3.30.97 \(\int \frac {1740 x^5-2388 x^6+1143 x^7-222 x^8+15 x^9+(-64 x+32 x^2) \log (\frac {-29+5 x}{-5+x})+(4640-4628 x+1240 x^2-100 x^3) \log ^2(\frac {-29+5 x}{-5+x})}{1740 x^5-2388 x^6+1143 x^7-222 x^8+15 x^9} \, dx\)

Optimal. Leaf size=28 \[ x-\frac {4 \log ^2\left (5-\frac {4}{-5+x}\right )}{3 (2-x) x^4} \]

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Rubi [C]  time = 3.03, antiderivative size = 998, normalized size of antiderivative = 35.64, number of steps used = 134, number of rules used = 27, integrand size = 107, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.252, Rules used = {6688, 2513, 2418, 2390, 2301, 2394, 2393, 2391, 2395, 44, 36, 31, 29, 2315, 893, 2514, 2490, 2503, 2502, 2492, 2488, 2411, 2343, 2333, 72, 2494, 2316}

result too large to display

Antiderivative was successfully verified.

[In]

Int[(1740*x^5 - 2388*x^6 + 1143*x^7 - 222*x^8 + 15*x^9 + (-64*x + 32*x^2)*Log[(-29 + 5*x)/(-5 + x)] + (4640 -
4628*x + 1240*x^2 - 100*x^3)*Log[(-29 + 5*x)/(-5 + x)]^2)/(1740*x^5 - 2388*x^6 + 1143*x^7 - 222*x^8 + 15*x^9),
x]

[Out]

x - (181796*Log[29 - 5*x])/45729375 - (16*Log[(29 - 5*x)/(5 - x)])/(1305*x^3) - (1012*Log[(29 - 5*x)/(5 - x)])
/(63075*x^2) - (181796*(5 - x)*Log[(29 - 5*x)/(5 - x)])/(45729375*x) - (30775*Log[4/(29 - 5*x)]*Log[(29 - 5*x)
/(5 - x)])/2121843 - (2*Log[(-4*(2 - x))/(3*(29 - 5*x))]*Log[(29 - 5*x)/(5 - x)])/171 + (13*Log[-4/(5*(5 - x))
]*Log[(29 - 5*x)/(5 - x)])/625 - ((5 - x)*Log[(29 - 5*x)/(5 - x)]^2)/(36*(2 - x)) - (2*Log[(29 - 5*x)/(5 - x)]
^2)/(3*x^4) - Log[(29 - 5*x)/(5 - x)]^2/(3*x^3) - Log[(29 - 5*x)/(5 - x)]^2/(6*x^2) - ((5 - x)*Log[(29 - 5*x)/
(5 - x)]^2)/(60*x) + (181796*Log[5 - x])/45729375 - (16*Log[-5 + x])/(1305*x^3) - (1012*Log[-5 + x])/(63075*x^
2) - (181796*Log[-5 + x])/(9145875*x) + (8349584*Log[5]*Log[-5 + x])/1326151875 - (25000*Log[(29 - 5*x)/4]*Log
[-5 + x])/40315017 + (4*Log[-5 + x]^2)/5625 - (2*Log[-5 + x]*Log[(-2 + x)/3])/171 + (14446834*Log[-5 + x]*Log[
x/5])/1326151875 + (8349584*Log[(29 - 5*x)/(5 - x)]*Log[x])/1326151875 + (2*Log[(29 - 5*x)/(5 - x)]*Log[(4*x)/
(5*(29 - 5*x))])/435 + (16*(Log[(29 - 5*x)/(5 - x)] + Log[-5 + x] - Log[-29 + 5*x]))/(1305*x^3) + (1012*(Log[(
29 - 5*x)/(5 - x)] + Log[-5 + x] - Log[-29 + 5*x]))/(63075*x^2) + (181796*(Log[(29 - 5*x)/(5 - x)] + Log[-5 +
x] - Log[-29 + 5*x]))/(9145875*x) + (25000*Log[29 - 5*x]*(Log[(29 - 5*x)/(5 - x)] + Log[-5 + x] - Log[-29 + 5*
x]))/40315017 + (2*Log[2 - x]*(Log[(29 - 5*x)/(5 - x)] + Log[-5 + x] - Log[-29 + 5*x]))/171 - (8*Log[5 - x]*(L
og[(29 - 5*x)/(5 - x)] + Log[-5 + x] - Log[-29 + 5*x]))/5625 - (14446834*Log[x]*(Log[(29 - 5*x)/(5 - x)] + Log
[-5 + x] - Log[-29 + 5*x]))/1326151875 + (16*Log[-29 + 5*x])/(1305*x^3) + (1012*Log[-29 + 5*x])/(63075*x^2) +
(181796*Log[-29 + 5*x])/(9145875*x) - (8349584*Log[29/5]*Log[-29 + 5*x])/1326151875 + (2*Log[(-5*(2 - x))/19]*
Log[-29 + 5*x])/171 - (8*Log[(-5*(5 - x))/4]*Log[-29 + 5*x])/5625 - (14446834*Log[(5*x)/29]*Log[-29 + 5*x])/13
26151875 + (12500*Log[-29 + 5*x]^2)/40315017 + (13*PolyLog[2, 1 + 4/(5*(5 - x))])/625 + (2*PolyLog[2, (29 - 5*
x)/19])/171 - (8*PolyLog[2, (29 - 5*x)/4])/5625 - (25000*PolyLog[2, (-5*(5 - x))/4])/40315017 - (2*PolyLog[2,
(5 - x)/3])/171 + (30775*PolyLog[2, (5*(5 - x))/(29 - 5*x)])/2121843 + (2*PolyLog[2, (19*(5 - x))/(3*(29 - 5*x
))])/171 + (2*PolyLog[2, 1 - x/5])/435 - (2*PolyLog[2, 1 - (5*x)/29])/435 - (2*PolyLog[2, 1 - (4*x)/(5*(29 - 5
*x))])/435

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 44

Int[((a_) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*
x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rule 72

Int[((e_.) + (f_.)*(x_))^(p_.)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Int[ExpandIntegrand[(
e + f*x)^p/((a + b*x)*(c + d*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IntegerQ[p]

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 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 2316

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

Rule 2333

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

Rule 2343

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

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 2395

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

Rule 2411

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

Rule 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 2488

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

Rule 2490

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

Rule 2492

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

Rule 2494

Int[Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.))^(r_.)]/((g_.) + (h_.)*(x_)), x_Sym
bol] :> Simp[(Log[g + h*x]*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/h, x] + (-Dist[(b*p*r)/h, Int[Log[g + h*x]/(a
+ b*x), x], x] - Dist[(d*q*r)/h, Int[Log[g + h*x]/(c + d*x), x], x]) /; FreeQ[{a, b, c, d, e, f, g, h, p, q,
r}, x] && NeQ[b*c - a*d, 0]

Rule 2502

Int[Log[((e_.)*((c_.) + (d_.)*(x_)))/((a_.) + (b_.)*(x_))]*(u_), x_Symbol] :> With[{g = Coeff[Simplify[1/(u*(a
+ b*x))], x, 0], h = Coeff[Simplify[1/(u*(a + b*x))], x, 1]}, -Dist[(b - d*e)/(h*(b*c - a*d)), Subst[Int[Log[
e*x]/(1 - e*x), x], x, (c + d*x)/(a + b*x)], x] /; EqQ[g*(b - d*e) - h*(a - c*e), 0]] /; FreeQ[{a, b, c, d, e}
, x] && NeQ[b*c - a*d, 0] && LinearQ[Simplify[1/(u*(a + b*x))], x]

Rule 2503

Int[Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.))^(r_.)]^(s_.)*(u_), x_Symbol] :> Wi
th[{g = Coeff[Simplify[1/(u*(a + b*x))], x, 0], h = Coeff[Simplify[1/(u*(a + b*x))], x, 1]}, -Simp[(Log[e*(f*(
a + b*x)^p*(c + d*x)^q)^r]^s*Log[-(((b*c - a*d)*(g + h*x))/((d*g - c*h)*(a + b*x)))])/(b*g - a*h), x] + Dist[(
p*r*s*(b*c - a*d))/(b*g - a*h), Int[(Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^(s - 1)*Log[-(((b*c - a*d)*(g + h*x)
)/((d*g - c*h)*(a + b*x)))])/((a + b*x)*(c + d*x)), x], x] /; NeQ[b*g - a*h, 0] && NeQ[d*g - c*h, 0]] /; FreeQ
[{a, b, c, d, e, f, p, q, r, s}, x] && NeQ[b*c - a*d, 0] && IGtQ[s, 0] && EqQ[p + q, 0] && LinearQ[Simplify[1/
(u*(a + b*x))], x]

Rule 2513

Int[Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.))^(r_.)]*(RFx_.), x_Symbol] :> Dist[
p*r, Int[RFx*Log[a + b*x], x], x] + (Dist[q*r, Int[RFx*Log[c + d*x], x], x] - Dist[p*r*Log[a + b*x] + q*r*Log[
c + d*x] - Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r], Int[RFx, x], x]) /; FreeQ[{a, b, c, d, e, f, p, q, r}, x] &&
RationalFunctionQ[RFx, x] && NeQ[b*c - a*d, 0] &&  !MatchQ[RFx, (u_.)*(a + b*x)^(m_.)*(c + d*x)^(n_.) /; Integ
ersQ[m, n]]

Rule 2514

Int[Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.))^(r_.)]^(s_.)*(RFx_), x_Symbol] :>
With[{u = ExpandIntegrand[Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^s, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a,
b, c, d, e, f, p, q, r, s}, x] && RationalFunctionQ[RFx, x] && IGtQ[s, 0]

Rule 6688

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (1+\frac {32 \log \left (\frac {-29+5 x}{-5+x}\right )}{3 (-2+x) x^4 \left (145-54 x+5 x^2\right )}-\frac {4 (-8+5 x) \log ^2\left (\frac {-29+5 x}{-5+x}\right )}{3 (-2+x)^2 x^5}\right ) \, dx\\ &=x-\frac {4}{3} \int \frac {(-8+5 x) \log ^2\left (\frac {-29+5 x}{-5+x}\right )}{(-2+x)^2 x^5} \, dx+\frac {32}{3} \int \frac {\log \left (\frac {-29+5 x}{-5+x}\right )}{(-2+x) x^4 \left (145-54 x+5 x^2\right )} \, dx\\ &=x-\frac {4}{3} \int \left (\frac {\log ^2\left (\frac {-29+5 x}{-5+x}\right )}{16 (-2+x)^2}-\frac {2 \log ^2\left (\frac {-29+5 x}{-5+x}\right )}{x^5}-\frac {3 \log ^2\left (\frac {-29+5 x}{-5+x}\right )}{4 x^4}-\frac {\log ^2\left (\frac {-29+5 x}{-5+x}\right )}{4 x^3}-\frac {\log ^2\left (\frac {-29+5 x}{-5+x}\right )}{16 x^2}\right ) \, dx-\frac {32}{3} \int \frac {\log (-5+x)}{(-2+x) x^4 \left (145-54 x+5 x^2\right )} \, dx+\frac {32}{3} \int \frac {\log (-29+5 x)}{(-2+x) x^4 \left (145-54 x+5 x^2\right )} \, dx-\frac {1}{3} \left (32 \left (-\log (-5+x)+\log (-29+5 x)-\log \left (\frac {-29+5 x}{-5+x}\right )\right )\right ) \int \frac {1}{(-2+x) x^4 \left (145-54 x+5 x^2\right )} \, dx\\ &=x-\frac {1}{12} \int \frac {\log ^2\left (\frac {-29+5 x}{-5+x}\right )}{(-2+x)^2} \, dx+\frac {1}{12} \int \frac {\log ^2\left (\frac {-29+5 x}{-5+x}\right )}{x^2} \, dx+\frac {1}{3} \int \frac {\log ^2\left (\frac {-29+5 x}{-5+x}\right )}{x^3} \, dx+\frac {8}{3} \int \frac {\log ^2\left (\frac {-29+5 x}{-5+x}\right )}{x^5} \, dx-\frac {32}{3} \int \left (-\frac {\log (-5+x)}{7500 (-5+x)}+\frac {\log (-5+x)}{912 (-2+x)}-\frac {\log (-5+x)}{290 x^4}-\frac {253 \log (-5+x)}{84100 x^3}-\frac {45449 \log (-5+x)}{24389000 x^2}-\frac {7223417 \log (-5+x)}{7072810000 x}+\frac {15625 \log (-5+x)}{53753356 (-29+5 x)}\right ) \, dx+\frac {32}{3} \int \left (-\frac {\log (-29+5 x)}{7500 (-5+x)}+\frac {\log (-29+5 x)}{912 (-2+x)}-\frac {\log (-29+5 x)}{290 x^4}-\frac {253 \log (-29+5 x)}{84100 x^3}-\frac {45449 \log (-29+5 x)}{24389000 x^2}-\frac {7223417 \log (-29+5 x)}{7072810000 x}+\frac {15625 \log (-29+5 x)}{53753356 (-29+5 x)}\right ) \, dx-\frac {1}{3} \left (32 \left (-\log (-5+x)+\log (-29+5 x)-\log \left (\frac {-29+5 x}{-5+x}\right )\right )\right ) \int \left (-\frac {1}{7500 (-5+x)}+\frac {1}{912 (-2+x)}-\frac {1}{290 x^4}-\frac {253}{84100 x^3}-\frac {45449}{24389000 x^2}-\frac {7223417}{7072810000 x}+\frac {15625}{53753356 (-29+5 x)}\right ) \, dx+\int \frac {\log ^2\left (\frac {-29+5 x}{-5+x}\right )}{x^4} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [C]  time = 29.91, size = 8290, normalized size = 296.07 \begin {gather*} \text {Result too large to show} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[(1740*x^5 - 2388*x^6 + 1143*x^7 - 222*x^8 + 15*x^9 + (-64*x + 32*x^2)*Log[(-29 + 5*x)/(-5 + x)] + (4
640 - 4628*x + 1240*x^2 - 100*x^3)*Log[(-29 + 5*x)/(-5 + x)]^2)/(1740*x^5 - 2388*x^6 + 1143*x^7 - 222*x^8 + 15
*x^9),x]

[Out]

Result too large to show

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fricas [A]  time = 0.65, size = 40, normalized size = 1.43 \begin {gather*} \frac {3 \, x^{6} - 6 \, x^{5} + 4 \, \log \left (\frac {5 \, x - 29}{x - 5}\right )^{2}}{3 \, {\left (x^{5} - 2 \, x^{4}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-100*x^3+1240*x^2-4628*x+4640)*log((5*x-29)/(x-5))^2+(32*x^2-64*x)*log((5*x-29)/(x-5))+15*x^9-222*
x^8+1143*x^7-2388*x^6+1740*x^5)/(15*x^9-222*x^8+1143*x^7-2388*x^6+1740*x^5),x, algorithm="fricas")

[Out]

1/3*(3*x^6 - 6*x^5 + 4*log((5*x - 29)/(x - 5))^2)/(x^5 - 2*x^4)

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giac [B]  time = 0.40, size = 156, normalized size = 5.57 \begin {gather*} -\frac {1}{5625} \, {\left (\frac {3 \, {\left (\frac {39625 \, {\left (5 \, x - 29\right )}^{3}}{{\left (x - 5\right )}^{3}} - \frac {673675 \, {\left (5 \, x - 29\right )}^{2}}{{\left (x - 5\right )}^{2}} + \frac {3819835 \, {\left (5 \, x - 29\right )}}{x - 5} - 7223417\right )}}{\frac {625 \, {\left (5 \, x - 29\right )}^{4}}{{\left (x - 5\right )}^{4}} - \frac {14500 \, {\left (5 \, x - 29\right )}^{3}}{{\left (x - 5\right )}^{3}} + \frac {126150 \, {\left (5 \, x - 29\right )}^{2}}{{\left (x - 5\right )}^{2}} - \frac {487780 \, {\left (5 \, x - 29\right )}}{x - 5} + 707281} - \frac {625}{\frac {3 \, {\left (5 \, x - 29\right )}}{x - 5} - 19} - 4\right )} \log \left (\frac {5 \, x - 29}{x - 5}\right )^{2} - \frac {4}{\frac {5 \, x - 29}{x - 5} - 5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-100*x^3+1240*x^2-4628*x+4640)*log((5*x-29)/(x-5))^2+(32*x^2-64*x)*log((5*x-29)/(x-5))+15*x^9-222*
x^8+1143*x^7-2388*x^6+1740*x^5)/(15*x^9-222*x^8+1143*x^7-2388*x^6+1740*x^5),x, algorithm="giac")

[Out]

-1/5625*(3*(39625*(5*x - 29)^3/(x - 5)^3 - 673675*(5*x - 29)^2/(x - 5)^2 + 3819835*(5*x - 29)/(x - 5) - 722341
7)/(625*(5*x - 29)^4/(x - 5)^4 - 14500*(5*x - 29)^3/(x - 5)^3 + 126150*(5*x - 29)^2/(x - 5)^2 - 487780*(5*x -
29)/(x - 5) + 707281) - 625/(3*(5*x - 29)/(x - 5) - 19) - 4)*log((5*x - 29)/(x - 5))^2 - 4/((5*x - 29)/(x - 5)
- 5)

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maple [A]  time = 0.57, size = 27, normalized size = 0.96

method	result	size
risch	\(\frac {4 \ln \left (\frac {5 x -29}{x -5}\right )^{2}}{3 x^{4} \left (x -2\right )}+x\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-100*x^3+1240*x^2-4628*x+4640)*ln((5*x-29)/(x-5))^2+(32*x^2-64*x)*ln((5*x-29)/(x-5))+15*x^9-222*x^8+1143
*x^7-2388*x^6+1740*x^5)/(15*x^9-222*x^8+1143*x^7-2388*x^6+1740*x^5),x,method=_RETURNVERBOSE)

[Out]

4/3/x^4/(x-2)*ln((5*x-29)/(x-5))^2+x

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maxima [A]  time = 1.04, size = 42, normalized size = 1.50 \begin {gather*} x + \frac {4 \, {\left (\log \left (5 \, x - 29\right )^{2} - 2 \, \log \left (5 \, x - 29\right ) \log \left (x - 5\right ) + \log \left (x - 5\right )^{2}\right )}}{3 \, {\left (x^{5} - 2 \, x^{4}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-100*x^3+1240*x^2-4628*x+4640)*log((5*x-29)/(x-5))^2+(32*x^2-64*x)*log((5*x-29)/(x-5))+15*x^9-222*
x^8+1143*x^7-2388*x^6+1740*x^5)/(15*x^9-222*x^8+1143*x^7-2388*x^6+1740*x^5),x, algorithm="maxima")

[Out]

x + 4/3*(log(5*x - 29)^2 - 2*log(5*x - 29)*log(x - 5) + log(x - 5)^2)/(x^5 - 2*x^4)

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mupad [B]  time = 2.14, size = 26, normalized size = 0.93 \begin {gather*} x+\frac {4\,{\ln \left (\frac {5\,x-29}{x-5}\right )}^2}{3\,x^4\,\left (x-2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log((5*x - 29)/(x - 5))*(64*x - 32*x^2) + log((5*x - 29)/(x - 5))^2*(4628*x - 1240*x^2 + 100*x^3 - 4640)
- 1740*x^5 + 2388*x^6 - 1143*x^7 + 222*x^8 - 15*x^9)/(1740*x^5 - 2388*x^6 + 1143*x^7 - 222*x^8 + 15*x^9),x)

[Out]

x + (4*log((5*x - 29)/(x - 5))^2)/(3*x^4*(x - 2))

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sympy [A]  time = 0.22, size = 24, normalized size = 0.86 \begin {gather*} x + \frac {4 \log {\left (\frac {5 x - 29}{x - 5} \right )}^{2}}{3 x^{5} - 6 x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-100*x**3+1240*x**2-4628*x+4640)*ln((5*x-29)/(x-5))**2+(32*x**2-64*x)*ln((5*x-29)/(x-5))+15*x**9-2
22*x**8+1143*x**7-2388*x**6+1740*x**5)/(15*x**9-222*x**8+1143*x**7-2388*x**6+1740*x**5),x)

[Out]

x + 4*log((5*x - 29)/(x - 5))**2/(3*x**5 - 6*x**4)

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