sqrt[19] - 3*x)]^2)/475 + (39*(152 - 29*Sqrt[19])*Log[-2*(7
+ Sqrt[19] - 3*x)]^2)/475 + ((171 - 17*Sqrt[19])*Log[-2*(7 + Sqrt[19] - 3*x)]^2)/95 - (3*(437 + 26*Sqrt[19])*
Log[-2*(7 + Sqrt[19] - 3*x)]^2)/475 - 2*Log[(7 - Sqrt[19] - 3*x)/(4 - Sqrt[19])]*Log[-1 + x] - 2*Log[(7 + Sqrt
[19] - 3*x)/(4 + Sqrt[19])]*Log[-1 + x] - Log[-1 + x]^2 + 2*Log[2]*Log[x] + 2*Log[1 + x/2]*Log[x] - 2*Log[-1 +
x]*Log[x] - Log[x]^2 + 2*Log[-1 + x]*Log[(2 + x)/3] + 2*Log[(7 - Sqrt[19] - 3*x)/(13 - Sqrt[19])]*Log[2 + x]
+ 2*Log[(7 + Sqrt[19] - 3*x)/(13 + Sqrt[19])]*Log[2 + x] + 2*Log[(1 - x)/3]*Log[2 + x] - Log[2 + x]^2 - 2*Log[
(7 - Sqrt[19])/3]*Log[-2*(7 - Sqrt[19]) + 6*x] - (6*(437 - 26*Sqrt[19])*Log[(7 + Sqrt[19] - 3*x)/(2*Sqrt[19])]
*Log[-2*(7 - Sqrt[19]) + 6*x])/475 + (2*(171 + 17*Sqrt[19])*Log[(7 + Sqrt[19] - 3*x)/(2*Sqrt[19])]*Log[-2*(7 -
Sqrt[19]) + 6*x])/95 + (78*(152 + 29*Sqrt[19])*Log[(7 + Sqrt[19] - 3*x)/(2*Sqrt[19])]*Log[-2*(7 - Sqrt[19]) +
6*x])/475 - (48*(323 + 71*Sqrt[19])*Log[(7 + Sqrt[19] - 3*x)/(2*Sqrt[19])]*Log[-2*(7 - Sqrt[19]) + 6*x])/475
+ (2*(361 + 82*Sqrt[19])*Log[(7 + Sqrt[19] - 3*x)/(2*Sqrt[19])]*Log[-2*(7 - Sqrt[19]) + 6*x])/95 - (6*(437 - 2
6*Sqrt[19])*Log[(-3*(1 - x))/(4 - Sqrt[19])]*Log[-2*(7 - Sqrt[19]) + 6*x])/475 + (2*(171 + 17*Sqrt[19])*Log[(-
3*(1 - x))/(4 - Sqrt[19])]*Log[-2*(7 - Sqrt[19]) + 6*x])/95 + (78*(152 + 29*Sqrt[19])*Log[(-3*(1 - x))/(4 - Sq
rt[19])]*Log[-2*(7 - Sqrt[19]) + 6*x])/475 - (48*(323 + 71*Sqrt[19])*Log[(-3*(1 - x))/(4 - Sqrt[19])]*Log[-2*(
7 - Sqrt[19]) + 6*x])/475 + (2*(361 + 82*Sqrt[19])*Log[(-3*(1 - x))/(4 - Sqrt[19])]*Log[-2*(7 - Sqrt[19]) + 6*
x])/95 - (6*(437 - 26*Sqrt[19])*Log[(3*x)/(7 - Sqrt[19])]*Log[-2*(7 - Sqrt[19]) + 6*x])/475 + (2*(171 + 17*Sqr
t[19])*Log[(3*x)/(7 - Sqrt[19])]*Log[-2*(7 - Sqrt[19]) + 6*x])/95 + (78*(152 + 29*Sqrt[19])*Log[(3*x)/(7 - Sqr
t[19])]*Log[-2*(7 - Sqrt[19]) + 6*x])/475 - (48*(323 + 71*Sqrt[19])*Log[(3*x)/(7 - Sqrt[19])]*Log[-2*(7 - Sqrt
[19]) + 6*x])/475 + (2*(361 + 82*Sqrt[19])*Log[(3*x)/(7 - Sqrt[19])]*Log[-2*(7 - Sqrt[19]) + 6*x])/95 + (6*(43
7 - 26*Sqrt[19])*Log[(3*(2 + x))/(13 - Sqrt[19])]*Log[-2*(7 - Sqrt[19]) + 6*x])/475 - (2*(171 + 17*Sqrt[19])*L
og[(3*(2 + x))/(13 - Sqrt[19])]*Log[-2*(7 - Sqrt[19]) + 6*x])/95 - (78*(152 + 29*Sqrt[19])*Log[(3*(2 + x))/(13
- Sqrt[19])]*Log[-2*(7 - Sqrt[19]) + 6*x])/475 + (48*(323 + 71*Sqrt[19])*Log[(3*(2 + x))/(13 - Sqrt[19])]*Log
[-2*(7 - Sqrt[19]) + 6*x])/475 - (2*(361 + 82*Sqrt[19])*Log[(3*(2 + x))/(13 - Sqrt[19])]*Log[-2*(7 - Sqrt[19])
+ 6*x])/95 - 2*Log[(7 + Sqrt[19])/3]*Log[-2*(7 + Sqrt[19]) + 6*x] + (2*(361 - 82*Sqrt[19])*Log[-1/2*(7 - Sqrt
[19] - 3*x)/Sqrt[19]]*Log[-2*(7 + Sqrt[19]) + 6*x])/95 - (48*(323 - 71*Sqrt[19])*Log[-1/2*(7 - Sqrt[19] - 3*x)
/Sqrt[19]]*Log[-2*(7 + Sqrt[19]) + 6*x])/475 + (78*(152 - 29*Sqrt[19])*Log[-1/2*(7 - Sqrt[19] - 3*x)/Sqrt[19]]
*Log[-2*(7 + Sqrt[19]) + 6*x])/475 + (2*(171 - 17*Sqrt[19])*Log[-1/2*(7 - Sqrt[19] - 3*x)/Sqrt[19]]*Log[-2*(7
+ Sqrt[19]) + 6*x])/95 - (6*(437 + 26*Sqrt[19])*Log[-1/2*(7 - Sqrt[19] - 3*x)/Sqrt[19]]*Log[-2*(7 + Sqrt[19])
+ 6*x])/475 + (2*(361 - 82*Sqrt[19])*Log[(-3*(1 - x))/(4 + Sqrt[19])]*Log[-2*(7 + Sqrt[19]) + 6*x])/95 - (48*(
323 - 71*Sqrt[19])*Log[(-3*(1 - x))/(4 + Sqrt[19])]*Log[-2*(7 + Sqrt[19]) + 6*x])/475 + (78*(152 - 29*Sqrt[19]
)*Log[(-3*(1 - x))/(4 + Sqrt[19])]*Log[-2*(7 + Sqrt[19]) + 6*x])/475 + (2*(171 - 17*Sqrt[19])*Log[(-3*(1 - x))
/(4 + Sqrt[19])]*Log[-2*(7 + Sqrt[19]) + 6*x])/95 - (6*(437 + 26*Sqrt[19])*Log[(-3*(1 - x))/(4 + Sqrt[19])]*Lo
g[-2*(7 + Sqrt[19]) + 6*x])/475 + (2*(361 - 82*...

Rule 12

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

Rule 2207

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(c + d*x)^m*
((b*F^(g*(e + f*x)))^n/(f*g*n*Log[F])), x] - Dist[d*(m/(f*g*n*Log[F])), Int[(c + d*x)^(m - 1)*(b*F^(g*(e + f*x
)))^n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && GtQ[m, 0] && IntegerQ[2*m] &&  !TrueQ[$UseGamma]

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 2225

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

Rule 2227

Int[(F_)^((c_.)*(v_))*(u_), x_Symbol] :> Int[ExpandIntegrand[F^(c*ExpandToSum[v, x]), u, x], x] /; FreeQ[{F, c
}, x] && PolynomialQ[u, x] && LinearQ[v, x] &&  !TrueQ[$UseGamma]

Rule 2302

Int[((F_)^((g_.)*((d_.) + (e_.)*(x_))^(n_.))*(u_)^(m_.))/((a_.) + (b_.)*(x_) + (c_)*(x_)^2), x_Symbol] :> Int[
ExpandIntegrand[F^(g*(d + e*x)^n), u^m/(a + b*x + c*x^2), x], x] /; FreeQ[{F, a, b, c, d, e, g, n}, x] && Poly
nomialQ[u, x] && IntegerQ[m]

Rule 2338

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 2352

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

Rule 2353

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 2354

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 2404

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

Rule 2437

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 2438

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 2439

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 2440

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 2441

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 2465

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 2604

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 2608

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 2634

Int[Log[u_]*(v_), x_Symbol] :> With[{w = IntHide[v, x]}, Dist[Log[u], w, x] - Int[SimplifyIntegrand[w*(D[u, x]
/u), x], x] /; InverseFunctionFreeQ[w, x]] /; InverseFunctionFreeQ[u, x]

Rule 6820

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

Rule 6860

Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a +
b*x^n + c*x^(2*n)), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]

Rule 6874

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 \left (20-\left (96-20 e^x\right ) x-6 \left (-13+3 e^x\right ) x^2-2 \left (-5+14 e^x\right ) x^3-\left (9-21 e^x\right ) x^4+8 e^x x^5-3 e^x x^6\right ) \left (e^x x+\log \left (\frac {x \left (10-24 x+17 x^2-3 x^3\right )}{2+x}\right )\right )}{x \left (20-38 x+10 x^2+11 x^3-3 x^4\right )} \, dx\\ &=2 \int \frac {\left (20-\left (96-20 e^x\right ) x-6 \left (-13+3 e^x\right ) x^2-2 \left (-5+14 e^x\right ) x^3-\left (9-21 e^x\right ) x^4+8 e^x x^5-3 e^x x^6\right ) \left (e^x x+\log \left (\frac {x \left (10-24 x+17 x^2-3 x^3\right )}{2+x}\right )\right )}{x \left (20-38 x+10 x^2+11 x^3-3 x^4\right )} \, dx\\ &=2 \int \left (e^{2 x} x (1+x)+\frac {96 \log \left (\frac {x \left (10-24 x+17 x^2-3 x^3\right )}{2+x}\right )}{(-1+x) (2+x) \left (10-14 x+3 x^2\right )}-\frac {20 \log \left (\frac {x \left (10-24 x+17 x^2-3 x^3\right )}{2+x}\right )}{(-1+x) x (2+x) \left (10-14 x+3 x^2\right )}-\frac {78 x \log \left (\frac {x \left (10-24 x+17 x^2-3 x^3\right )}{2+x}\right )}{(-1+x) (2+x) \left (10-14 x+3 x^2\right )}-\frac {10 x^2 \log \left (\frac {x \left (10-24 x+17 x^2-3 x^3\right )}{2+x}\right )}{(-1+x) (2+x) \left (10-14 x+3 x^2\right )}+\frac {9 x^3 \log \left (\frac {x \left (10-24 x+17 x^2-3 x^3\right )}{2+x}\right )}{(-1+x) (2+x) \left (10-14 x+3 x^2\right )}+\frac {e^x \left (-20+96 x-78 x^2-10 x^3+9 x^4-20 \log \left (\frac {x \left (10-24 x+17 x^2-3 x^3\right )}{2+x}\right )+18 x \log \left (\frac {x \left (10-24 x+17 x^2-3 x^3\right )}{2+x}\right )+28 x^2 \log \left (\frac {x \left (10-24 x+17 x^2-3 x^3\right )}{2+x}\right )-21 x^3 \log \left (\frac {x \left (10-24 x+17 x^2-3 x^3\right )}{2+x}\right )-8 x^4 \log \left (\frac {x \left (10-24 x+17 x^2-3 x^3\right )}{2+x}\right )+3 x^5 \log \left (\frac {x \left (10-24 x+17 x^2-3 x^3\right )}{2+x}\right )\right )}{(-1+x) (2+x) \left (10-14 x+3 x^2\right )}\right ) \, dx\\ &=2 \int e^{2 x} x (1+x) \, dx+2 \int \frac {e^x \left (-20+96 x-78 x^2-10 x^3+9 x^4-20 \log \left (\frac {x \left (10-24 x+17 x^2-3 x^3\right )}{2+x}\right )+18 x \log \left (\frac {x \left (10-24 x+17 x^2-3 x^3\right )}{2+x}\right )+28 x^2 \log \left (\frac {x \left (10-24 x+17 x^2-3 x^3\right )}{2+x}\right )-21 x^3 \log \left (\frac {x \left (10-24 x+17 x^2-3 x^3\right )}{2+x}\right )-8 x^4 \log \left (\frac {x \left (10-24 x+17 x^2-3 x^3\right )}{2+x}\right )+3 x^5 \log \left (\frac {x \left (10-24 x+17 x^2-3 x^3\right )}{2+x}\right )\right )}{(-1+x) (2+x) \left (10-14 x+3 x^2\right )} \, dx+18 \int \frac {x^3 \log \left (\frac {x \left (10-24 x+17 x^2-3 x^3\right )}{2+x}\right )}{(-1+x) (2+x) \left (10-14 x+3 x^2\right )} \, dx-20 \int \frac {x^2 \log \left (\frac {x \left (10-24 x+17 x^2-3 x^3\right )}{2+x}\right )}{(-1+x) (2+x) \left (10-14 x+3 x^2\right )} \, dx-40 \int \frac {\log \left (\frac {x \left (10-24 x+17 x^2-3 x^3\right )}{2+x}\right )}{(-1+x) x (2+x) \left (10-14 x+3 x^2\right )} \, dx-156 \int \frac {x \log \left (\frac {x \left (10-24 x+17 x^2-3 x^3\right )}{2+x}\right )}{(-1+x) (2+x) \left (10-14 x+3 x^2\right )} \, dx+192 \int \frac {\log \left (\frac {x \left (10-24 x+17 x^2-3 x^3\right )}{2+x}\right )}{(-1+x) (2+x) \left (10-14 x+3 x^2\right )} \, dx\\ &=2 \int \left (e^{2 x} x+e^{2 x} x^2\right ) \, dx+2 \int \frac {e^x \left (20-96 x+78 x^2+10 x^3-9 x^4-\left (-20+18 x+28 x^2-21 x^3-8 x^4+3 x^5\right ) \log \left (\frac {x \left (10-24 x+17 x^2-3 x^3\right )}{2+x}\right )\right )}{(1-x) (2+x) \left (10-14 x+3 x^2\right )} \, dx+18 \int \left (-\frac {\log \left (\frac {x \left (10-24 x+17 x^2-3 x^3\right )}{2+x}\right )}{3 (-1+x)}+\frac {4 \log \left (\frac {x \left (10-24 x+17 x^2-3 x^3\right )}{2+x}\right )}{75 (2+x)}+\frac {2 (-45+23 x) \log \left (\frac {x \left (10-24 x+17 x^2-3 x^3\right )}{2+x}\right )}{25 \left (10-14 x+3 x^2\right )}\right ) \, dx-20 \int \left (-\frac {\log \left (\frac {x \left (10-24 x+17 x^2-3 x^3\right )}{2+x}\right )}{3 (-1+x)}-\frac {2 \log \left (\frac {x \left (10-24 x+17 x^2-3 x^3\right )}{2+x}\right )}{75 (2+x)}+\frac {(-80+27 x) \log \left (\frac {x \left (10-24 x+17 x^2-3 x^3\right )}{2+x}\right )}{25 \left (10-14 x+3 x^2\right )}\right ) \, dx-40 \int \left (-\frac {\log \left (\frac {x \left (10-24 x+17 x^2-3 x^3\right )}{2+x}\right )}{3 (-1+x)}-\frac {\log \left (\frac {x \left (10-24 x+17 x^2-3 x^3\right )}{2+x}\right )}{20 x}+\frac {\log \left (\frac {x \left (10-24 x+17 x^2-3 x^3\right )}{2+x}\right )}{300 (2+x)}+\frac {(-215+57 x) \log \left (\frac {x \left (10-24 x+17 x^2-3 x^3\right )}{2+x}\right )}{50 \left (10-14 x+3 x^2\right )}\right ) \, dx-156 \int \left (-\frac {\log \left (\frac {x \left (10-24 x+17 x^2-3 x^3\right )}{2+x}\right )}{3 (-1+x)}+\frac {\log \left (\frac {x \left (10-24 x+17 x^2-3 x^3\right )}{2+x}\right )}{75 (2+x)}+\frac {(-85+24 x) \log \left (\frac {x \left (10-24 x+17 x^2-3 x^3\right )}{2+x}\right )}{25 \left (10-14 x+3 x^2\right )}\right ) \, dx+192 \int \left (-\frac {\log \left (\frac {x \left (10-24 x+17 x^2-3 x^3\right )}{2+x}\right )}{3 (-1+x)}-\frac {\log \left (\frac {x \left (10-24 x+17 x^2-3 x^3\right )}{2+x}\right )}{150 (2+x)}+\frac {(-190+51 x) \log \left (\frac {x \left (10-24 x+17 x^2-3 x^3\right )}{2+x}\right )}{50 \left (10-14 x+3 x^2\right )}\right ) \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.10, size = 31, normalized size = 1.03 \begin {gather*} \left (e^x x+\log \left (\frac {x \left (10-24 x+17 x^2-3 x^3\right )}{2+x}\right )\right )^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^x*(-40*x + 192*x^2 - 156*x^3 - 20*x^4 + 18*x^5) + E^(2*x)*(-40*x^2 + 36*x^3 + 56*x^4 - 42*x^5 - 1
6*x^6 + 6*x^7) + (-40 + 192*x - 156*x^2 - 20*x^3 + 18*x^4 + E^x*(-40*x + 36*x^2 + 56*x^3 - 42*x^4 - 16*x^5 + 6
*x^6))*Log[(10*x - 24*x^2 + 17*x^3 - 3*x^4)/(2 + x)])/(-20*x + 38*x^2 - 10*x^3 - 11*x^4 + 3*x^5),x]

[Out]

(E^x*x + Log[(x*(10 - 24*x + 17*x^2 - 3*x^3))/(2 + x)])^2

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Maple [F]
time = 0.07, size = 0, normalized size = 0.00 \[\int \frac {\left (\left (6 x^{6}-16 x^{5}-42 x^{4}+56 x^{3}+36 x^{2}-40 x \right ) {\mathrm e}^{x}+18 x^{4}-20 x^{3}-156 x^{2}+192 x -40\right ) \ln \left (\frac {-3 x^{4}+17 x^{3}-24 x^{2}+10 x}{2+x}\right )+\left (6 x^{7}-16 x^{6}-42 x^{5}+56 x^{4}+36 x^{3}-40 x^{2}\right ) {\mathrm e}^{2 x}+\left (18 x^{5}-20 x^{4}-156 x^{3}+192 x^{2}-40 x \right ) {\mathrm e}^{x}}{3 x^{5}-11 x^{4}-10 x^{3}+38 x^{2}-20 x}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((6*x^6-16*x^5-42*x^4+56*x^3+36*x^2-40*x)*exp(x)+18*x^4-20*x^3-156*x^2+192*x-40)*ln((-3*x^4+17*x^3-24*x^2
+10*x)/(2+x))+(6*x^7-16*x^6-42*x^5+56*x^4+36*x^3-40*x^2)*exp(x)^2+(18*x^5-20*x^4-156*x^3+192*x^2-40*x)*exp(x))
/(3*x^5-11*x^4-10*x^3+38*x^2-20*x),x)

[Out]

int((((6*x^6-16*x^5-42*x^4+56*x^3+36*x^2-40*x)*exp(x)+18*x^4-20*x^3-156*x^2+192*x-40)*ln((-3*x^4+17*x^3-24*x^2
+10*x)/(2+x))+(6*x^7-16*x^6-42*x^5+56*x^4+36*x^3-40*x^2)*exp(x)^2+(18*x^5-20*x^4-156*x^3+192*x^2-40*x)*exp(x))
/(3*x^5-11*x^4-10*x^3+38*x^2-20*x),x)

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 105 vs. \(2 (30) = 60\).
time = 0.33, size = 105, normalized size = 3.50 \begin {gather*} x^{2} e^{\left (2 \, x\right )} + 2 \, x e^{x} \log \left (x\right ) + 2 \, {\left (x e^{x} - \log \left (x + 2\right ) + \log \left (x - 1\right ) + \log \left (x\right )\right )} \log \left (-3 \, x^{2} + 14 \, x - 10\right ) + \log \left (-3 \, x^{2} + 14 \, x - 10\right )^{2} - 2 \, {\left (x e^{x} + \log \left (x - 1\right ) + \log \left (x\right )\right )} \log \left (x + 2\right ) + \log \left (x + 2\right )^{2} + 2 \, {\left (x e^{x} + \log \left (x\right )\right )} \log \left (x - 1\right ) + \log \left (x - 1\right )^{2} + \log \left (x\right )^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((6*x^6-16*x^5-42*x^4+56*x^3+36*x^2-40*x)*exp(x)+18*x^4-20*x^3-156*x^2+192*x-40)*log((-3*x^4+17*x^3
-24*x^2+10*x)/(2+x))+(6*x^7-16*x^6-42*x^5+56*x^4+36*x^3-40*x^2)*exp(x)^2+(18*x^5-20*x^4-156*x^3+192*x^2-40*x)*
exp(x))/(3*x^5-11*x^4-10*x^3+38*x^2-20*x),x, algorithm="maxima")

[Out]

x^2*e^(2*x) + 2*x*e^x*log(x) + 2*(x*e^x - log(x + 2) + log(x - 1) + log(x))*log(-3*x^2 + 14*x - 10) + log(-3*x
^2 + 14*x - 10)^2 - 2*(x*e^x + log(x - 1) + log(x))*log(x + 2) + log(x + 2)^2 + 2*(x*e^x + log(x))*log(x - 1)
+ log(x - 1)^2 + log(x)^2

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 70 vs. \(2 (30) = 60\).
time = 0.43, size = 70, normalized size = 2.33 \begin {gather*} x^{2} e^{\left (2 \, x\right )} + 2 \, x e^{x} \log \left (-\frac {3 \, x^{4} - 17 \, x^{3} + 24 \, x^{2} - 10 \, x}{x + 2}\right ) + \log \left (-\frac {3 \, x^{4} - 17 \, x^{3} + 24 \, x^{2} - 10 \, x}{x + 2}\right )^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((6*x^6-16*x^5-42*x^4+56*x^3+36*x^2-40*x)*exp(x)+18*x^4-20*x^3-156*x^2+192*x-40)*log((-3*x^4+17*x^3
-24*x^2+10*x)/(2+x))+(6*x^7-16*x^6-42*x^5+56*x^4+36*x^3-40*x^2)*exp(x)^2+(18*x^5-20*x^4-156*x^3+192*x^2-40*x)*
exp(x))/(3*x^5-11*x^4-10*x^3+38*x^2-20*x),x, algorithm="fricas")

[Out]

x^2*e^(2*x) + 2*x*e^x*log(-(3*x^4 - 17*x^3 + 24*x^2 - 10*x)/(x + 2)) + log(-(3*x^4 - 17*x^3 + 24*x^2 - 10*x)/(
x + 2))^2

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 63 vs. \(2 (26) = 52\).
time = 0.63, size = 63, normalized size = 2.10 \begin {gather*} x^{2} e^{2 x} + 2 x e^{x} \log {\left (\frac {- 3 x^{4} + 17 x^{3} - 24 x^{2} + 10 x}{x + 2} \right )} + \log {\left (\frac {- 3 x^{4} + 17 x^{3} - 24 x^{2} + 10 x}{x + 2} \right )}^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((6*x**6-16*x**5-42*x**4+56*x**3+36*x**2-40*x)*exp(x)+18*x**4-20*x**3-156*x**2+192*x-40)*ln((-3*x**
4+17*x**3-24*x**2+10*x)/(2+x))+(6*x**7-16*x**6-42*x**5+56*x**4+36*x**3-40*x**2)*exp(x)**2+(18*x**5-20*x**4-156
*x**3+192*x**2-40*x)*exp(x))/(3*x**5-11*x**4-10*x**3+38*x**2-20*x),x)

[Out]

x**2*exp(2*x) + 2*x*exp(x)*log((-3*x**4 + 17*x**3 - 24*x**2 + 10*x)/(x + 2)) + log((-3*x**4 + 17*x**3 - 24*x**
2 + 10*x)/(x + 2))**2

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((6*x^6-16*x^5-42*x^4+56*x^3+36*x^2-40*x)*exp(x)+18*x^4-20*x^3-156*x^2+192*x-40)*log((-3*x^4+17*x^3
-24*x^2+10*x)/(2+x))+(6*x^7-16*x^6-42*x^5+56*x^4+36*x^3-40*x^2)*exp(x)^2+(18*x^5-20*x^4-156*x^3+192*x^2-40*x)*
exp(x))/(3*x^5-11*x^4-10*x^3+38*x^2-20*x),x, algorithm="giac")

[Out]

integrate(2*((3*x^7 - 8*x^6 - 21*x^5 + 28*x^4 + 18*x^3 - 20*x^2)*e^(2*x) + (9*x^5 - 10*x^4 - 78*x^3 + 96*x^2 -
20*x)*e^x + (9*x^4 - 10*x^3 - 78*x^2 + (3*x^6 - 8*x^5 - 21*x^4 + 28*x^3 + 18*x^2 - 20*x)*e^x + 96*x - 20)*log
(-(3*x^4 - 17*x^3 + 24*x^2 - 10*x)/(x + 2)))/(3*x^5 - 11*x^4 - 10*x^3 + 38*x^2 - 20*x), x)

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Mupad [B]
time = 7.13, size = 33, normalized size = 1.10 \begin {gather*} {\left (\ln \left (\frac {-3\,x^4+17\,x^3-24\,x^2+10\,x}{x+2}\right )+x\,{\mathrm {e}}^x\right )}^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(2*x)*(40*x^2 - 36*x^3 - 56*x^4 + 42*x^5 + 16*x^6 - 6*x^7) + exp(x)*(40*x - 192*x^2 + 156*x^3 + 20*x^4
- 18*x^5) + log((10*x - 24*x^2 + 17*x^3 - 3*x^4)/(x + 2))*(exp(x)*(40*x - 36*x^2 - 56*x^3 + 42*x^4 + 16*x^5 -
6*x^6) - 192*x + 156*x^2 + 20*x^3 - 18*x^4 + 40))/(20*x - 38*x^2 + 10*x^3 + 11*x^4 - 3*x^5),x)

[Out]

(log((10*x - 24*x^2 + 17*x^3 - 3*x^4)/(x + 2)) + x*exp(x))^2

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