3.63.81 \(\int \frac {137 x^2+54 x^3+5 x^4-8 x \log (\frac {29+5 x}{5+x})+(-145-54 x-5 x^2) \log ^2(\frac {29+5 x}{5+x})}{290 x^2+108 x^3+10 x^4} \, dx\)

Optimal. Leaf size=23 \[ \frac {\left (x+\log \left (4+\frac {9+x}{5+x}\right )\right )^2}{2 x} \]

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Rubi [C]  time = 0.85, antiderivative size = 246, normalized size of antiderivative = 10.70, number of steps used = 29, number of rules used = 17, integrand size = 75, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {1594, 6728, 1657, 616, 31, 2514, 2494, 2317, 2391, 2488, 2411, 2343, 2333, 2315, 2490, 2503, 2502} \begin {gather*} -\frac {4 \text {Li}_2\left (-\frac {x}{5}\right )}{145}+\frac {4}{145} \text {Li}_2\left (-\frac {5 x}{29}\right )-\frac {5}{29} \text {Li}_2\left (\frac {5 (x+5)}{5 x+29}\right )-\frac {1}{5} \text {Li}_2\left (1+\frac {4}{5 (x+5)}\right )-\frac {4}{145} \text {Li}_2\left (\frac {4 x}{5 (5 x+29)}+1\right )+\frac {x}{2}+\frac {(x+5) \log ^2\left (\frac {5 x+29}{x+5}\right )}{10 x}-\frac {4}{145} \log (x) \log \left (\frac {5 x+29}{x+5}\right )-\frac {1}{5} \log \left (-\frac {4}{5 (x+5)}\right ) \log \left (\frac {5 x+29}{x+5}\right )+\frac {5}{29} \log \left (\frac {4}{5 x+29}\right ) \log \left (\frac {5 x+29}{x+5}\right )+\frac {4}{145} \log \left (-\frac {4 x}{5 (5 x+29)}\right ) \log \left (\frac {5 x+29}{x+5}\right )+\frac {4}{145} \log \left (\frac {5 x}{29}+1\right ) \log (x)-\frac {4}{145} \log \left (\frac {x}{5}+1\right ) \log (x)-\log (x+5)+\log (5 x+29) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(137*x^2 + 54*x^3 + 5*x^4 - 8*x*Log[(29 + 5*x)/(5 + x)] + (-145 - 54*x - 5*x^2)*Log[(29 + 5*x)/(5 + x)]^2)
/(290*x^2 + 108*x^3 + 10*x^4),x]

[Out]

x/2 + (4*Log[1 + (5*x)/29]*Log[x])/145 - (4*Log[1 + x/5]*Log[x])/145 - Log[5 + x] + Log[29 + 5*x] - (4*Log[x]*
Log[(29 + 5*x)/(5 + x)])/145 - (Log[-4/(5*(5 + x))]*Log[(29 + 5*x)/(5 + x)])/5 + (5*Log[4/(29 + 5*x)]*Log[(29
+ 5*x)/(5 + x)])/29 + (4*Log[(-4*x)/(5*(29 + 5*x))]*Log[(29 + 5*x)/(5 + x)])/145 + ((5 + x)*Log[(29 + 5*x)/(5
+ x)]^2)/(10*x) - (4*PolyLog[2, -1/5*x])/145 + (4*PolyLog[2, (-5*x)/29])/145 - (5*PolyLog[2, (5*(5 + x))/(29 +
5*x)])/29 - PolyLog[2, 1 + 4/(5*(5 + x))]/5 - (4*PolyLog[2, 1 + (4*x)/(5*(29 + 5*x))])/145

Rule 31

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

Rule 616

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[c/q, Int[1/Simp
[b/2 - q/2 + c*x, x], x], x] - Dist[c/q, Int[1/Simp[b/2 + q/2 + c*x, x], x], x]] /; FreeQ[{a, b, c}, x] && NeQ
[b^2 - 4*a*c, 0] && PosQ[b^2 - 4*a*c] && PerfectSquareQ[b^2 - 4*a*c]

Rule 1594

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.) + (c_.)*(x_)^(r_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^
(q - p) + c*x^(r - p))^n, x] /; FreeQ[{a, b, c, p, q, r}, x] && IntegerQ[n] && PosQ[q - p] && PosQ[r - p]

Rule 1657

Int[(Pq_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq*(a + b*x + c*x^2)^p, x
], x] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

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 2317

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 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 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 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 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 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 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 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 \frac {137 x^2+54 x^3+5 x^4-8 x \log \left (\frac {29+5 x}{5+x}\right )+\left (-145-54 x-5 x^2\right ) \log ^2\left (\frac {29+5 x}{5+x}\right )}{x^2 \left (290+108 x+10 x^2\right )} \, dx\\ &=\int \left (\frac {137+54 x+5 x^2}{2 \left (145+54 x+5 x^2\right )}-\frac {4 \log \left (\frac {29+5 x}{5+x}\right )}{x (5+x) (29+5 x)}-\frac {\log ^2\left (\frac {29+5 x}{5+x}\right )}{2 x^2}\right ) \, dx\\ &=\frac {1}{2} \int \frac {137+54 x+5 x^2}{145+54 x+5 x^2} \, dx-\frac {1}{2} \int \frac {\log ^2\left (\frac {29+5 x}{5+x}\right )}{x^2} \, dx-4 \int \frac {\log \left (\frac {29+5 x}{5+x}\right )}{x (5+x) (29+5 x)} \, dx\\ &=\frac {(5+x) \log ^2\left (\frac {29+5 x}{5+x}\right )}{10 x}+\frac {1}{2} \int \left (1-\frac {8}{145+54 x+5 x^2}\right ) \, dx+\frac {4}{5} \int \frac {\log \left (\frac {29+5 x}{5+x}\right )}{x (29+5 x)} \, dx-4 \int \left (\frac {\log \left (\frac {29+5 x}{5+x}\right )}{145 x}-\frac {\log \left (\frac {29+5 x}{5+x}\right )}{20 (5+x)}+\frac {25 \log \left (\frac {29+5 x}{5+x}\right )}{116 (29+5 x)}\right ) \, dx\\ &=\frac {x}{2}+\frac {4}{145} \log \left (-\frac {4 x}{5 (29+5 x)}\right ) \log \left (\frac {29+5 x}{5+x}\right )+\frac {(5+x) \log ^2\left (\frac {29+5 x}{5+x}\right )}{10 x}-\frac {4}{145} \int \frac {\log \left (\frac {29+5 x}{5+x}\right )}{x} \, dx+\frac {16}{145} \int \frac {\log \left (-\frac {4 x}{5 (29+5 x)}\right )}{(5+x) (29+5 x)} \, dx+\frac {1}{5} \int \frac {\log \left (\frac {29+5 x}{5+x}\right )}{5+x} \, dx-\frac {25}{29} \int \frac {\log \left (\frac {29+5 x}{5+x}\right )}{29+5 x} \, dx-4 \int \frac {1}{145+54 x+5 x^2} \, dx\\ &=\frac {x}{2}-\frac {4}{145} \log (x) \log \left (\frac {29+5 x}{5+x}\right )-\frac {1}{5} \log \left (-\frac {4}{5 (5+x)}\right ) \log \left (\frac {29+5 x}{5+x}\right )+\frac {5}{29} \log \left (\frac {4}{29+5 x}\right ) \log \left (\frac {29+5 x}{5+x}\right )+\frac {4}{145} \log \left (-\frac {4 x}{5 (29+5 x)}\right ) \log \left (\frac {29+5 x}{5+x}\right )+\frac {(5+x) \log ^2\left (\frac {29+5 x}{5+x}\right )}{10 x}+\frac {16}{725} \operatorname {Subst}\left (\int \frac {\log \left (-\frac {4 x}{5}\right )}{1+\frac {4 x}{5}} \, dx,x,\frac {x}{29+5 x}\right )-\frac {4}{145} \int \frac {\log (x)}{5+x} \, dx+\frac {4}{29} \int \frac {\log (x)}{29+5 x} \, dx+\frac {20}{29} \int \frac {\log \left (\frac {4}{29+5 x}\right )}{(5+x) (29+5 x)} \, dx-\frac {4}{5} \int \frac {\log \left (-\frac {4}{5 (5+x)}\right )}{(5+x) (29+5 x)} \, dx-5 \int \frac {1}{25+5 x} \, dx+5 \int \frac {1}{29+5 x} \, dx\\ &=\frac {x}{2}+\frac {4}{145} \log \left (1+\frac {5 x}{29}\right ) \log (x)-\frac {4}{145} \log \left (1+\frac {x}{5}\right ) \log (x)-\log (5+x)+\log (29+5 x)-\frac {4}{145} \log (x) \log \left (\frac {29+5 x}{5+x}\right )-\frac {1}{5} \log \left (-\frac {4}{5 (5+x)}\right ) \log \left (\frac {29+5 x}{5+x}\right )+\frac {5}{29} \log \left (\frac {4}{29+5 x}\right ) \log \left (\frac {29+5 x}{5+x}\right )+\frac {4}{145} \log \left (-\frac {4 x}{5 (29+5 x)}\right ) \log \left (\frac {29+5 x}{5+x}\right )+\frac {(5+x) \log ^2\left (\frac {29+5 x}{5+x}\right )}{10 x}-\frac {4}{145} \text {Li}_2\left (1+\frac {4 x}{5 (29+5 x)}\right )-\frac {4}{145} \int \frac {\log \left (1+\frac {5 x}{29}\right )}{x} \, dx+\frac {4}{145} \int \frac {\log \left (1+\frac {x}{5}\right )}{x} \, dx+\frac {4}{29} \operatorname {Subst}\left (\int \frac {\log \left (\frac {4}{x}\right )}{\left (-\frac {4}{5}+\frac {x}{5}\right ) x} \, dx,x,29+5 x\right )-\frac {4}{5} \operatorname {Subst}\left (\int \frac {\log \left (-\frac {4}{5 x}\right )}{x (4+5 x)} \, dx,x,5+x\right )\\ &=\frac {x}{2}+\frac {4}{145} \log \left (1+\frac {5 x}{29}\right ) \log (x)-\frac {4}{145} \log \left (1+\frac {x}{5}\right ) \log (x)-\log (5+x)+\log (29+5 x)-\frac {4}{145} \log (x) \log \left (\frac {29+5 x}{5+x}\right )-\frac {1}{5} \log \left (-\frac {4}{5 (5+x)}\right ) \log \left (\frac {29+5 x}{5+x}\right )+\frac {5}{29} \log \left (\frac {4}{29+5 x}\right ) \log \left (\frac {29+5 x}{5+x}\right )+\frac {4}{145} \log \left (-\frac {4 x}{5 (29+5 x)}\right ) \log \left (\frac {29+5 x}{5+x}\right )+\frac {(5+x) \log ^2\left (\frac {29+5 x}{5+x}\right )}{10 x}-\frac {4 \text {Li}_2\left (-\frac {x}{5}\right )}{145}+\frac {4}{145} \text {Li}_2\left (-\frac {5 x}{29}\right )-\frac {4}{145} \text {Li}_2\left (1+\frac {4 x}{5 (29+5 x)}\right )-\frac {4}{29} \operatorname {Subst}\left (\int \frac {\log (4 x)}{\left (-\frac {4}{5}+\frac {1}{5 x}\right ) x} \, dx,x,\frac {1}{29+5 x}\right )+\frac {4}{5} \operatorname {Subst}\left (\int \frac {\log \left (-\frac {4 x}{5}\right )}{\left (4+\frac {5}{x}\right ) x} \, dx,x,\frac {1}{5+x}\right )\\ &=\frac {x}{2}+\frac {4}{145} \log \left (1+\frac {5 x}{29}\right ) \log (x)-\frac {4}{145} \log \left (1+\frac {x}{5}\right ) \log (x)-\log (5+x)+\log (29+5 x)-\frac {4}{145} \log (x) \log \left (\frac {29+5 x}{5+x}\right )-\frac {1}{5} \log \left (-\frac {4}{5 (5+x)}\right ) \log \left (\frac {29+5 x}{5+x}\right )+\frac {5}{29} \log \left (\frac {4}{29+5 x}\right ) \log \left (\frac {29+5 x}{5+x}\right )+\frac {4}{145} \log \left (-\frac {4 x}{5 (29+5 x)}\right ) \log \left (\frac {29+5 x}{5+x}\right )+\frac {(5+x) \log ^2\left (\frac {29+5 x}{5+x}\right )}{10 x}-\frac {4 \text {Li}_2\left (-\frac {x}{5}\right )}{145}+\frac {4}{145} \text {Li}_2\left (-\frac {5 x}{29}\right )-\frac {4}{145} \text {Li}_2\left (1+\frac {4 x}{5 (29+5 x)}\right )-\frac {4}{29} \operatorname {Subst}\left (\int \frac {\log (4 x)}{\frac {1}{5}-\frac {4 x}{5}} \, dx,x,\frac {1}{29+5 x}\right )+\frac {4}{5} \operatorname {Subst}\left (\int \frac {\log \left (-\frac {4 x}{5}\right )}{5+4 x} \, dx,x,\frac {1}{5+x}\right )\\ &=\frac {x}{2}+\frac {4}{145} \log \left (1+\frac {5 x}{29}\right ) \log (x)-\frac {4}{145} \log \left (1+\frac {x}{5}\right ) \log (x)-\log (5+x)+\log (29+5 x)-\frac {4}{145} \log (x) \log \left (\frac {29+5 x}{5+x}\right )-\frac {1}{5} \log \left (-\frac {4}{5 (5+x)}\right ) \log \left (\frac {29+5 x}{5+x}\right )+\frac {5}{29} \log \left (\frac {4}{29+5 x}\right ) \log \left (\frac {29+5 x}{5+x}\right )+\frac {4}{145} \log \left (-\frac {4 x}{5 (29+5 x)}\right ) \log \left (\frac {29+5 x}{5+x}\right )+\frac {(5+x) \log ^2\left (\frac {29+5 x}{5+x}\right )}{10 x}-\frac {4 \text {Li}_2\left (-\frac {x}{5}\right )}{145}+\frac {4}{145} \text {Li}_2\left (-\frac {5 x}{29}\right )-\frac {1}{5} \text {Li}_2\left (1+\frac {4}{5 (5+x)}\right )-\frac {5}{29} \text {Li}_2\left (1-\frac {4}{29+5 x}\right )-\frac {4}{145} \text {Li}_2\left (1+\frac {4 x}{5 (29+5 x)}\right )\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.14, size = 152, normalized size = 6.61 \begin {gather*} \frac {1}{2} \left (x-\frac {1}{5} \log ^2\left (-\frac {4}{5 (5+x)}\right )-2 \log (5+x)+\frac {1}{5} \log ^2(5+x)-\frac {2}{5} \log \left (-\frac {4}{5 (5+x)}\right ) \log \left (\frac {1}{4} (29+5 x)\right )-\frac {2}{5} \log (5+x) \log \left (\frac {1}{4} (29+5 x)\right )+2 \log (29+5 x)+\frac {2}{5} \log \left (-\frac {4}{5 (5+x)}\right ) \log \left (\frac {29+5 x}{5+x}\right )+\frac {2}{5} \log (5+x) \log \left (\frac {29+5 x}{5+x}\right )+\frac {\log ^2\left (\frac {29+5 x}{5+x}\right )}{x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(137*x^2 + 54*x^3 + 5*x^4 - 8*x*Log[(29 + 5*x)/(5 + x)] + (-145 - 54*x - 5*x^2)*Log[(29 + 5*x)/(5 +
x)]^2)/(290*x^2 + 108*x^3 + 10*x^4),x]

[Out]

(x - Log[-4/(5*(5 + x))]^2/5 - 2*Log[5 + x] + Log[5 + x]^2/5 - (2*Log[-4/(5*(5 + x))]*Log[(29 + 5*x)/4])/5 - (
2*Log[5 + x]*Log[(29 + 5*x)/4])/5 + 2*Log[29 + 5*x] + (2*Log[-4/(5*(5 + x))]*Log[(29 + 5*x)/(5 + x)])/5 + (2*L
og[5 + x]*Log[(29 + 5*x)/(5 + x)])/5 + Log[(29 + 5*x)/(5 + x)]^2/x)/2

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fricas [A]  time = 0.54, size = 38, normalized size = 1.65 \begin {gather*} \frac {x^{2} + 2 \, x \log \left (\frac {5 \, x + 29}{x + 5}\right ) + \log \left (\frac {5 \, x + 29}{x + 5}\right )^{2}}{2 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-5*x^2-54*x-145)*log((5*x+29)/(5+x))^2-8*x*log((5*x+29)/(5+x))+5*x^4+54*x^3+137*x^2)/(10*x^4+108*x
^3+290*x^2),x, algorithm="fricas")

[Out]

1/2*(x^2 + 2*x*log((5*x + 29)/(x + 5)) + log((5*x + 29)/(x + 5))^2)/x

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giac [B]  time = 0.20, size = 66, normalized size = 2.87 \begin {gather*} -\frac {1}{10} \, {\left (\frac {4}{\frac {5 \, {\left (5 \, x + 29\right )}}{x + 5} - 29} + 1\right )} \log \left (\frac {5 \, x + 29}{x + 5}\right )^{2} + \frac {2}{\frac {5 \, x + 29}{x + 5} - 5} + \log \left (\frac {5 \, x + 29}{x + 5}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-5*x^2-54*x-145)*log((5*x+29)/(5+x))^2-8*x*log((5*x+29)/(5+x))+5*x^4+54*x^3+137*x^2)/(10*x^4+108*x
^3+290*x^2),x, algorithm="giac")

[Out]

-1/10*(4/(5*(5*x + 29)/(x + 5) - 29) + 1)*log((5*x + 29)/(x + 5))^2 + 2/((5*x + 29)/(x + 5) - 5) + log((5*x +
29)/(x + 5))

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maple [A]  time = 0.20, size = 36, normalized size = 1.57

method	result	size
risch	\(\frac {\ln \left (\frac {5 x +29}{5+x}\right )^{2}}{2 x}+\frac {x}{2}+\ln \left (5 x +29\right )-\ln \left (5+x \right )\)	\(36\)
norman	\(\frac {x \ln \left (\frac {5 x +29}{5+x}\right )+\frac {x^{2}}{2}+\frac {\ln \left (\frac {5 x +29}{5+x}\right )^{2}}{2}}{x}\)	\(41\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-5*x^2-54*x-145)*ln((5*x+29)/(5+x))^2-8*x*ln((5*x+29)/(5+x))+5*x^4+54*x^3+137*x^2)/(10*x^4+108*x^3+290*x
^2),x,method=_RETURNVERBOSE)

[Out]

1/2/x*ln((5*x+29)/(5+x))^2+1/2*x+ln(5*x+29)-ln(5+x)

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maxima [B]  time = 0.42, size = 48, normalized size = 2.09 \begin {gather*} \frac {1}{2} \, x + \frac {\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}}{2 \, x} + \log \left (5 \, x + 29\right ) - \log \left (x + 5\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-5*x^2-54*x-145)*log((5*x+29)/(5+x))^2-8*x*log((5*x+29)/(5+x))+5*x^4+54*x^3+137*x^2)/(10*x^4+108*x
^3+290*x^2),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 5.19, size = 31, normalized size = 1.35 \begin {gather*} \frac {x}{2}+2\,\mathrm {atanh}\left (\frac {5\,x}{2}+\frac {27}{2}\right )+\frac {{\ln \left (\frac {5\,x+29}{x+5}\right )}^2}{2\,x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((137*x^2 - log((5*x + 29)/(x + 5))^2*(54*x + 5*x^2 + 145) - 8*x*log((5*x + 29)/(x + 5)) + 54*x^3 + 5*x^4)/
(290*x^2 + 108*x^3 + 10*x^4),x)

[Out]

x/2 + 2*atanh((5*x)/2 + 27/2) + log((5*x + 29)/(x + 5))^2/(2*x)

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sympy [A]  time = 0.20, size = 29, normalized size = 1.26 \begin {gather*} \frac {x}{2} - \log {\left (x + 5 \right )} + \log {\left (x + \frac {29}{5} \right )} + \frac {\log {\left (\frac {5 x + 29}{x + 5} \right )}^{2}}{2 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-5*x**2-54*x-145)*ln((5*x+29)/(5+x))**2-8*x*ln((5*x+29)/(5+x))+5*x**4+54*x**3+137*x**2)/(10*x**4+1
08*x**3+290*x**2),x)

[Out]

x/2 - log(x + 5) + log(x + 29/5) + log((5*x + 29)/(x + 5))**2/(2*x)

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