3.18.13 \(\int \frac {360+818 x-320 x^2-18 x^3+e^x (1800+980 x-560 x^2+70 x^3+10 x^4)+e^{2 x} (450 x+200 x^2-200 x^3+50 x^4)+(-1800+820 x-350 x^2+80 x^3+e^x (-900 x+50 x^2+150 x^3-50 x^4)) \log ((9-5 x+x^2) \log (5))+(450 x-250 x^2+50 x^3) \log ^2((9-5 x+x^2) \log (5))}{9-5 x+x^2} \, dx\)
Optimal. Leaf size=29 \[ 2+\left (x-5 \left (-4+x \left (-e^x+\log \left (\left ((-3+x)^2+x\right ) \log (5)\right )\right )\right )\right )^2 \]
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Rubi [C] time = 6.93, antiderivative size = 1593, normalized size of antiderivative = 54.93,
number of steps used = 170, number of rules used = 32, integrand size = 155, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.206, Rules
used = {6728, 2196, 2176, 2194, 618, 204, 634, 628, 703, 701, 2527, 12, 5057, 4920, 4854, 2402,
2315, 2528, 2524, 2418, 2390, 2301, 2394, 2393, 2391, 2523, 773, 2525, 800, 6688, 2178,
2554}
result too large to display
Antiderivative was successfully verified.
[In]
Int[(360 + 818*x - 320*x^2 - 18*x^3 + E^x*(1800 + 980*x - 560*x^2 + 70*x^3 + 10*x^4) + E^(2*x)*(450*x + 200*x^
2 - 200*x^3 + 50*x^4) + (-1800 + 820*x - 350*x^2 + 80*x^3 + E^x*(-900*x + 50*x^2 + 150*x^3 - 50*x^4))*Log[(9 -
5*x + x^2)*Log[5]] + (450*x - 250*x^2 + 50*x^3)*Log[(9 - 5*x + x^2)*Log[5]]^2)/(9 - 5*x + x^2),x]
[Out]
40*x + 200*E^x*x + x^2 + 10*E^x*x^2 + 25*E^(2*x)*x^2 + ((3600*I)*ArcTan[(5 - 2*x)/Sqrt[11]]^2)/Sqrt[11] + (720
0*ArcTan[(5 - 2*x)/Sqrt[11]]*Log[(2*Sqrt[11])/(Sqrt[11] + I*(5 - 2*x))])/Sqrt[11] - (410*(11 - (5*I)*Sqrt[11])
*Log[((I/2)*(5 - I*Sqrt[11] - 2*x))/Sqrt[11]]*Log[-5 - I*Sqrt[11] + 2*x])/11 + 25*(7 + (5*I)*Sqrt[11])*Log[((I
/2)*(5 - I*Sqrt[11] - 2*x))/Sqrt[11]]*Log[-5 - I*Sqrt[11] + 2*x] - (80*(88 + (5*I)*Sqrt[11])*Log[((I/2)*(5 - I
*Sqrt[11] - 2*x))/Sqrt[11]]*Log[-5 - I*Sqrt[11] + 2*x])/11 + (175*(55 - (7*I)*Sqrt[11])*Log[((I/2)*(5 - I*Sqrt
[11] - 2*x))/Sqrt[11]]*Log[-5 - I*Sqrt[11] + 2*x])/11 - (205*(11 - (5*I)*Sqrt[11])*Log[-5 - I*Sqrt[11] + 2*x]^
2)/11 + (25*(7 + (5*I)*Sqrt[11])*Log[-5 - I*Sqrt[11] + 2*x]^2)/2 - (40*(88 + (5*I)*Sqrt[11])*Log[-5 - I*Sqrt[1
1] + 2*x]^2)/11 + (175*(55 - (7*I)*Sqrt[11])*Log[-5 - I*Sqrt[11] + 2*x]^2)/22 + 25*(7 - (5*I)*Sqrt[11])*Log[((
-1/2*I)*(5 + I*Sqrt[11] - 2*x))/Sqrt[11]]*Log[-5 + I*Sqrt[11] + 2*x] - (80*(88 - (5*I)*Sqrt[11])*Log[((-1/2*I)
*(5 + I*Sqrt[11] - 2*x))/Sqrt[11]]*Log[-5 + I*Sqrt[11] + 2*x])/11 - (410*(11 + (5*I)*Sqrt[11])*Log[((-1/2*I)*(
5 + I*Sqrt[11] - 2*x))/Sqrt[11]]*Log[-5 + I*Sqrt[11] + 2*x])/11 + (175*(55 + (7*I)*Sqrt[11])*Log[((-1/2*I)*(5
+ I*Sqrt[11] - 2*x))/Sqrt[11]]*Log[-5 + I*Sqrt[11] + 2*x])/11 + (25*(7 - (5*I)*Sqrt[11])*Log[-5 + I*Sqrt[11] +
2*x]^2)/2 - (40*(88 - (5*I)*Sqrt[11])*Log[-5 + I*Sqrt[11] + 2*x]^2)/11 - (205*(11 + (5*I)*Sqrt[11])*Log[-5 +
I*Sqrt[11] + 2*x]^2)/11 + (175*(55 + (7*I)*Sqrt[11])*Log[-5 + I*Sqrt[11] + 2*x]^2)/22 - 200*x*Log[(9 - 5*x + x
^2)*Log[5]] - 10*x^2*Log[(9 - 5*x + x^2)*Log[5]] - 50*E^x*x^2*Log[(9 - 5*x + x^2)*Log[5]] + (3600*ArcTan[(5 -
2*x)/Sqrt[11]]*Log[(9 - 5*x + x^2)*Log[5]])/Sqrt[11] + (410*(11 - (5*I)*Sqrt[11])*Log[-5 - I*Sqrt[11] + 2*x]*L
og[(9 - 5*x + x^2)*Log[5]])/11 - 25*(7 + (5*I)*Sqrt[11])*Log[-5 - I*Sqrt[11] + 2*x]*Log[(9 - 5*x + x^2)*Log[5]
] + (80*(88 + (5*I)*Sqrt[11])*Log[-5 - I*Sqrt[11] + 2*x]*Log[(9 - 5*x + x^2)*Log[5]])/11 - (175*(55 - (7*I)*Sq
rt[11])*Log[-5 - I*Sqrt[11] + 2*x]*Log[(9 - 5*x + x^2)*Log[5]])/11 - 25*(7 - (5*I)*Sqrt[11])*Log[-5 + I*Sqrt[1
1] + 2*x]*Log[(9 - 5*x + x^2)*Log[5]] + (80*(88 - (5*I)*Sqrt[11])*Log[-5 + I*Sqrt[11] + 2*x]*Log[(9 - 5*x + x^
2)*Log[5]])/11 + (410*(11 + (5*I)*Sqrt[11])*Log[-5 + I*Sqrt[11] + 2*x]*Log[(9 - 5*x + x^2)*Log[5]])/11 - (175*
(55 + (7*I)*Sqrt[11])*Log[-5 + I*Sqrt[11] + 2*x]*Log[(9 - 5*x + x^2)*Log[5]])/11 + 25*x^2*Log[(9 - 5*x + x^2)*
Log[5]]^2 + ((3600*I)*PolyLog[2, 1 - (2*Sqrt[11])/(Sqrt[11] + I*(5 - 2*x))])/Sqrt[11] - (410*(11 - (5*I)*Sqrt[
11])*PolyLog[2, -1/2*(5*I - Sqrt[11] - (2*I)*x)/Sqrt[11]])/11 + 25*(7 + (5*I)*Sqrt[11])*PolyLog[2, -1/2*(5*I -
Sqrt[11] - (2*I)*x)/Sqrt[11]] - (80*(88 + (5*I)*Sqrt[11])*PolyLog[2, -1/2*(5*I - Sqrt[11] - (2*I)*x)/Sqrt[11]
])/11 + (175*(55 - (7*I)*Sqrt[11])*PolyLog[2, -1/2*(5*I - Sqrt[11] - (2*I)*x)/Sqrt[11]])/11 + 25*(7 - (5*I)*Sq
rt[11])*PolyLog[2, (5*I + Sqrt[11] - (2*I)*x)/(2*Sqrt[11])] - (80*(88 - (5*I)*Sqrt[11])*PolyLog[2, (5*I + Sqrt
[11] - (2*I)*x)/(2*Sqrt[11])])/11 - (410*(11 + (5*I)*Sqrt[11])*PolyLog[2, (5*I + Sqrt[11] - (2*I)*x)/(2*Sqrt[1
1])])/11 + (175*(55 + (7*I)*Sqrt[11])*PolyLog[2, (5*I + Sqrt[11] - (2*I)*x)/(2*Sqrt[11])])/11
Rule 12
Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] && !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]
Rule 204
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])
Rule 618
Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]
Rule 628
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Rule 634
Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] && !NiceSqrtQ[b^2 - 4*a*c]
Rule 701
Int[((d_.) + (e_.)*(x_))^(m_)/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Int[PolynomialDivide[(d + e*x)
^m, a + b*x + c*x^2, x], x] /; FreeQ[{a, b, c, d, e}, 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] && IGtQ[m, 1] && (NeQ[d, 0] || GtQ[m, 2])
Rule 703
Int[((d_.) + (e_.)*(x_))^(m_)/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(e*(d + e*x)^(m - 1))/(c*
(m - 1)), x] + Dist[1/c, Int[((d + e*x)^(m - 2)*Simp[c*d^2 - a*e^2 + e*(2*c*d - b*e)*x, x])/(a + b*x + c*x^2),
x], x] /; FreeQ[{a, b, c, d, e}, 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] && GtQ[m, 1]
Rule 773
Int[(((d_.) + (e_.)*(x_))*((f_) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(e*g*x)/
c, x] + Dist[1/c, Int[(c*d*f - a*e*g + (c*e*f + c*d*g - b*e*g)*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c,
d, e, f, g}, x] && NeQ[b^2 - 4*a*c, 0]
Rule 800
Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Int[Exp
andIntegrand[((d + e*x)^m*(f + g*x))/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 -
4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[m]
Rule 2176
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] && !$UseGamma === True
Rule 2178
Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(F^(g*(e - (c*f)/d))*ExpIntegral
Ei[(f*g*(c + d*x)*Log[F])/d])/d, x] /; FreeQ[{F, c, d, e, f, g}, x] && !$UseGamma === True
Rule 2194
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 2196
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] && !$UseGamma === True
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 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 2402
Int[Log[(c_.)/((d_) + (e_.)*(x_))]/((f_) + (g_.)*(x_)^2), x_Symbol] :> -Dist[e/g, Subst[Int[Log[2*d*x]/(1 - 2*
d*x), x], x, 1/(d + e*x)], x] /; FreeQ[{c, d, e, f, g}, x] && EqQ[c, 2*d] && EqQ[e^2*f + d^2*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 2527
Int[Log[(c_.)*(Px_)^(n_.)]/(Qx_), x_Symbol] :> With[{u = IntHide[1/Qx, x]}, Simp[u*Log[c*Px^n], x] - Dist[n, I
nt[SimplifyIntegrand[(u*D[Px, x])/Px, x], x], x]] /; FreeQ[{c, n}, x] && QuadraticQ[{Qx, Px}, x] && EqQ[D[Px/Q
x, x], 0]
Rule 2528
Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)*(RGx_), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*Log[c*
RFx^p])^n, RGx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, p}, x] && RationalFunctionQ[RFx, x] && RationalF
unctionQ[RGx, x] && IGtQ[n, 0]
Rule 2554
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 4854
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[((a + b*ArcTan[c*x])^p*Lo
g[2/(1 + (e*x)/d)])/e, x] + Dist[(b*c*p)/e, Int[((a + b*ArcTan[c*x])^(p - 1)*Log[2/(1 + (e*x)/d)])/(1 + c^2*x^
2), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d^2 + e^2, 0]
Rule 4920
Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> -Simp[(I*(a + b*ArcTan
[c*x])^(p + 1))/(b*e*(p + 1)), x] - Dist[1/(c*d), Int[(a + b*ArcTan[c*x])^p/(I - c*x), x], x] /; FreeQ[{a, b,
c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[p, 0]
Rule 5057
Int[((a_.) + ArcTan[(c_) + (d_.)*(x_)]*(b_.))^(p_.)*((e_.) + (f_.)*(x_))^(m_.)*((A_.) + (B_.)*(x_) + (C_.)*(x_
)^2)^(q_.), x_Symbol] :> Dist[1/d, Subst[Int[((d*e - c*f)/d + (f*x)/d)^m*(C/d^2 + (C*x^2)/d^2)^q*(a + b*ArcTan
[x])^p, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, m, p, q}, x] && EqQ[B*(1 + c^2) - 2*A*c*d, 0]
&& EqQ[2*c*C - B*d, 0]
Rule 6688
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]
Rule 6728
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]
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (50 e^{2 x} x (1+x)+\frac {360}{9-5 x+x^2}+\frac {818 x}{9-5 x+x^2}-\frac {320 x^2}{9-5 x+x^2}-\frac {18 x^3}{9-5 x+x^2}-\frac {1800 \log \left (\left (9-5 x+x^2\right ) \log (5)\right )}{9-5 x+x^2}+\frac {820 x \log \left (\left (9-5 x+x^2\right ) \log (5)\right )}{9-5 x+x^2}-\frac {350 x^2 \log \left (\left (9-5 x+x^2\right ) \log (5)\right )}{9-5 x+x^2}+\frac {80 x^3 \log \left (\left (9-5 x+x^2\right ) \log (5)\right )}{9-5 x+x^2}+50 x \log ^2\left (\left (9-5 x+x^2\right ) \log (5)\right )-\frac {10 e^x \left (-180-98 x+56 x^2-7 x^3-x^4+90 x \log \left (\left (9-5 x+x^2\right ) \log (5)\right )-5 x^2 \log \left (\left (9-5 x+x^2\right ) \log (5)\right )-15 x^3 \log \left (\left (9-5 x+x^2\right ) \log (5)\right )+5 x^4 \log \left (\left (9-5 x+x^2\right ) \log (5)\right )\right )}{9-5 x+x^2}\right ) \, dx\\ &=-\left (10 \int \frac {e^x \left (-180-98 x+56 x^2-7 x^3-x^4+90 x \log \left (\left (9-5 x+x^2\right ) \log (5)\right )-5 x^2 \log \left (\left (9-5 x+x^2\right ) \log (5)\right )-15 x^3 \log \left (\left (9-5 x+x^2\right ) \log (5)\right )+5 x^4 \log \left (\left (9-5 x+x^2\right ) \log (5)\right )\right )}{9-5 x+x^2} \, dx\right )-18 \int \frac {x^3}{9-5 x+x^2} \, dx+50 \int e^{2 x} x (1+x) \, dx+50 \int x \log ^2\left (\left (9-5 x+x^2\right ) \log (5)\right ) \, dx+80 \int \frac {x^3 \log \left (\left (9-5 x+x^2\right ) \log (5)\right )}{9-5 x+x^2} \, dx-320 \int \frac {x^2}{9-5 x+x^2} \, dx-350 \int \frac {x^2 \log \left (\left (9-5 x+x^2\right ) \log (5)\right )}{9-5 x+x^2} \, dx+360 \int \frac {1}{9-5 x+x^2} \, dx+818 \int \frac {x}{9-5 x+x^2} \, dx+820 \int \frac {x \log \left (\left (9-5 x+x^2\right ) \log (5)\right )}{9-5 x+x^2} \, dx-1800 \int \frac {\log \left (\left (9-5 x+x^2\right ) \log (5)\right )}{9-5 x+x^2} \, dx\\ &=-320 x+\frac {3600 \tan ^{-1}\left (\frac {5-2 x}{\sqrt {11}}\right ) \log \left (\left (9-5 x+x^2\right ) \log (5)\right )}{\sqrt {11}}+25 x^2 \log ^2\left (\left (9-5 x+x^2\right ) \log (5)\right )-10 \int \frac {e^x \left (-180-98 x+56 x^2-7 x^3-x^4+5 x \left (18-x-3 x^2+x^3\right ) \log \left (\left (9-5 x+x^2\right ) \log (5)\right )\right )}{9-5 x+x^2} \, dx-18 \int \left (5+x-\frac {45-16 x}{9-5 x+x^2}\right ) \, dx+50 \int \left (e^{2 x} x+e^{2 x} x^2\right ) \, dx-50 \int \frac {x^2 (-5+2 x) \log \left (\left (9-5 x+x^2\right ) \log (5)\right )}{9-5 x+x^2} \, dx+80 \int \left (5 \log \left (\left (9-5 x+x^2\right ) \log (5)\right )+x \log \left (\left (9-5 x+x^2\right ) \log (5)\right )-\frac {(45-16 x) \log \left (\left (9-5 x+x^2\right ) \log (5)\right )}{9-5 x+x^2}\right ) \, dx-320 \int \frac {-9+5 x}{9-5 x+x^2} \, dx-350 \int \left (\log \left (\left (9-5 x+x^2\right ) \log (5)\right )-\frac {(9-5 x) \log \left (\left (9-5 x+x^2\right ) \log (5)\right )}{9-5 x+x^2}\right ) \, dx+409 \int \frac {-5+2 x}{9-5 x+x^2} \, dx-720 \operatorname {Subst}\left (\int \frac {1}{-11-x^2} \, dx,x,-5+2 x\right )+820 \int \left (\frac {\left (1-\frac {5 i}{\sqrt {11}}\right ) \log \left (\left (9-5 x+x^2\right ) \log (5)\right )}{-5-i \sqrt {11}+2 x}+\frac {\left (1+\frac {5 i}{\sqrt {11}}\right ) \log \left (\left (9-5 x+x^2\right ) \log (5)\right )}{-5+i \sqrt {11}+2 x}\right ) \, dx+1800 \int \frac {2 (5-2 x) \tan ^{-1}\left (\frac {5}{\sqrt {11}}-\frac {2 x}{\sqrt {11}}\right )}{\sqrt {11} \left (9-5 x+x^2\right )} \, dx+2045 \int \frac {1}{9-5 x+x^2} \, dx\\ &=-410 x-9 x^2-\frac {720 \tan ^{-1}\left (\frac {5-2 x}{\sqrt {11}}\right )}{\sqrt {11}}+409 \log \left (9-5 x+x^2\right )+\frac {3600 \tan ^{-1}\left (\frac {5-2 x}{\sqrt {11}}\right ) \log \left (\left (9-5 x+x^2\right ) \log (5)\right )}{\sqrt {11}}+25 x^2 \log ^2\left (\left (9-5 x+x^2\right ) \log (5)\right )-10 \int \left (\frac {e^x \left (-180-98 x+56 x^2-7 x^3-x^4\right )}{9-5 x+x^2}+5 e^x x (2+x) \log \left (\left (9-5 x+x^2\right ) \log (5)\right )\right ) \, dx+18 \int \frac {45-16 x}{9-5 x+x^2} \, dx+50 \int e^{2 x} x \, dx+50 \int e^{2 x} x^2 \, dx-50 \int \left (5 \log \left (\left (9-5 x+x^2\right ) \log (5)\right )+2 x \log \left (\left (9-5 x+x^2\right ) \log (5)\right )-\frac {(45-7 x) \log \left (\left (9-5 x+x^2\right ) \log (5)\right )}{9-5 x+x^2}\right ) \, dx+80 \int x \log \left (\left (9-5 x+x^2\right ) \log (5)\right ) \, dx-80 \int \frac {(45-16 x) \log \left (\left (9-5 x+x^2\right ) \log (5)\right )}{9-5 x+x^2} \, dx-350 \int \log \left (\left (9-5 x+x^2\right ) \log (5)\right ) \, dx+350 \int \frac {(9-5 x) \log \left (\left (9-5 x+x^2\right ) \log (5)\right )}{9-5 x+x^2} \, dx+400 \int \log \left (\left (9-5 x+x^2\right ) \log (5)\right ) \, dx-800 \int \frac {-5+2 x}{9-5 x+x^2} \, dx-1120 \int \frac {1}{9-5 x+x^2} \, dx-4090 \operatorname {Subst}\left (\int \frac {1}{-11-x^2} \, dx,x,-5+2 x\right )+\frac {3600 \int \frac {(5-2 x) \tan ^{-1}\left (\frac {5}{\sqrt {11}}-\frac {2 x}{\sqrt {11}}\right )}{9-5 x+x^2} \, dx}{\sqrt {11}}+\frac {1}{11} \left (820 \left (11-5 i \sqrt {11}\right )\right ) \int \frac {\log \left (\left (9-5 x+x^2\right ) \log (5)\right )}{-5-i \sqrt {11}+2 x} \, dx+\frac {1}{11} \left (820 \left (11+5 i \sqrt {11}\right )\right ) \int \frac {\log \left (\left (9-5 x+x^2\right ) \log (5)\right )}{-5+i \sqrt {11}+2 x} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}
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Mathematica [A] time = 0.12, size = 47, normalized size = 1.62 \begin {gather*} x \left (1+5 e^x-5 \log \left (\left (9-5 x+x^2\right ) \log (5)\right )\right ) \left (40+x+5 e^x x-5 x \log \left (\left (9-5 x+x^2\right ) \log (5)\right )\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(360 + 818*x - 320*x^2 - 18*x^3 + E^x*(1800 + 980*x - 560*x^2 + 70*x^3 + 10*x^4) + E^(2*x)*(450*x +
200*x^2 - 200*x^3 + 50*x^4) + (-1800 + 820*x - 350*x^2 + 80*x^3 + E^x*(-900*x + 50*x^2 + 150*x^3 - 50*x^4))*Lo
g[(9 - 5*x + x^2)*Log[5]] + (450*x - 250*x^2 + 50*x^3)*Log[(9 - 5*x + x^2)*Log[5]]^2)/(9 - 5*x + x^2),x]
[Out]
x*(1 + 5*E^x - 5*Log[(9 - 5*x + x^2)*Log[5]])*(40 + x + 5*E^x*x - 5*x*Log[(9 - 5*x + x^2)*Log[5]])
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fricas [B] time = 0.75, size = 74, normalized size = 2.55 \begin {gather*} 25 \, x^{2} \log \left ({\left (x^{2} - 5 \, x + 9\right )} \log \relax (5)\right )^{2} + 25 \, x^{2} e^{\left (2 \, x\right )} + x^{2} + 10 \, {\left (x^{2} + 20 \, x\right )} e^{x} - 10 \, {\left (5 \, x^{2} e^{x} + x^{2} + 20 \, x\right )} \log \left ({\left (x^{2} - 5 \, x + 9\right )} \log \relax (5)\right ) + 40 \, x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(((50*x^3-250*x^2+450*x)*log((x^2-5*x+9)*log(5))^2+((-50*x^4+150*x^3+50*x^2-900*x)*exp(x)+80*x^3-350*
x^2+820*x-1800)*log((x^2-5*x+9)*log(5))+(50*x^4-200*x^3+200*x^2+450*x)*exp(x)^2+(10*x^4+70*x^3-560*x^2+980*x+1
800)*exp(x)-18*x^3-320*x^2+818*x+360)/(x^2-5*x+9),x, algorithm="fricas")
[Out]
25*x^2*log((x^2 - 5*x + 9)*log(5))^2 + 25*x^2*e^(2*x) + x^2 + 10*(x^2 + 20*x)*e^x - 10*(5*x^2*e^x + x^2 + 20*x
)*log((x^2 - 5*x + 9)*log(5)) + 40*x
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giac [B] time = 0.25, size = 137, normalized size = 4.72 \begin {gather*} -50 \, x^{2} e^{x} \log \left (x^{2} - 5 \, x + 9\right ) + 25 \, x^{2} \log \left (x^{2} - 5 \, x + 9\right )^{2} - 50 \, x^{2} e^{x} \log \left (\log \relax (5)\right ) + 50 \, x^{2} \log \left (x^{2} - 5 \, x + 9\right ) \log \left (\log \relax (5)\right ) + 25 \, x^{2} \log \left (\log \relax (5)\right )^{2} + 25 \, x^{2} e^{\left (2 \, x\right )} + 10 \, x^{2} e^{x} - 10 \, x^{2} \log \left (x^{2} - 5 \, x + 9\right ) - 10 \, x^{2} \log \left (\log \relax (5)\right ) + x^{2} + 200 \, x e^{x} - 200 \, x \log \left (x^{2} - 5 \, x + 9\right ) - 200 \, x \log \left (\log \relax (5)\right ) + 40 \, x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(((50*x^3-250*x^2+450*x)*log((x^2-5*x+9)*log(5))^2+((-50*x^4+150*x^3+50*x^2-900*x)*exp(x)+80*x^3-350*
x^2+820*x-1800)*log((x^2-5*x+9)*log(5))+(50*x^4-200*x^3+200*x^2+450*x)*exp(x)^2+(10*x^4+70*x^3-560*x^2+980*x+1
800)*exp(x)-18*x^3-320*x^2+818*x+360)/(x^2-5*x+9),x, algorithm="giac")
[Out]
-50*x^2*e^x*log(x^2 - 5*x + 9) + 25*x^2*log(x^2 - 5*x + 9)^2 - 50*x^2*e^x*log(log(5)) + 50*x^2*log(x^2 - 5*x +
9)*log(log(5)) + 25*x^2*log(log(5))^2 + 25*x^2*e^(2*x) + 10*x^2*e^x - 10*x^2*log(x^2 - 5*x + 9) - 10*x^2*log(
log(5)) + x^2 + 200*x*e^x - 200*x*log(x^2 - 5*x + 9) - 200*x*log(log(5)) + 40*x
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maple [B] time = 0.62, size = 77, normalized size = 2.66
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\(25 \ln \left (\left (x^{2}-5 x +9\right ) \ln \relax (5)\right )^{2} x^{2}+\left (-50 \,{\mathrm e}^{x} x^{2}-10 x^{2}-200 x \right ) \ln \left (\left (x^{2}-5 x +9\right ) \ln \relax (5)\right )+25 \,{\mathrm e}^{2 x} x^{2}+10 \,{\mathrm e}^{x} x^{2}+x^{2}+200 \,{\mathrm e}^{x} x +40 x\) |
\(77\) |
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Verification of antiderivative is not currently implemented for this CAS.
[In]
int(((50*x^3-250*x^2+450*x)*ln((x^2-5*x+9)*ln(5))^2+((-50*x^4+150*x^3+50*x^2-900*x)*exp(x)+80*x^3-350*x^2+820*
x-1800)*ln((x^2-5*x+9)*ln(5))+(50*x^4-200*x^3+200*x^2+450*x)*exp(x)^2+(10*x^4+70*x^3-560*x^2+980*x+1800)*exp(x
)-18*x^3-320*x^2+818*x+360)/(x^2-5*x+9),x,method=_RETURNVERBOSE)
[Out]
25*ln((x^2-5*x+9)*ln(5))^2*x^2+(-50*exp(x)*x^2-10*x^2-200*x)*ln((x^2-5*x+9)*ln(5))+25*exp(2*x)*x^2+10*exp(x)*x
^2+x^2+200*exp(x)*x+40*x
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maxima [B] time = 0.79, size = 128, normalized size = 4.41 \begin {gather*} 25 \, x^{2} \log \left (x^{2} - 5 \, x + 9\right )^{2} + 5 \, {\left (5 \, \log \left (\log \relax (5)\right )^{2} - 2 \, \log \left (\log \relax (5)\right ) + 2\right )} x^{2} + 25 \, x^{2} e^{\left (2 \, x\right )} - 9 \, x^{2} - 50 \, x {\left (4 \, \log \left (\log \relax (5)\right ) - 9\right )} - 10 \, {\left (x^{2} {\left (5 \, \log \left (\log \relax (5)\right ) - 1\right )} - 20 \, x\right )} e^{x} + 5 \, {\left (2 \, x^{2} {\left (5 \, \log \left (\log \relax (5)\right ) - 1\right )} - 10 \, x^{2} e^{x} - 40 \, x + 107\right )} \log \left (x^{2} - 5 \, x + 9\right ) - 410 \, x - 535 \, \log \left (x^{2} - 5 \, x + 9\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(((50*x^3-250*x^2+450*x)*log((x^2-5*x+9)*log(5))^2+((-50*x^4+150*x^3+50*x^2-900*x)*exp(x)+80*x^3-350*
x^2+820*x-1800)*log((x^2-5*x+9)*log(5))+(50*x^4-200*x^3+200*x^2+450*x)*exp(x)^2+(10*x^4+70*x^3-560*x^2+980*x+1
800)*exp(x)-18*x^3-320*x^2+818*x+360)/(x^2-5*x+9),x, algorithm="maxima")
[Out]
25*x^2*log(x^2 - 5*x + 9)^2 + 5*(5*log(log(5))^2 - 2*log(log(5)) + 2)*x^2 + 25*x^2*e^(2*x) - 9*x^2 - 50*x*(4*l
og(log(5)) - 9) - 10*(x^2*(5*log(log(5)) - 1) - 20*x)*e^x + 5*(2*x^2*(5*log(log(5)) - 1) - 10*x^2*e^x - 40*x +
107)*log(x^2 - 5*x + 9) - 410*x - 535*log(x^2 - 5*x + 9)
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mupad [B] time = 1.28, size = 77, normalized size = 2.66 \begin {gather*} 40\,x-\ln \left (\ln \relax (5)\,\left (x^2-5\,x+9\right )\right )\,\left (200\,x+50\,x^2\,{\mathrm {e}}^x+10\,x^2\right )+25\,x^2\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^x\,\left (10\,x^2+200\,x\right )+x^2+25\,x^2\,{\ln \left (\ln \relax (5)\,\left (x^2-5\,x+9\right )\right )}^2 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((818*x + exp(x)*(980*x - 560*x^2 + 70*x^3 + 10*x^4 + 1800) - log(log(5)*(x^2 - 5*x + 9))*(exp(x)*(900*x -
50*x^2 - 150*x^3 + 50*x^4) - 820*x + 350*x^2 - 80*x^3 + 1800) + exp(2*x)*(450*x + 200*x^2 - 200*x^3 + 50*x^4)
+ log(log(5)*(x^2 - 5*x + 9))^2*(450*x - 250*x^2 + 50*x^3) - 320*x^2 - 18*x^3 + 360)/(x^2 - 5*x + 9),x)
[Out]
40*x - log(log(5)*(x^2 - 5*x + 9))*(200*x + 50*x^2*exp(x) + 10*x^2) + 25*x^2*exp(2*x) + exp(x)*(200*x + 10*x^2
) + x^2 + 25*x^2*log(log(5)*(x^2 - 5*x + 9))^2
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sympy [B] time = 0.74, size = 90, normalized size = 3.10 \begin {gather*} 25 x^{2} e^{2 x} + 25 x^{2} \log {\left (\left (x^{2} - 5 x + 9\right ) \log {\relax (5 )} \right )}^{2} + x^{2} + 40 x + \left (- 10 x^{2} - 200 x\right ) \log {\left (\left (x^{2} - 5 x + 9\right ) \log {\relax (5 )} \right )} + \left (- 50 x^{2} \log {\left (\left (x^{2} - 5 x + 9\right ) \log {\relax (5 )} \right )} + 10 x^{2} + 200 x\right ) e^{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(((50*x**3-250*x**2+450*x)*ln((x**2-5*x+9)*ln(5))**2+((-50*x**4+150*x**3+50*x**2-900*x)*exp(x)+80*x**
3-350*x**2+820*x-1800)*ln((x**2-5*x+9)*ln(5))+(50*x**4-200*x**3+200*x**2+450*x)*exp(x)**2+(10*x**4+70*x**3-560
*x**2+980*x+1800)*exp(x)-18*x**3-320*x**2+818*x+360)/(x**2-5*x+9),x)
[Out]
25*x**2*exp(2*x) + 25*x**2*log((x**2 - 5*x + 9)*log(5))**2 + x**2 + 40*x + (-10*x**2 - 200*x)*log((x**2 - 5*x
+ 9)*log(5)) + (-50*x**2*log((x**2 - 5*x + 9)*log(5)) + 10*x**2 + 200*x)*exp(x)
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