\(\int \frac {137 x^2+54 x^3+5 x^4-8 x \log (\frac {29+5 x}{5+x})+(-145-54 x-5 x^2) \log ^2(\frac {29+5 x}{5+x})}{290 x^2+108 x^3+10 x^4} \, dx\) [6281]

Optimal result

Integrand size = 75, antiderivative size = 23 \[ \int \frac {137 x^2+54 x^3+5 x^4-8 x \log \left (\frac {29+5 x}{5+x}\right )+\left (-145-54 x-5 x^2\right ) \log ^2\left (\frac {29+5 x}{5+x}\right )}{290 x^2+108 x^3+10 x^4} \, dx=\frac {\left (x+\log \left (4+\frac {9+x}{5+x}\right )\right )^2}{2 x} \]

[Out]

1/2*(ln(4+(x+9)/(5+x))+x)^2/x

Rubi [C] (verified)

Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.

Time = 0.51 (sec) , antiderivative size = 231, normalized size of antiderivative = 10.04, number of steps used = 29, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {1608, 6860, 1671, 630, 31, 2594, 2545, 2354, 2438, 2543, 2458, 2378, 2370, 2352, 2541, 2553, 2355, 2353} \[ \int \frac {137 x^2+54 x^3+5 x^4-8 x \log \left (\frac {29+5 x}{5+x}\right )+\left (-145-54 x-5 x^2\right ) \log ^2\left (\frac {29+5 x}{5+x}\right )}{290 x^2+108 x^3+10 x^4} \, dx=-\frac {4 \operatorname {PolyLog}\left (2,-\frac {x}{5}\right )}{145}+\frac {4}{145} \operatorname {PolyLog}\left (2,-\frac {5 x}{29}\right )-\frac {4}{145} \operatorname {PolyLog}\left (2,\frac {4 x}{29 (x+5)}\right )-\frac {5}{29} \operatorname {PolyLog}\left (2,\frac {5 (x+5)}{5 x+29}\right )-\frac {1}{5} \operatorname {PolyLog}\left (2,1+\frac {4}{5 (x+5)}\right )+\frac {x}{2}+\frac {(5 x+29) \log ^2\left (\frac {5 x+29}{x+5}\right )}{58 x}-\frac {4}{145} \log (x) \log \left (\frac {5 x+29}{x+5}\right )-\frac {1}{5} \log \left (-\frac {4}{5 (x+5)}\right ) \log \left (\frac {5 x+29}{x+5}\right )+\frac {5}{29} \log \left (\frac {4}{5 x+29}\right ) \log \left (\frac {5 x+29}{x+5}\right )+\frac {4}{145} \log \left (\frac {5 x}{29}+1\right ) \log (x)-\frac {4}{145} \log \left (\frac {x}{5}+1\right ) \log (x)+\frac {4}{145} \log \left (\frac {29}{5}\right ) \log \left (\frac {x}{x+5}\right )-\log (x+5)+\log (5 x+29) \]

[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 + (4*Log[29/5]*Log[x/(5 + 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 + ((29 + 5*x)*Log[(29 + 5*x)/(5 + x)]^2)/(58*x) - (4*Po
lyLog[2, -1/5*x])/145 + (4*PolyLog[2, (-5*x)/29])/145 - (4*PolyLog[2, (4*x)/(29*(5 + x))])/145 - (5*PolyLog[2,
(5*(5 + x))/(29 + 5*x)])/29 - PolyLog[2, 1 + 4/(5*(5 + x))]/5

Rule 31

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

Rule 630

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 1608

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 1671

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 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 2355

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((d_) + (e_.)*(x_))^2, x_Symbol] :> Simp[x*((a + b*Log[c*x^n])
^p/(d*(d + e*x))), x] - Dist[b*n*(p/d), Int[(a + b*Log[c*x^n])^(p - 1)/(d + e*x), x], x] /; FreeQ[{a, b, c, d,
e, n, p}, x] && GtQ[p, 0]

Rule 2370

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 2378

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 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 2458

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 2541

Int[((A_.) + Log[(e_.)*(((a_.) + (b_.)*(x_))/((c_.) + (d_.)*(x_)))^(n_.)]*(B_.))/((f_.) + (g_.)*(x_)), x_Symbo
l] :> Simp[(-Log[-(b*c - a*d)/(d*(a + b*x))])*((A + B*Log[e*((a + b*x)/(c + d*x))^n])/g), x] + Dist[B*n*((b*c
- a*d)/g), Int[Log[-(b*c - a*d)/(d*(a + b*x))]/((a + b*x)*(c + d*x)), x], x] /; FreeQ[{a, b, c, d, e, f, g, A,
B, n}, x] && NeQ[b*c - a*d, 0] && EqQ[b*f - a*g, 0]

Rule 2543

Int[((A_.) + Log[(e_.)*(((a_.) + (b_.)*(x_))/((c_.) + (d_.)*(x_)))^(n_.)]*(B_.))/((f_.) + (g_.)*(x_)), x_Symbo
l] :> Simp[(-Log[(b*c - a*d)/(b*(c + d*x))])*((A + B*Log[e*((a + b*x)/(c + d*x))^n])/g), x] + Dist[B*n*((b*c -
a*d)/g), Int[Log[(b*c - a*d)/(b*(c + d*x))]/((a + b*x)*(c + d*x)), x], x] /; FreeQ[{a, b, c, d, e, f, g, A, B
, n}, x] && NeQ[b*c - a*d, 0] && EqQ[d*f - c*g, 0]

Rule 2545

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

Rule 2553

Int[((A_.) + Log[(e_.)*(((a_.) + (b_.)*(x_))/((c_.) + (d_.)*(x_)))^(n_.)]*(B_.))^(p_.)*((f_.) + (g_.)*(x_))^(m
_.), x_Symbol] :> Dist[b*c - a*d, Subst[Int[(b*f - a*g - (d*f - c*g)*x)^m*((A + B*Log[e*x^n])^p/(b - d*x)^(m +
2)), x], x, (a + b*x)/(c + d*x)], x] /; FreeQ[{a, b, c, d, e, f, g, A, B, n}, x] && NeQ[b*c - a*d, 0] && Inte
gerQ[m] && IGtQ[p, 0]

Rule 2594

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 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]

Rubi steps \begin{align*} \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 {1}{2} \int \left (1-\frac {8}{145+54 x+5 x^2}\right ) \, dx+2 \text {Subst}\left (\int \frac {\log ^2(x)}{(-29+5 x)^2} \, dx,x,\frac {29+5 x}{5+x}\right )-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 {(29+5 x) \log ^2\left (\frac {29+5 x}{5+x}\right )}{58 x}-\frac {4}{145} \int \frac {\log \left (\frac {29+5 x}{5+x}\right )}{x} \, dx+\frac {4}{29} \text {Subst}\left (\int \frac {\log (x)}{-29+5 x} \, dx,x,\frac {29+5 x}{5+x}\right )+\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 \left (\frac {29}{5}\right ) \log \left (\frac {x}{5+x}\right )-\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 {(29+5 x) \log ^2\left (\frac {29+5 x}{5+x}\right )}{58 x}-\frac {4}{145} \int \frac {\log (x)}{5+x} \, dx+\frac {4}{29} \int \frac {\log (x)}{29+5 x} \, dx+\frac {4}{29} \text {Subst}\left (\int \frac {\log \left (\frac {5 x}{29}\right )}{-29+5 x} \, dx,x,\frac {29+5 x}{5+x}\right )+\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)+\frac {4}{145} \log \left (\frac {29}{5}\right ) \log \left (\frac {x}{5+x}\right )-\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 {(29+5 x) \log ^2\left (\frac {29+5 x}{5+x}\right )}{58 x}-\frac {4}{145} \text {Li}_2\left (\frac {4 x}{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} \text {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} \text {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)+\frac {4}{145} \log \left (\frac {29}{5}\right ) \log \left (\frac {x}{5+x}\right )-\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 {(29+5 x) \log ^2\left (\frac {29+5 x}{5+x}\right )}{58 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 (\frac {4 x}{29 (5+x)}\right )-\frac {4}{29} \text {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} \text {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)+\frac {4}{145} \log \left (\frac {29}{5}\right ) \log \left (\frac {x}{5+x}\right )-\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 {(29+5 x) \log ^2\left (\frac {29+5 x}{5+x}\right )}{58 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 (\frac {4 x}{29 (5+x)}\right )-\frac {4}{29} \text {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} \text {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)+\frac {4}{145} \log \left (\frac {29}{5}\right ) \log \left (\frac {x}{5+x}\right )-\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 {(29+5 x) \log ^2\left (\frac {29+5 x}{5+x}\right )}{58 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 (\frac {4 x}{29 (5+x)}\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 ) \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(152\) vs. \(2(23)=46\).

Time = 0.15 (sec) , antiderivative size = 152, normalized size of antiderivative = 6.61 \[ \int \frac {137 x^2+54 x^3+5 x^4-8 x \log \left (\frac {29+5 x}{5+x}\right )+\left (-145-54 x-5 x^2\right ) \log ^2\left (\frac {29+5 x}{5+x}\right )}{290 x^2+108 x^3+10 x^4} \, dx=\frac {1}{2} \left (x-\frac {1}{5} \log ^2\left (-\frac {4}{5 (5+x)}\right )-2 \log (5+x)+\frac {1}{5} \log ^2(5+x)-\frac {2}{5} \log \left (-\frac {4}{5 (5+x)}\right ) \log \left (\frac {1}{4} (29+5 x)\right )-\frac {2}{5} \log (5+x) \log \left (\frac {1}{4} (29+5 x)\right )+2 \log (29+5 x)+\frac {2}{5} \log \left (-\frac {4}{5 (5+x)}\right ) \log \left (\frac {29+5 x}{5+x}\right )+\frac {2}{5} \log (5+x) \log \left (\frac {29+5 x}{5+x}\right )+\frac {\log ^2\left (\frac {29+5 x}{5+x}\right )}{x}\right ) \]

[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

Maple [A] (verified)

Time = 0.35 (sec) , antiderivative size = 36, normalized size of antiderivative = 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	\(\frac {x \ln \left (\frac {5 x +29}{5+x}\right )+\frac {x^{2}}{2}+\frac {\ln \left (\frac {5 x +29}{5+x}\right )^{2}}{2}}{x}\)	\(41\)
parallelrisch	\(-\frac {-25 x^{2}-50 x \ln \left (\frac {5 x +29}{5+x}\right )-25 \ln \left (\frac {5 x +29}{5+x}\right )^{2}+540 x}{50 x}\)	\(46\)

[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)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.65 \[ \int \frac {137 x^2+54 x^3+5 x^4-8 x \log \left (\frac {29+5 x}{5+x}\right )+\left (-145-54 x-5 x^2\right ) \log ^2\left (\frac {29+5 x}{5+x}\right )}{290 x^2+108 x^3+10 x^4} \, dx=\frac {x^{2} + 2 \, x \log \left (\frac {5 \, x + 29}{x + 5}\right ) + \log \left (\frac {5 \, x + 29}{x + 5}\right )^{2}}{2 \, x} \]

[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

Sympy [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.26 \[ \int \frac {137 x^2+54 x^3+5 x^4-8 x \log \left (\frac {29+5 x}{5+x}\right )+\left (-145-54 x-5 x^2\right ) \log ^2\left (\frac {29+5 x}{5+x}\right )}{290 x^2+108 x^3+10 x^4} \, dx=\frac {x}{2} - \log {\left (x + 5 \right )} + \log {\left (x + \frac {29}{5} \right )} + \frac {\log {\left (\frac {5 x + 29}{x + 5} \right )}^{2}}{2 x} \]

[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)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 48 vs. \(2 (21) = 42\).

Time = 0.26 (sec) , antiderivative size = 48, normalized size of antiderivative = 2.09 \[ \int \frac {137 x^2+54 x^3+5 x^4-8 x \log \left (\frac {29+5 x}{5+x}\right )+\left (-145-54 x-5 x^2\right ) \log ^2\left (\frac {29+5 x}{5+x}\right )}{290 x^2+108 x^3+10 x^4} \, dx=\frac {1}{2} \, x + \frac {\log \left (5 \, x + 29\right )^{2} - 2 \, \log \left (5 \, x + 29\right ) \log \left (x + 5\right ) + \log \left (x + 5\right )^{2}}{2 \, x} + \log \left (5 \, x + 29\right ) - \log \left (x + 5\right ) \]

[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)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 66 vs. \(2 (21) = 42\).

Time = 0.29 (sec) , antiderivative size = 66, normalized size of antiderivative = 2.87 \[ \int \frac {137 x^2+54 x^3+5 x^4-8 x \log \left (\frac {29+5 x}{5+x}\right )+\left (-145-54 x-5 x^2\right ) \log ^2\left (\frac {29+5 x}{5+x}\right )}{290 x^2+108 x^3+10 x^4} \, dx=-\frac {1}{10} \, {\left (\frac {4}{\frac {5 \, {\left (5 \, x + 29\right )}}{x + 5} - 29} + 1\right )} \log \left (\frac {5 \, x + 29}{x + 5}\right )^{2} + \frac {2}{\frac {5 \, x + 29}{x + 5} - 5} + \log \left (\frac {5 \, x + 29}{x + 5}\right ) \]

[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))

Mupad [B] (verification not implemented)

Time = 12.44 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.35 \[ \int \frac {137 x^2+54 x^3+5 x^4-8 x \log \left (\frac {29+5 x}{5+x}\right )+\left (-145-54 x-5 x^2\right ) \log ^2\left (\frac {29+5 x}{5+x}\right )}{290 x^2+108 x^3+10 x^4} \, dx=\frac {x}{2}+2\,\mathrm {atanh}\left (\frac {5\,x}{2}+\frac {27}{2}\right )+\frac {{\ln \left (\frac {5\,x+29}{x+5}\right )}^2}{2\,x} \]

[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)