3.244 \(\int \frac {(A+B \log (\frac {e (a+b x)}{c+d x}))^2}{f+g x} \, dx\)

Optimal. Leaf size=277 \[ \frac {2 B \text {Li}_2\left (\frac {(d f-c g) (a+b x)}{(b f-a g) (c+d x)}\right ) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{g}+\frac {\log \left (1-\frac {(a+b x) (d f-c g)}{(c+d x) (b f-a g)}\right ) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{g}-\frac {2 B \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right ) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{g}-\frac {\log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{g}-\frac {2 B^2 \text {Li}_3\left (\frac {(d f-c g) (a+b x)}{(b f-a g) (c+d x)}\right )}{g}+\frac {2 B^2 \text {Li}_3\left (\frac {d (a+b x)}{b (c+d x)}\right )}{g} \]

[Out]

-ln((-a*d+b*c)/b/(d*x+c))*(A+B*ln(e*(b*x+a)/(d*x+c)))^2/g+(A+B*ln(e*(b*x+a)/(d*x+c)))^2*ln(1-(-c*g+d*f)*(b*x+a
)/(-a*g+b*f)/(d*x+c))/g-2*B*(A+B*ln(e*(b*x+a)/(d*x+c)))*polylog(2,d*(b*x+a)/b/(d*x+c))/g+2*B*(A+B*ln(e*(b*x+a)
/(d*x+c)))*polylog(2,(-c*g+d*f)*(b*x+a)/(-a*g+b*f)/(d*x+c))/g+2*B^2*polylog(3,d*(b*x+a)/b/(d*x+c))/g-2*B^2*pol
ylog(3,(-c*g+d*f)*(b*x+a)/(-a*g+b*f)/(d*x+c))/g

________________________________________________________________________________________

Rubi [B]  time = 4.90, antiderivative size = 1998, normalized size of antiderivative = 7.21, number of steps used = 41, number of rules used = 21, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.724, Rules used = {2524, 12, 2528, 2418, 2390, 2301, 2394, 2393, 2391, 6688, 6742, 2500, 2433, 2375, 2317, 2374, 6589, 2440, 2437, 2435, 2315} \[ \text {result too large to display} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*Log[(e*(a + b*x))/(c + d*x)])^2/(f + g*x),x]

[Out]

-((B^2*Log[a + b*x]^2*Log[f + g*x])/g) - (2*A*B*Log[-((g*(a + b*x))/(b*f - a*g))]*Log[f + g*x])/g - (B^2*Log[(
c + d*x)^(-1)]^2*Log[f + g*x])/g + (2*B^2*Log[-((g*(a + b*x))/(b*f - a*g))]*(Log[a + b*x] + Log[(c + d*x)^(-1)
] - Log[(e*(a + b*x))/(c + d*x)])*Log[f + g*x])/g + ((A + B*Log[(e*(a + b*x))/(c + d*x)])^2*Log[f + g*x])/g +
(2*B^2*Log[-((d*(a + b*x))/(b*c - a*d))]*Log[c + d*x]*Log[f + g*x])/g - (2*B^2*Log[-((g*(a + b*x))/(b*f - a*g)
)]*(Log[(c + d*x)^(-1)] + Log[c + d*x])*Log[f + g*x])/g + (2*B^2*Log[a + b*x]*Log[(b*(c + d*x))/(b*c - a*d)]*L
og[f + g*x])/g + (2*A*B*Log[-((g*(c + d*x))/(d*f - c*g))]*Log[f + g*x])/g - (2*B^2*(Log[a + b*x] + Log[(c + d*
x)^(-1)] - Log[(e*(a + b*x))/(c + d*x)])*Log[-((g*(c + d*x))/(d*f - c*g))]*Log[f + g*x])/g + (B^2*Log[a + b*x]
^2*Log[(b*(f + g*x))/(b*f - a*g)])/g + (B^2*Log[(c + d*x)^(-1)]^2*Log[(d*(f + g*x))/(d*f - c*g)])/g + (B^2*(Lo
g[(b*(c + d*x))/(b*c - a*d)] + Log[(b*f - a*g)/(b*(f + g*x))] - Log[((b*f - a*g)*(c + d*x))/((b*c - a*d)*(f +
g*x))])*Log[-(((b*c - a*d)*(f + g*x))/((d*f - c*g)*(a + b*x)))]^2)/g - (B^2*(Log[(b*(c + d*x))/(b*c - a*d)] -
Log[-((g*(c + d*x))/(d*f - c*g))])*(Log[a + b*x] + Log[-(((b*c - a*d)*(f + g*x))/((d*f - c*g)*(a + b*x)))])^2)
/g + (B^2*(Log[-((d*(a + b*x))/(b*c - a*d))] + Log[(d*f - c*g)/(d*(f + g*x))] - Log[-(((d*f - c*g)*(a + b*x))/
((b*c - a*d)*(f + g*x)))])*Log[((b*c - a*d)*(f + g*x))/((b*f - a*g)*(c + d*x))]^2)/g - (B^2*(Log[-((d*(a + b*x
))/(b*c - a*d))] - Log[-((g*(a + b*x))/(b*f - a*g))])*(Log[c + d*x] + Log[((b*c - a*d)*(f + g*x))/((b*f - a*g)
*(c + d*x))])^2)/g + (2*B^2*(Log[f + g*x] - Log[-(((b*c - a*d)*(f + g*x))/((d*f - c*g)*(a + b*x)))])*PolyLog[2
, -((d*(a + b*x))/(b*c - a*d))])/g + (2*B^2*Log[a + b*x]*PolyLog[2, -((g*(a + b*x))/(b*f - a*g))])/g + (2*B^2*
(Log[f + g*x] - Log[((b*c - a*d)*(f + g*x))/((b*f - a*g)*(c + d*x))])*PolyLog[2, (b*(c + d*x))/(b*c - a*d)])/g
- (2*B^2*Log[(c + d*x)^(-1)]*PolyLog[2, -((g*(c + d*x))/(d*f - c*g))])/g - (2*B^2*Log[-(((b*c - a*d)*(f + g*x
))/((d*f - c*g)*(a + b*x)))]*PolyLog[2, (g*(a + b*x))/(b*(f + g*x))])/g + (2*B^2*Log[-(((b*c - a*d)*(f + g*x))
/((d*f - c*g)*(a + b*x)))]*PolyLog[2, -(((d*f - c*g)*(a + b*x))/((b*c - a*d)*(f + g*x)))])/g - (2*B^2*Log[((b*
c - a*d)*(f + g*x))/((b*f - a*g)*(c + d*x))]*PolyLog[2, (g*(c + d*x))/(d*(f + g*x))])/g + (2*B^2*Log[((b*c - a
*d)*(f + g*x))/((b*f - a*g)*(c + d*x))]*PolyLog[2, ((b*f - a*g)*(c + d*x))/((b*c - a*d)*(f + g*x))])/g - (2*A*
B*PolyLog[2, (b*(f + g*x))/(b*f - a*g)])/g + (2*B^2*(Log[a + b*x] + Log[(c + d*x)^(-1)] - Log[(e*(a + b*x))/(c
+ d*x)])*PolyLog[2, (b*(f + g*x))/(b*f - a*g)])/g - (2*B^2*(Log[(c + d*x)^(-1)] + Log[c + d*x])*PolyLog[2, (b
*(f + g*x))/(b*f - a*g)])/g + (2*B^2*(Log[c + d*x] + Log[((b*c - a*d)*(f + g*x))/((b*f - a*g)*(c + d*x))])*Pol
yLog[2, (b*(f + g*x))/(b*f - a*g)])/g + (2*A*B*PolyLog[2, (d*(f + g*x))/(d*f - c*g)])/g - (2*B^2*(Log[a + b*x]
+ Log[(c + d*x)^(-1)] - Log[(e*(a + b*x))/(c + d*x)])*PolyLog[2, (d*(f + g*x))/(d*f - c*g)])/g + (2*B^2*(Log[
a + b*x] + Log[-(((b*c - a*d)*(f + g*x))/((d*f - c*g)*(a + b*x)))])*PolyLog[2, (d*(f + g*x))/(d*f - c*g)])/g -
(2*B^2*PolyLog[3, -((d*(a + b*x))/(b*c - a*d))])/g - (2*B^2*PolyLog[3, -((g*(a + b*x))/(b*f - a*g))])/g - (2*
B^2*PolyLog[3, (b*(c + d*x))/(b*c - a*d)])/g - (2*B^2*PolyLog[3, -((g*(c + d*x))/(d*f - c*g))])/g - (2*B^2*Pol
yLog[3, (g*(a + b*x))/(b*(f + g*x))])/g + (2*B^2*PolyLog[3, -(((d*f - c*g)*(a + b*x))/((b*c - a*d)*(f + g*x)))
])/g - (2*B^2*PolyLog[3, (g*(c + d*x))/(d*(f + g*x))])/g + (2*B^2*PolyLog[3, ((b*f - a*g)*(c + d*x))/((b*c - a
*d)*(f + g*x))])/g - (2*B^2*PolyLog[3, (b*(f + g*x))/(b*f - a*g)])/g - (2*B^2*PolyLog[3, (d*(f + g*x))/(d*f -
c*g)])/g

Rule 12

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

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 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 2374

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

Rule 2375

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

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 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 2433

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + Log[(h_.)*((i_.) + (j_.)*(x_))^(m_.)]*
(g_.))*((k_.) + (l_.)*(x_))^(r_.), x_Symbol] :> Dist[1/e, Subst[Int[((k*x)/d)^r*(a + b*Log[c*x^n])^p*(f + g*Lo
g[h*((e*i - d*j)/e + (j*x)/e)^m]), x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, j, k, l, n, p, r},
x] && EqQ[e*k - d*l, 0]

Rule 2435

Int[(Log[(a_) + (b_.)*(x_)]*Log[(c_) + (d_.)*(x_)])/(x_), x_Symbol] :> Simp[Log[-((b*x)/a)]*Log[a + b*x]*Log[c
+ d*x], x] + (Simp[(1*(Log[-((b*x)/a)] - Log[-(((b*c - a*d)*x)/(a*(c + d*x)))] + Log[(b*c - a*d)/(b*(c + d*x)
)])*Log[(a*(c + d*x))/(c*(a + b*x))]^2)/2, x] - Simp[(1*(Log[-((b*x)/a)] - Log[-((d*x)/c)])*(Log[a + b*x] + Lo
g[(a*(c + d*x))/(c*(a + b*x))])^2)/2, x] + Simp[(Log[c + d*x] - Log[(a*(c + d*x))/(c*(a + b*x))])*PolyLog[2, 1
+ (b*x)/a], x] + Simp[(Log[a + b*x] + Log[(a*(c + d*x))/(c*(a + b*x))])*PolyLog[2, 1 + (d*x)/c], x] + Simp[Lo
g[(a*(c + d*x))/(c*(a + b*x))]*PolyLog[2, (c*(a + b*x))/(a*(c + d*x))], x] - Simp[Log[(a*(c + d*x))/(c*(a + b*
x))]*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))], x] - Simp[PolyLog[3, 1 + (b*x)/a], x] - Simp[PolyLog[3, 1 + (d*x
)/c], x] + Simp[PolyLog[3, (c*(a + b*x))/(a*(c + d*x))], x] - Simp[PolyLog[3, (d*(a + b*x))/(b*(c + d*x))], x]
) /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 2437

Int[(Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*Log[(h_.)*((i_.) + (j_.)*(x_))^(m_.)])/(x_), x_Symbol] :> Dist[m, In
t[(Log[i + j*x]*Log[c*(d + e*x)^n])/x, x], x] - Dist[m*Log[i + j*x] - Log[h*(i + j*x)^m], Int[Log[c*(d + e*x)^
n]/x, x], x] /; FreeQ[{c, d, e, h, i, j, m, n}, x] && NeQ[e*i - d*j, 0] && NeQ[i + j*x, h*(i + j*x)^m]

Rule 2440

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))*((f_.) + Log[(h_.)*((i_.) + (j_.)*(x_))^(m_.)]*(g_.))
*((k_) + (l_.)*(x_))^(r_.), x_Symbol] :> Dist[1/l, Subst[Int[x^r*(a + b*Log[c*(-((e*k - d*l)/l) + (e*x)/l)^n])
*(f + g*Log[h*(-((j*k - i*l)/l) + (j*x)/l)^m]), x], x, k + l*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, j, k,
l, m, n}, x] && IntegerQ[r]

Rule 2500

Int[(Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.))^(r_.)]*((s_.) + Log[(i_.)*((g_.)
+ (h_.)*(x_))^(n_.)]*(t_.)))/((j_.) + (k_.)*(x_)), x_Symbol] :> Dist[Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r] - Lo
g[(a + b*x)^(p*r)] - Log[(c + d*x)^(q*r)], Int[(s + t*Log[i*(g + h*x)^n])/(j + k*x), x], x] + (Int[(Log[(a + b
*x)^(p*r)]*(s + t*Log[i*(g + h*x)^n]))/(j + k*x), x] + Int[(Log[(c + d*x)^(q*r)]*(s + t*Log[i*(g + h*x)^n]))/(
j + k*x), x]) /; FreeQ[{a, b, c, d, e, f, g, h, i, j, k, s, t, n, p, q, r}, x] && NeQ[b*c - a*d, 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 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 6589

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

Rule 6688

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

Rule 6742

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

Rubi steps

\begin {align*} \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{f+g x} \, dx &=\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (f+g x)}{g}-\frac {(2 B) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (f+g x)}{e (a+b x)} \, dx}{g}\\ &=\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (f+g x)}{g}-\frac {(2 B) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (f+g x)}{a+b x} \, dx}{e g}\\ &=\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (f+g x)}{g}-\frac {(2 B) \int \frac {(b c-a d) e \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (f+g x)}{(a+b x) (c+d x)} \, dx}{e g}\\ &=\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (f+g x)}{g}-\frac {(2 B (b c-a d)) \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (f+g x)}{(a+b x) (c+d x)} \, dx}{g}\\ &=\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (f+g x)}{g}-\frac {(2 B (b c-a d)) \int \left (\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (f+g x)}{(b c-a d) (a+b x)}-\frac {d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (f+g x)}{(b c-a d) (c+d x)}\right ) \, dx}{g}\\ &=\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (f+g x)}{g}-\frac {(2 b B) \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (f+g x)}{a+b x} \, dx}{g}+\frac {(2 B d) \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (f+g x)}{c+d x} \, dx}{g}\\ &=\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (f+g x)}{g}-\frac {(2 b B) \int \left (\frac {A \log (f+g x)}{a+b x}+\frac {B \log \left (\frac {e (a+b x)}{c+d x}\right ) \log (f+g x)}{a+b x}\right ) \, dx}{g}+\frac {(2 B d) \int \left (\frac {A \log (f+g x)}{c+d x}+\frac {B \log \left (\frac {e (a+b x)}{c+d x}\right ) \log (f+g x)}{c+d x}\right ) \, dx}{g}\\ &=\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (f+g x)}{g}-\frac {(2 A b B) \int \frac {\log (f+g x)}{a+b x} \, dx}{g}-\frac {\left (2 b B^2\right ) \int \frac {\log \left (\frac {e (a+b x)}{c+d x}\right ) \log (f+g x)}{a+b x} \, dx}{g}+\frac {(2 A B d) \int \frac {\log (f+g x)}{c+d x} \, dx}{g}+\frac {\left (2 B^2 d\right ) \int \frac {\log \left (\frac {e (a+b x)}{c+d x}\right ) \log (f+g x)}{c+d x} \, dx}{g}\\ &=-\frac {2 A B \log \left (-\frac {g (a+b x)}{b f-a g}\right ) \log (f+g x)}{g}+\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (f+g x)}{g}+\frac {2 A B \log \left (-\frac {g (c+d x)}{d f-c g}\right ) \log (f+g x)}{g}+(2 A B) \int \frac {\log \left (\frac {g (a+b x)}{-b f+a g}\right )}{f+g x} \, dx-(2 A B) \int \frac {\log \left (\frac {g (c+d x)}{-d f+c g}\right )}{f+g x} \, dx-\frac {\left (2 b B^2\right ) \int \frac {\log (a+b x) \log (f+g x)}{a+b x} \, dx}{g}-\frac {\left (2 b B^2\right ) \int \frac {\log \left (\frac {1}{c+d x}\right ) \log (f+g x)}{a+b x} \, dx}{g}+\frac {\left (2 B^2 d\right ) \int \frac {\log (a+b x) \log (f+g x)}{c+d x} \, dx}{g}+\frac {\left (2 B^2 d\right ) \int \frac {\log \left (\frac {1}{c+d x}\right ) \log (f+g x)}{c+d x} \, dx}{g}-\frac {\left (2 b B^2 \left (-\log (a+b x)-\log \left (\frac {1}{c+d x}\right )+\log \left (\frac {e (a+b x)}{c+d x}\right )\right )\right ) \int \frac {\log (f+g x)}{a+b x} \, dx}{g}+\frac {\left (2 B^2 d \left (-\log (a+b x)-\log \left (\frac {1}{c+d x}\right )+\log \left (\frac {e (a+b x)}{c+d x}\right )\right )\right ) \int \frac {\log (f+g x)}{c+d x} \, dx}{g}\\ &=-\frac {2 A B \log \left (-\frac {g (a+b x)}{b f-a g}\right ) \log (f+g x)}{g}+\frac {2 B^2 \log \left (-\frac {g (a+b x)}{b f-a g}\right ) \left (\log (a+b x)+\log \left (\frac {1}{c+d x}\right )-\log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (f+g x)}{g}+\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (f+g x)}{g}+\frac {2 A B \log \left (-\frac {g (c+d x)}{d f-c g}\right ) \log (f+g x)}{g}-\frac {2 B^2 \left (\log (a+b x)+\log \left (\frac {1}{c+d x}\right )-\log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log \left (-\frac {g (c+d x)}{d f-c g}\right ) \log (f+g x)}{g}+\frac {(2 A B) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b f+a g}\right )}{x} \, dx,x,f+g x\right )}{g}-\frac {(2 A B) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {d x}{-d f+c g}\right )}{x} \, dx,x,f+g x\right )}{g}-\frac {\left (2 B^2\right ) \operatorname {Subst}\left (\int \frac {\log (x) \log \left (\frac {b f-a g}{b}+\frac {g x}{b}\right )}{x} \, dx,x,a+b x\right )}{g}-\frac {\left (2 B^2\right ) \operatorname {Subst}\left (\int \frac {\log \left (\frac {1}{-\frac {-b c+a d}{b}+\frac {d x}{b}}\right ) \log \left (-\frac {-b f+a g}{b}+\frac {g x}{b}\right )}{x} \, dx,x,a+b x\right )}{g}+\frac {\left (2 B^2\right ) \operatorname {Subst}\left (\int \frac {\log \left (\frac {1}{x}\right ) \log \left (\frac {d f-c g}{d}+\frac {g x}{d}\right )}{x} \, dx,x,c+d x\right )}{g}+\frac {\left (2 B^2\right ) \operatorname {Subst}\left (\int \frac {\log \left (-\frac {b c-a d}{d}+\frac {b x}{d}\right ) \log \left (-\frac {-d f+c g}{d}+\frac {g x}{d}\right )}{x} \, dx,x,c+d x\right )}{g}+\left (2 B^2 \left (-\log (a+b x)-\log \left (\frac {1}{c+d x}\right )+\log \left (\frac {e (a+b x)}{c+d x}\right )\right )\right ) \int \frac {\log \left (\frac {g (a+b x)}{-b f+a g}\right )}{f+g x} \, dx-\left (2 B^2 \left (-\log (a+b x)-\log \left (\frac {1}{c+d x}\right )+\log \left (\frac {e (a+b x)}{c+d x}\right )\right )\right ) \int \frac {\log \left (\frac {g (c+d x)}{-d f+c g}\right )}{f+g x} \, dx\\ &=-\frac {B^2 \log ^2(a+b x) \log (f+g x)}{g}-\frac {2 A B \log \left (-\frac {g (a+b x)}{b f-a g}\right ) \log (f+g x)}{g}-\frac {B^2 \log ^2\left (\frac {1}{c+d x}\right ) \log (f+g x)}{g}+\frac {2 B^2 \log \left (-\frac {g (a+b x)}{b f-a g}\right ) \left (\log (a+b x)+\log \left (\frac {1}{c+d x}\right )-\log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (f+g x)}{g}+\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (f+g x)}{g}+\frac {2 B^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right ) \log (f+g x)}{g}+\frac {2 A B \log \left (-\frac {g (c+d x)}{d f-c g}\right ) \log (f+g x)}{g}-\frac {2 B^2 \left (\log (a+b x)+\log \left (\frac {1}{c+d x}\right )-\log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log \left (-\frac {g (c+d x)}{d f-c g}\right ) \log (f+g x)}{g}+\frac {B^2 \left (\log \left (\frac {b (c+d x)}{b c-a d}\right )+\log \left (\frac {b f-a g}{b (f+g x)}\right )-\log \left (\frac {(b f-a g) (c+d x)}{(b c-a d) (f+g x)}\right )\right ) \log ^2\left (-\frac {(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{g}-\frac {B^2 \left (\log \left (\frac {b (c+d x)}{b c-a d}\right )-\log \left (-\frac {g (c+d x)}{d f-c g}\right )\right ) \left (\log (a+b x)+\log \left (-\frac {(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )\right )^2}{g}+\frac {2 B^2 \left (\log (f+g x)-\log \left (-\frac {(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )\right ) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{g}-\frac {2 B^2 \log \left (-\frac {(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right ) \text {Li}_2\left (\frac {g (a+b x)}{b (f+g x)}\right )}{g}+\frac {2 B^2 \log \left (-\frac {(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right ) \text {Li}_2\left (-\frac {(d f-c g) (a+b x)}{(b c-a d) (f+g x)}\right )}{g}-\frac {2 A B \text {Li}_2\left (\frac {b (f+g x)}{b f-a g}\right )}{g}+\frac {2 A B \text {Li}_2\left (\frac {d (f+g x)}{d f-c g}\right )}{g}+\frac {2 B^2 \left (\log (a+b x)+\log \left (-\frac {(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )\right ) \text {Li}_2\left (\frac {d (f+g x)}{d f-c g}\right )}{g}-\frac {2 B^2 \text {Li}_3\left (-\frac {d (a+b x)}{b c-a d}\right )}{g}-\frac {2 B^2 \text {Li}_3\left (\frac {g (a+b x)}{b (f+g x)}\right )}{g}+\frac {2 B^2 \text {Li}_3\left (-\frac {(d f-c g) (a+b x)}{(b c-a d) (f+g x)}\right )}{g}-\frac {2 B^2 \text {Li}_3\left (\frac {d (f+g x)}{d f-c g}\right )}{g}+\frac {B^2 \operatorname {Subst}\left (\int \frac {\log ^2(x)}{\frac {b f-a g}{b}+\frac {g x}{b}} \, dx,x,a+b x\right )}{b}+\frac {B^2 \operatorname {Subst}\left (\int \frac {\log ^2\left (\frac {1}{x}\right )}{\frac {d f-c g}{d}+\frac {g x}{d}} \, dx,x,c+d x\right )}{d}+\frac {\left (2 B^2\right ) \operatorname {Subst}\left (\int \frac {\log \left (-\frac {-b c+a d}{b}+\frac {d x}{b}\right ) \log \left (-\frac {-b f+a g}{b}+\frac {g x}{b}\right )}{x} \, dx,x,a+b x\right )}{g}+\frac {\left (2 B^2 \left (-\log (a+b x)-\log \left (\frac {1}{c+d x}\right )+\log \left (\frac {e (a+b x)}{c+d x}\right )\right )\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b f+a g}\right )}{x} \, dx,x,f+g x\right )}{g}-\frac {\left (2 B^2 \left (-\log (a+b x)-\log \left (\frac {1}{c+d x}\right )+\log \left (\frac {e (a+b x)}{c+d x}\right )\right )\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {d x}{-d f+c g}\right )}{x} \, dx,x,f+g x\right )}{g}+\frac {\left (2 B^2 \left (-\log \left (\frac {1}{c+d x}\right )-\log (c+d x)\right )\right ) \operatorname {Subst}\left (\int \frac {\log \left (-\frac {-b f+a g}{b}+\frac {g x}{b}\right )}{x} \, dx,x,a+b x\right )}{g}\\ &=-\frac {B^2 \log ^2(a+b x) \log (f+g x)}{g}-\frac {2 A B \log \left (-\frac {g (a+b x)}{b f-a g}\right ) \log (f+g x)}{g}-\frac {B^2 \log ^2\left (\frac {1}{c+d x}\right ) \log (f+g x)}{g}+\frac {2 B^2 \log \left (-\frac {g (a+b x)}{b f-a g}\right ) \left (\log (a+b x)+\log \left (\frac {1}{c+d x}\right )-\log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (f+g x)}{g}+\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (f+g x)}{g}+\frac {2 B^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x) \log (f+g x)}{g}-\frac {2 B^2 \log \left (-\frac {g (a+b x)}{b f-a g}\right ) \left (\log \left (\frac {1}{c+d x}\right )+\log (c+d x)\right ) \log (f+g x)}{g}+\frac {2 B^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right ) \log (f+g x)}{g}+\frac {2 A B \log \left (-\frac {g (c+d x)}{d f-c g}\right ) \log (f+g x)}{g}-\frac {2 B^2 \left (\log (a+b x)+\log \left (\frac {1}{c+d x}\right )-\log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log \left (-\frac {g (c+d x)}{d f-c g}\right ) \log (f+g x)}{g}+\frac {B^2 \log ^2(a+b x) \log \left (\frac {b (f+g x)}{b f-a g}\right )}{g}+\frac {B^2 \log ^2\left (\frac {1}{c+d x}\right ) \log \left (\frac {d (f+g x)}{d f-c g}\right )}{g}+\frac {B^2 \left (\log \left (\frac {b (c+d x)}{b c-a d}\right )+\log \left (\frac {b f-a g}{b (f+g x)}\right )-\log \left (\frac {(b f-a g) (c+d x)}{(b c-a d) (f+g x)}\right )\right ) \log ^2\left (-\frac {(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{g}-\frac {B^2 \left (\log \left (\frac {b (c+d x)}{b c-a d}\right )-\log \left (-\frac {g (c+d x)}{d f-c g}\right )\right ) \left (\log (a+b x)+\log \left (-\frac {(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )\right )^2}{g}+\frac {B^2 \left (\log \left (-\frac {d (a+b x)}{b c-a d}\right )+\log \left (\frac {d f-c g}{d (f+g x)}\right )-\log \left (-\frac {(d f-c g) (a+b x)}{(b c-a d) (f+g x)}\right )\right ) \log ^2\left (\frac {(b c-a d) (f+g x)}{(b f-a g) (c+d x)}\right )}{g}-\frac {B^2 \left (\log \left (-\frac {d (a+b x)}{b c-a d}\right )-\log \left (-\frac {g (a+b x)}{b f-a g}\right )\right ) \left (\log (c+d x)+\log \left (\frac {(b c-a d) (f+g x)}{(b f-a g) (c+d x)}\right )\right )^2}{g}+\frac {2 B^2 \left (\log (f+g x)-\log \left (-\frac {(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )\right ) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{g}+\frac {2 B^2 \left (\log (f+g x)-\log \left (\frac {(b c-a d) (f+g x)}{(b f-a g) (c+d x)}\right )\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{g}-\frac {2 B^2 \log \left (-\frac {(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right ) \text {Li}_2\left (\frac {g (a+b x)}{b (f+g x)}\right )}{g}+\frac {2 B^2 \log \left (-\frac {(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right ) \text {Li}_2\left (-\frac {(d f-c g) (a+b x)}{(b c-a d) (f+g x)}\right )}{g}-\frac {2 B^2 \log \left (\frac {(b c-a d) (f+g x)}{(b f-a g) (c+d x)}\right ) \text {Li}_2\left (\frac {g (c+d x)}{d (f+g x)}\right )}{g}+\frac {2 B^2 \log \left (\frac {(b c-a d) (f+g x)}{(b f-a g) (c+d x)}\right ) \text {Li}_2\left (\frac {(b f-a g) (c+d x)}{(b c-a d) (f+g x)}\right )}{g}-\frac {2 A B \text {Li}_2\left (\frac {b (f+g x)}{b f-a g}\right )}{g}+\frac {2 B^2 \left (\log (a+b x)+\log \left (\frac {1}{c+d x}\right )-\log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \text {Li}_2\left (\frac {b (f+g x)}{b f-a g}\right )}{g}+\frac {2 B^2 \left (\log (c+d x)+\log \left (\frac {(b c-a d) (f+g x)}{(b f-a g) (c+d x)}\right )\right ) \text {Li}_2\left (\frac {b (f+g x)}{b f-a g}\right )}{g}+\frac {2 A B \text {Li}_2\left (\frac {d (f+g x)}{d f-c g}\right )}{g}-\frac {2 B^2 \left (\log (a+b x)+\log \left (\frac {1}{c+d x}\right )-\log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \text {Li}_2\left (\frac {d (f+g x)}{d f-c g}\right )}{g}+\frac {2 B^2 \left (\log (a+b x)+\log \left (-\frac {(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )\right ) \text {Li}_2\left (\frac {d (f+g x)}{d f-c g}\right )}{g}-\frac {2 B^2 \text {Li}_3\left (-\frac {d (a+b x)}{b c-a d}\right )}{g}-\frac {2 B^2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{g}-\frac {2 B^2 \text {Li}_3\left (\frac {g (a+b x)}{b (f+g x)}\right )}{g}+\frac {2 B^2 \text {Li}_3\left (-\frac {(d f-c g) (a+b x)}{(b c-a d) (f+g x)}\right )}{g}-\frac {2 B^2 \text {Li}_3\left (\frac {g (c+d x)}{d (f+g x)}\right )}{g}+\frac {2 B^2 \text {Li}_3\left (\frac {(b f-a g) (c+d x)}{(b c-a d) (f+g x)}\right )}{g}-\frac {2 B^2 \text {Li}_3\left (\frac {b (f+g x)}{b f-a g}\right )}{g}-\frac {2 B^2 \text {Li}_3\left (\frac {d (f+g x)}{d f-c g}\right )}{g}-\frac {\left (2 B^2\right ) \operatorname {Subst}\left (\int \frac {\log (x) \log \left (1+\frac {g x}{b f-a g}\right )}{x} \, dx,x,a+b x\right )}{g}+\frac {\left (2 B^2\right ) \operatorname {Subst}\left (\int \frac {\log \left (\frac {1}{x}\right ) \log \left (1+\frac {g x}{d f-c g}\right )}{x} \, dx,x,c+d x\right )}{g}-\frac {\left (2 B^2 \left (-\log \left (\frac {1}{c+d x}\right )-\log (c+d x)\right )\right ) \operatorname {Subst}\left (\int \frac {\log \left (\frac {g x}{-b f+a g}\right )}{-\frac {-b f+a g}{b}+\frac {g x}{b}} \, dx,x,a+b x\right )}{b}\\ &=-\frac {B^2 \log ^2(a+b x) \log (f+g x)}{g}-\frac {2 A B \log \left (-\frac {g (a+b x)}{b f-a g}\right ) \log (f+g x)}{g}-\frac {B^2 \log ^2\left (\frac {1}{c+d x}\right ) \log (f+g x)}{g}+\frac {2 B^2 \log \left (-\frac {g (a+b x)}{b f-a g}\right ) \left (\log (a+b x)+\log \left (\frac {1}{c+d x}\right )-\log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (f+g x)}{g}+\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (f+g x)}{g}+\frac {2 B^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x) \log (f+g x)}{g}-\frac {2 B^2 \log \left (-\frac {g (a+b x)}{b f-a g}\right ) \left (\log \left (\frac {1}{c+d x}\right )+\log (c+d x)\right ) \log (f+g x)}{g}+\frac {2 B^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right ) \log (f+g x)}{g}+\frac {2 A B \log \left (-\frac {g (c+d x)}{d f-c g}\right ) \log (f+g x)}{g}-\frac {2 B^2 \left (\log (a+b x)+\log \left (\frac {1}{c+d x}\right )-\log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log \left (-\frac {g (c+d x)}{d f-c g}\right ) \log (f+g x)}{g}+\frac {B^2 \log ^2(a+b x) \log \left (\frac {b (f+g x)}{b f-a g}\right )}{g}+\frac {B^2 \log ^2\left (\frac {1}{c+d x}\right ) \log \left (\frac {d (f+g x)}{d f-c g}\right )}{g}+\frac {B^2 \left (\log \left (\frac {b (c+d x)}{b c-a d}\right )+\log \left (\frac {b f-a g}{b (f+g x)}\right )-\log \left (\frac {(b f-a g) (c+d x)}{(b c-a d) (f+g x)}\right )\right ) \log ^2\left (-\frac {(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{g}-\frac {B^2 \left (\log \left (\frac {b (c+d x)}{b c-a d}\right )-\log \left (-\frac {g (c+d x)}{d f-c g}\right )\right ) \left (\log (a+b x)+\log \left (-\frac {(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )\right )^2}{g}+\frac {B^2 \left (\log \left (-\frac {d (a+b x)}{b c-a d}\right )+\log \left (\frac {d f-c g}{d (f+g x)}\right )-\log \left (-\frac {(d f-c g) (a+b x)}{(b c-a d) (f+g x)}\right )\right ) \log ^2\left (\frac {(b c-a d) (f+g x)}{(b f-a g) (c+d x)}\right )}{g}-\frac {B^2 \left (\log \left (-\frac {d (a+b x)}{b c-a d}\right )-\log \left (-\frac {g (a+b x)}{b f-a g}\right )\right ) \left (\log (c+d x)+\log \left (\frac {(b c-a d) (f+g x)}{(b f-a g) (c+d x)}\right )\right )^2}{g}+\frac {2 B^2 \left (\log (f+g x)-\log \left (-\frac {(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )\right ) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{g}+\frac {2 B^2 \log (a+b x) \text {Li}_2\left (-\frac {g (a+b x)}{b f-a g}\right )}{g}+\frac {2 B^2 \left (\log (f+g x)-\log \left (\frac {(b c-a d) (f+g x)}{(b f-a g) (c+d x)}\right )\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{g}-\frac {2 B^2 \log \left (\frac {1}{c+d x}\right ) \text {Li}_2\left (-\frac {g (c+d x)}{d f-c g}\right )}{g}-\frac {2 B^2 \log \left (-\frac {(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right ) \text {Li}_2\left (\frac {g (a+b x)}{b (f+g x)}\right )}{g}+\frac {2 B^2 \log \left (-\frac {(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right ) \text {Li}_2\left (-\frac {(d f-c g) (a+b x)}{(b c-a d) (f+g x)}\right )}{g}-\frac {2 B^2 \log \left (\frac {(b c-a d) (f+g x)}{(b f-a g) (c+d x)}\right ) \text {Li}_2\left (\frac {g (c+d x)}{d (f+g x)}\right )}{g}+\frac {2 B^2 \log \left (\frac {(b c-a d) (f+g x)}{(b f-a g) (c+d x)}\right ) \text {Li}_2\left (\frac {(b f-a g) (c+d x)}{(b c-a d) (f+g x)}\right )}{g}-\frac {2 A B \text {Li}_2\left (\frac {b (f+g x)}{b f-a g}\right )}{g}+\frac {2 B^2 \left (\log (a+b x)+\log \left (\frac {1}{c+d x}\right )-\log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \text {Li}_2\left (\frac {b (f+g x)}{b f-a g}\right )}{g}-\frac {2 B^2 \left (\log \left (\frac {1}{c+d x}\right )+\log (c+d x)\right ) \text {Li}_2\left (\frac {b (f+g x)}{b f-a g}\right )}{g}+\frac {2 B^2 \left (\log (c+d x)+\log \left (\frac {(b c-a d) (f+g x)}{(b f-a g) (c+d x)}\right )\right ) \text {Li}_2\left (\frac {b (f+g x)}{b f-a g}\right )}{g}+\frac {2 A B \text {Li}_2\left (\frac {d (f+g x)}{d f-c g}\right )}{g}-\frac {2 B^2 \left (\log (a+b x)+\log \left (\frac {1}{c+d x}\right )-\log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \text {Li}_2\left (\frac {d (f+g x)}{d f-c g}\right )}{g}+\frac {2 B^2 \left (\log (a+b x)+\log \left (-\frac {(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )\right ) \text {Li}_2\left (\frac {d (f+g x)}{d f-c g}\right )}{g}-\frac {2 B^2 \text {Li}_3\left (-\frac {d (a+b x)}{b c-a d}\right )}{g}-\frac {2 B^2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{g}-\frac {2 B^2 \text {Li}_3\left (\frac {g (a+b x)}{b (f+g x)}\right )}{g}+\frac {2 B^2 \text {Li}_3\left (-\frac {(d f-c g) (a+b x)}{(b c-a d) (f+g x)}\right )}{g}-\frac {2 B^2 \text {Li}_3\left (\frac {g (c+d x)}{d (f+g x)}\right )}{g}+\frac {2 B^2 \text {Li}_3\left (\frac {(b f-a g) (c+d x)}{(b c-a d) (f+g x)}\right )}{g}-\frac {2 B^2 \text {Li}_3\left (\frac {b (f+g x)}{b f-a g}\right )}{g}-\frac {2 B^2 \text {Li}_3\left (\frac {d (f+g x)}{d f-c g}\right )}{g}-\frac {\left (2 B^2\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {g x}{b f-a g}\right )}{x} \, dx,x,a+b x\right )}{g}-\frac {\left (2 B^2\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {g x}{d f-c g}\right )}{x} \, dx,x,c+d x\right )}{g}\\ &=-\frac {B^2 \log ^2(a+b x) \log (f+g x)}{g}-\frac {2 A B \log \left (-\frac {g (a+b x)}{b f-a g}\right ) \log (f+g x)}{g}-\frac {B^2 \log ^2\left (\frac {1}{c+d x}\right ) \log (f+g x)}{g}+\frac {2 B^2 \log \left (-\frac {g (a+b x)}{b f-a g}\right ) \left (\log (a+b x)+\log \left (\frac {1}{c+d x}\right )-\log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (f+g x)}{g}+\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (f+g x)}{g}+\frac {2 B^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x) \log (f+g x)}{g}-\frac {2 B^2 \log \left (-\frac {g (a+b x)}{b f-a g}\right ) \left (\log \left (\frac {1}{c+d x}\right )+\log (c+d x)\right ) \log (f+g x)}{g}+\frac {2 B^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right ) \log (f+g x)}{g}+\frac {2 A B \log \left (-\frac {g (c+d x)}{d f-c g}\right ) \log (f+g x)}{g}-\frac {2 B^2 \left (\log (a+b x)+\log \left (\frac {1}{c+d x}\right )-\log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log \left (-\frac {g (c+d x)}{d f-c g}\right ) \log (f+g x)}{g}+\frac {B^2 \log ^2(a+b x) \log \left (\frac {b (f+g x)}{b f-a g}\right )}{g}+\frac {B^2 \log ^2\left (\frac {1}{c+d x}\right ) \log \left (\frac {d (f+g x)}{d f-c g}\right )}{g}+\frac {B^2 \left (\log \left (\frac {b (c+d x)}{b c-a d}\right )+\log \left (\frac {b f-a g}{b (f+g x)}\right )-\log \left (\frac {(b f-a g) (c+d x)}{(b c-a d) (f+g x)}\right )\right ) \log ^2\left (-\frac {(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{g}-\frac {B^2 \left (\log \left (\frac {b (c+d x)}{b c-a d}\right )-\log \left (-\frac {g (c+d x)}{d f-c g}\right )\right ) \left (\log (a+b x)+\log \left (-\frac {(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )\right )^2}{g}+\frac {B^2 \left (\log \left (-\frac {d (a+b x)}{b c-a d}\right )+\log \left (\frac {d f-c g}{d (f+g x)}\right )-\log \left (-\frac {(d f-c g) (a+b x)}{(b c-a d) (f+g x)}\right )\right ) \log ^2\left (\frac {(b c-a d) (f+g x)}{(b f-a g) (c+d x)}\right )}{g}-\frac {B^2 \left (\log \left (-\frac {d (a+b x)}{b c-a d}\right )-\log \left (-\frac {g (a+b x)}{b f-a g}\right )\right ) \left (\log (c+d x)+\log \left (\frac {(b c-a d) (f+g x)}{(b f-a g) (c+d x)}\right )\right )^2}{g}+\frac {2 B^2 \left (\log (f+g x)-\log \left (-\frac {(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )\right ) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{g}+\frac {2 B^2 \log (a+b x) \text {Li}_2\left (-\frac {g (a+b x)}{b f-a g}\right )}{g}+\frac {2 B^2 \left (\log (f+g x)-\log \left (\frac {(b c-a d) (f+g x)}{(b f-a g) (c+d x)}\right )\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{g}-\frac {2 B^2 \log \left (\frac {1}{c+d x}\right ) \text {Li}_2\left (-\frac {g (c+d x)}{d f-c g}\right )}{g}-\frac {2 B^2 \log \left (-\frac {(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right ) \text {Li}_2\left (\frac {g (a+b x)}{b (f+g x)}\right )}{g}+\frac {2 B^2 \log \left (-\frac {(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right ) \text {Li}_2\left (-\frac {(d f-c g) (a+b x)}{(b c-a d) (f+g x)}\right )}{g}-\frac {2 B^2 \log \left (\frac {(b c-a d) (f+g x)}{(b f-a g) (c+d x)}\right ) \text {Li}_2\left (\frac {g (c+d x)}{d (f+g x)}\right )}{g}+\frac {2 B^2 \log \left (\frac {(b c-a d) (f+g x)}{(b f-a g) (c+d x)}\right ) \text {Li}_2\left (\frac {(b f-a g) (c+d x)}{(b c-a d) (f+g x)}\right )}{g}-\frac {2 A B \text {Li}_2\left (\frac {b (f+g x)}{b f-a g}\right )}{g}+\frac {2 B^2 \left (\log (a+b x)+\log \left (\frac {1}{c+d x}\right )-\log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \text {Li}_2\left (\frac {b (f+g x)}{b f-a g}\right )}{g}-\frac {2 B^2 \left (\log \left (\frac {1}{c+d x}\right )+\log (c+d x)\right ) \text {Li}_2\left (\frac {b (f+g x)}{b f-a g}\right )}{g}+\frac {2 B^2 \left (\log (c+d x)+\log \left (\frac {(b c-a d) (f+g x)}{(b f-a g) (c+d x)}\right )\right ) \text {Li}_2\left (\frac {b (f+g x)}{b f-a g}\right )}{g}+\frac {2 A B \text {Li}_2\left (\frac {d (f+g x)}{d f-c g}\right )}{g}-\frac {2 B^2 \left (\log (a+b x)+\log \left (\frac {1}{c+d x}\right )-\log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \text {Li}_2\left (\frac {d (f+g x)}{d f-c g}\right )}{g}+\frac {2 B^2 \left (\log (a+b x)+\log \left (-\frac {(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )\right ) \text {Li}_2\left (\frac {d (f+g x)}{d f-c g}\right )}{g}-\frac {2 B^2 \text {Li}_3\left (-\frac {d (a+b x)}{b c-a d}\right )}{g}-\frac {2 B^2 \text {Li}_3\left (-\frac {g (a+b x)}{b f-a g}\right )}{g}-\frac {2 B^2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{g}-\frac {2 B^2 \text {Li}_3\left (-\frac {g (c+d x)}{d f-c g}\right )}{g}-\frac {2 B^2 \text {Li}_3\left (\frac {g (a+b x)}{b (f+g x)}\right )}{g}+\frac {2 B^2 \text {Li}_3\left (-\frac {(d f-c g) (a+b x)}{(b c-a d) (f+g x)}\right )}{g}-\frac {2 B^2 \text {Li}_3\left (\frac {g (c+d x)}{d (f+g x)}\right )}{g}+\frac {2 B^2 \text {Li}_3\left (\frac {(b f-a g) (c+d x)}{(b c-a d) (f+g x)}\right )}{g}-\frac {2 B^2 \text {Li}_3\left (\frac {b (f+g x)}{b f-a g}\right )}{g}-\frac {2 B^2 \text {Li}_3\left (\frac {d (f+g x)}{d f-c g}\right )}{g}\\ \end {align*}

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Mathematica [A]  time = 0.85, size = 431, normalized size = 1.56 \[ \frac {2 A B \log (f+g x) \log \left (\frac {e (a+b x)}{c+d x}\right )+2 A B \text {Li}_2\left (\frac {g (a+b x)}{a g-b f}\right )-2 A B \log \left (\frac {a}{b}+x\right ) \log (f+g x)+2 A B \log \left (\frac {a}{b}+x\right ) \log \left (\frac {b (f+g x)}{b f-a g}\right )+2 B^2 \log \left (\frac {e (a+b x)}{c+d x}\right ) \text {Li}_2\left (\frac {(d f-c g) (a+b x)}{(b f-a g) (c+d x)}\right )+B^2 \log ^2\left (\frac {e (a+b x)}{c+d x}\right ) \log \left (\frac {(f+g x) (b c-a d)}{(c+d x) (b f-a g)}\right )-2 B^2 \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right ) \log \left (\frac {e (a+b x)}{c+d x}\right )-B^2 \log \left (\frac {b c-a d}{b c+b d x}\right ) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )-2 B^2 \text {Li}_3\left (\frac {(d f-c g) (a+b x)}{(b f-a g) (c+d x)}\right )+2 B^2 \text {Li}_3\left (\frac {d (a+b x)}{b (c+d x)}\right )+A^2 \log (f+g x)-2 A B \text {Li}_2\left (\frac {g (c+d x)}{c g-d f}\right )+2 A B \log \left (\frac {c}{d}+x\right ) \log (f+g x)-2 A B \log \left (\frac {c}{d}+x\right ) \log \left (\frac {d (f+g x)}{d f-c g}\right )}{g} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*Log[(e*(a + b*x))/(c + d*x)])^2/(f + g*x),x]

[Out]

(-(B^2*Log[(e*(a + b*x))/(c + d*x)]^2*Log[(b*c - a*d)/(b*c + b*d*x)]) + A^2*Log[f + g*x] - 2*A*B*Log[a/b + x]*
Log[f + g*x] + 2*A*B*Log[c/d + x]*Log[f + g*x] + 2*A*B*Log[(e*(a + b*x))/(c + d*x)]*Log[f + g*x] + 2*A*B*Log[a
/b + x]*Log[(b*(f + g*x))/(b*f - a*g)] - 2*A*B*Log[c/d + x]*Log[(d*(f + g*x))/(d*f - c*g)] + B^2*Log[(e*(a + b
*x))/(c + d*x)]^2*Log[((b*c - a*d)*(f + g*x))/((b*f - a*g)*(c + d*x))] + 2*A*B*PolyLog[2, (g*(a + b*x))/(-(b*f
) + a*g)] - 2*B^2*Log[(e*(a + b*x))/(c + d*x)]*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))] + 2*B^2*Log[(e*(a + b*x
))/(c + d*x)]*PolyLog[2, ((d*f - c*g)*(a + b*x))/((b*f - a*g)*(c + d*x))] - 2*A*B*PolyLog[2, (g*(c + d*x))/(-(
d*f) + c*g)] + 2*B^2*PolyLog[3, (d*(a + b*x))/(b*(c + d*x))] - 2*B^2*PolyLog[3, ((d*f - c*g)*(a + b*x))/((b*f
- a*g)*(c + d*x))])/g

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fricas [F]  time = 1.08, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {B^{2} \log \left (\frac {b e x + a e}{d x + c}\right )^{2} + 2 \, A B \log \left (\frac {b e x + a e}{d x + c}\right ) + A^{2}}{g x + f}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*log(e*(b*x+a)/(d*x+c)))^2/(g*x+f),x, algorithm="fricas")

[Out]

integral((B^2*log((b*e*x + a*e)/(d*x + c))^2 + 2*A*B*log((b*e*x + a*e)/(d*x + c)) + A^2)/(g*x + f), x)

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*log(e*(b*x+a)/(d*x+c)))^2/(g*x+f),x, algorithm="giac")

[Out]

Timed out

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maple [B]  time = 0.08, size = 2428, normalized size = 8.77 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*ln((b*x+a)/(d*x+c)*e)+A)^2/(g*x+f),x)

[Out]

2*d*A*B/g/(a*d-b*c)*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)*ln((-a*e*g+b*e*f+(c*g-d*f)*(b/d*e+(a*d-b*c)/(d*x+c)/d*e))/
(-a*e*g+b*e*f))/(c*g-d*f)*f*b*c+2*A*B/g/(a*d-b*c)*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)*ln(-(-b*e+(b/d*e+(a*d-b*c)/(
d*x+c)/d*e)*d)/b/e)*b*c-2*d*A*B/g/(a*d-b*c)*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)*ln(-(-b*e+(b/d*e+(a*d-b*c)/(d*x+c)
/d*e)*d)/b/e)*a+2*d*A*B/(a*d-b*c)*dilog((-a*e*g+b*e*f+(c*g-d*f)*(b/d*e+(a*d-b*c)/(d*x+c)/d*e))/(-a*e*g+b*e*f))
/(c*g-d*f)*c*a-2*B^2/g/(a*d-b*c)*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)*polylog(2,-(c*g-d*f)/(-a*e*g+b*e*f)*(b/d*e+(a
*d-b*c)/(d*x+c)/d*e))*b*c+2*A*B/g/(a*d-b*c)*dilog(-(-b*e+(b/d*e+(a*d-b*c)/(d*x+c)/d*e)*d)/b/e)*b*c+B^2/g/(a*d-
b*c)*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)^2*ln(-(b/d*e+(a*d-b*c)/(d*x+c)/d*e)/b*d/e+1)*b*c+2*B^2/g/(a*d-b*c)*ln(b/d
*e+(a*d-b*c)/(d*x+c)/d*e)*polylog(2,(b/d*e+(a*d-b*c)/(d*x+c)/d*e)/b*d/e)*b*c+d*B^2/g/(a*d-b*c)*ln(b/d*e+(a*d-b
*c)/(d*x+c)/d*e)^2*ln(1+(c*g-d*f)/(-a*e*g+b*e*f)*(b/d*e+(a*d-b*c)/(d*x+c)/d*e))*a-2*d*B^2/g/(a*d-b*c)*ln(b/d*e
+(a*d-b*c)/(d*x+c)/d*e)*polylog(2,(b/d*e+(a*d-b*c)/(d*x+c)/d*e)/b*d/e)*a-d*B^2/g/(a*d-b*c)*ln(b/d*e+(a*d-b*c)/
(d*x+c)/d*e)^2*ln(-(b/d*e+(a*d-b*c)/(d*x+c)/d*e)/b*d/e+1)*a-2*d^2*A*B/g/(a*d-b*c)*dilog((-a*e*g+b*e*f+(c*g-d*f
)*(b/d*e+(a*d-b*c)/(d*x+c)/d*e))/(-a*e*g+b*e*f))/(c*g-d*f)*f*a+2*d*A*B/(a*d-b*c)*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*
e)*ln((-a*e*g+b*e*f+(c*g-d*f)*(b/d*e+(a*d-b*c)/(d*x+c)/d*e))/(-a*e*g+b*e*f))/(c*g-d*f)*c*a-2*A*B/(a*d-b*c)*ln(
b/d*e+(a*d-b*c)/(d*x+c)/d*e)*ln((-a*e*g+b*e*f+(c*g-d*f)*(b/d*e+(a*d-b*c)/(d*x+c)/d*e))/(-a*e*g+b*e*f))/(c*g-d*
f)*c^2*b-2*d*B^2/g/(a*d-b*c)*polylog(3,-(c*g-d*f)/(-a*e*g+b*e*f)*(b/d*e+(a*d-b*c)/(d*x+c)/d*e))*a-d*A^2/g/(a*d
-b*c)*ln(-b*e+(b/d*e+(a*d-b*c)/(d*x+c)/d*e)*d)*a+d*A^2/g/(a*d-b*c)*ln(-a*e*g+b*e*f+(b/d*e+(a*d-b*c)/(d*x+c)/d*
e)*c*g-(b/d*e+(a*d-b*c)/(d*x+c)/d*e)*d*f)*a+2*d*B^2/g/(a*d-b*c)*polylog(3,(b/d*e+(a*d-b*c)/(d*x+c)/d*e)/b*d/e)
*a-A^2/g/(a*d-b*c)*ln(-a*e*g+b*e*f+(b/d*e+(a*d-b*c)/(d*x+c)/d*e)*c*g-(b/d*e+(a*d-b*c)/(d*x+c)/d*e)*d*f)*b*c+A^
2/g/(a*d-b*c)*ln(-b*e+(b/d*e+(a*d-b*c)/(d*x+c)/d*e)*d)*b*c+2*d*B^2/g/(a*d-b*c)*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)
*polylog(2,-(c*g-d*f)/(-a*e*g+b*e*f)*(b/d*e+(a*d-b*c)/(d*x+c)/d*e))*a-2*A*B/(a*d-b*c)*dilog((-a*e*g+b*e*f+(c*g
-d*f)*(b/d*e+(a*d-b*c)/(d*x+c)/d*e))/(-a*e*g+b*e*f))/(c*g-d*f)*c^2*b-2*B^2/g/(a*d-b*c)*polylog(3,(b/d*e+(a*d-b
*c)/(d*x+c)/d*e)/b*d/e)*b*c+2*B^2/g/(a*d-b*c)*polylog(3,-(c*g-d*f)/(-a*e*g+b*e*f)*(b/d*e+(a*d-b*c)/(d*x+c)/d*e
))*b*c-B^2/g/(a*d-b*c)*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)^2*ln(1+(c*g-d*f)/(-a*e*g+b*e*f)*(b/d*e+(a*d-b*c)/(d*x+c
)/d*e))*b*c-2*d*A*B/g/(a*d-b*c)*dilog(-(-b*e+(b/d*e+(a*d-b*c)/(d*x+c)/d*e)*d)/b/e)*a-2*d^2*A*B/g/(a*d-b*c)*ln(
b/d*e+(a*d-b*c)/(d*x+c)/d*e)*ln((-a*e*g+b*e*f+(c*g-d*f)*(b/d*e+(a*d-b*c)/(d*x+c)/d*e))/(-a*e*g+b*e*f))/(c*g-d*
f)*f*a+2*d*A*B/g/(a*d-b*c)*dilog((-a*e*g+b*e*f+(c*g-d*f)*(b/d*e+(a*d-b*c)/(d*x+c)/d*e))/(-a*e*g+b*e*f))/(c*g-d
*f)*f*b*c

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {A^{2} \log \left (g x + f\right )}{g} - \int -\frac {B^{2} \log \left (b x + a\right )^{2} + B^{2} \log \relax (e)^{2} + 2 \, A B \log \relax (e) + 2 \, {\left (B^{2} \log \relax (e) + A B\right )} \log \left (b x + a\right ) - 2 \, {\left (B^{2} \log \left (b x + a\right ) + B^{2} \log \relax (e) + A B\right )} \log \left (d x + c\right )}{g x + f}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*log(e*(b*x+a)/(d*x+c)))^2/(g*x+f),x, algorithm="maxima")

[Out]

A^2*log(g*x + f)/g - integrate(-(B^2*log(b*x + a)^2 + B^2*log(e)^2 + 2*A*B*log(e) + 2*(B^2*log(e) + A*B)*log(b
*x + a) - 2*(B^2*log(b*x + a) + B^2*log(e) + A*B)*log(d*x + c))/(g*x + f), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (A+B\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )\right )}^2}{f+g\,x} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*log((e*(a + b*x))/(c + d*x)))^2/(f + g*x),x)

[Out]

int((A + B*log((e*(a + b*x))/(c + d*x)))^2/(f + g*x), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (A + B \log {\left (\frac {a e}{c + d x} + \frac {b e x}{c + d x} \right )}\right )^{2}}{f + g x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*ln(e*(b*x+a)/(d*x+c)))**2/(g*x+f),x)

[Out]

Integral((A + B*log(a*e/(c + d*x) + b*e*x/(c + d*x)))**2/(f + g*x), x)

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