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

Optimal. Leaf size=128 \[ \frac {2 B \text {Li}_2\left (\frac {b (c+d x)}{d (a+b x)}\right ) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{b g}-\frac {\log \left (1-\frac {b (c+d x)}{d (a+b x)}\right ) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{b g}+\frac {2 B^2 \text {Li}_3\left (\frac {b (c+d x)}{d (a+b x)}\right )}{b g} \]

[Out]

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

________________________________________________________________________________________

Rubi [B]  time = 3.42, antiderivative size = 728, normalized size of antiderivative = 5.69, number of steps used = 46, number of rules used = 23, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.719, Rules used = {2524, 12, 2528, 2418, 2390, 2301, 2394, 2393, 2391, 6688, 6742, 2500, 2440, 2434, 2433, 2375, 2317, 2374, 6589, 2499, 2302, 30, 2396} \[ \frac {2 A B \text {PolyLog}\left (2,-\frac {d (a+b x)}{b c-a d}\right )}{b g}-\frac {2 B^2 \text {PolyLog}\left (2,-\frac {d (a+b x)}{b c-a d}\right ) \left (-\log \left (\frac {e (a+b x)}{c+d x}\right )+\log (a+b x)+\log \left (\frac {1}{c+d x}\right )\right )}{b g}-\frac {2 B^2 \text {PolyLog}\left (3,-\frac {d (a+b x)}{b c-a d}\right )}{b g}-\frac {2 B^2 \text {PolyLog}\left (3,\frac {b (c+d x)}{b c-a d}\right )}{b g}+\frac {2 B^2 \log (a+b x) \text {PolyLog}\left (2,-\frac {d (a+b x)}{b c-a d}\right )}{b g}-\frac {2 B^2 \log \left (\frac {1}{c+d x}\right ) \text {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )}{b g}+\frac {\log (a g+b g x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{b g}+\frac {2 A B \log (a g+b g x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b g}-\frac {A B \log ^2(g (a+b x))}{b g}-\frac {B^2 \log ^2(a g+b g x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{b g}-\frac {2 B^2 \log (a g+b g x) \log \left (\frac {b (c+d x)}{b c-a d}\right ) \left (-\log \left (\frac {e (a+b x)}{c+d x}\right )+\log (a+b x)+\log \left (\frac {1}{c+d x}\right )\right )}{b g}-\frac {B^2 \log (-c-d x) \log ^2(g (a+b x))}{b g}+\frac {B^2 \log ^2(g (a+b x)) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b g}-\frac {B^2 \log ^2\left (\frac {1}{c+d x}\right ) \log (g (a+b x))}{b g}+\frac {B^2 \log ^2\left (\frac {1}{c+d x}\right ) \log \left (-\frac {d (a+b x)}{b c-a d}\right )}{b g}-\frac {B^2 \log ^2(a g+b g x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b g}-\frac {B^2 \log ^2(a+b x) \log (-c-d x)}{b g}+\frac {B^2 \log ^2(a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b g}+\frac {2 B^2 \log (a+b x) \log (-c-d x) \log (g (a+b x))}{b g}+\frac {B^2 \log ^3(g (a+b x))}{3 b g} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((A*B*Log[g*(a + b*x)]^2)/(b*g)) + (B^2*Log[g*(a + b*x)]^3)/(3*b*g) - (B^2*Log[a + b*x]^2*Log[-c - d*x])/(b*g
) + (2*B^2*Log[a + b*x]*Log[g*(a + b*x)]*Log[-c - d*x])/(b*g) - (B^2*Log[g*(a + b*x)]^2*Log[-c - d*x])/(b*g) +
(B^2*Log[-((d*(a + b*x))/(b*c - a*d))]*Log[(c + d*x)^(-1)]^2)/(b*g) - (B^2*Log[g*(a + b*x)]*Log[(c + d*x)^(-1
)]^2)/(b*g) + (B^2*Log[a + b*x]^2*Log[(b*(c + d*x))/(b*c - a*d)])/(b*g) + (B^2*Log[g*(a + b*x)]^2*Log[(b*(c +
d*x))/(b*c - a*d)])/(b*g) + ((A + B*Log[(e*(a + b*x))/(c + d*x)])^2*Log[a*g + b*g*x])/(b*g) + (2*A*B*Log[(b*(c
+ d*x))/(b*c - a*d)]*Log[a*g + b*g*x])/(b*g) - (2*B^2*(Log[a + b*x] + Log[(c + d*x)^(-1)] - Log[(e*(a + b*x))
/(c + d*x)])*Log[(b*(c + d*x))/(b*c - a*d)]*Log[a*g + b*g*x])/(b*g) - (B^2*Log[(e*(a + b*x))/(c + d*x)]*Log[a*
g + b*g*x]^2)/(b*g) - (B^2*Log[(b*(c + d*x))/(b*c - a*d)]*Log[a*g + b*g*x]^2)/(b*g) + (2*A*B*PolyLog[2, -((d*(
a + b*x))/(b*c - a*d))])/(b*g) + (2*B^2*Log[a + b*x]*PolyLog[2, -((d*(a + b*x))/(b*c - a*d))])/(b*g) - (2*B^2*
(Log[a + b*x] + Log[(c + d*x)^(-1)] - Log[(e*(a + b*x))/(c + d*x)])*PolyLog[2, -((d*(a + b*x))/(b*c - a*d))])/
(b*g) - (2*B^2*Log[(c + d*x)^(-1)]*PolyLog[2, (b*(c + d*x))/(b*c - a*d)])/(b*g) - (2*B^2*PolyLog[3, -((d*(a +
b*x))/(b*c - a*d))])/(b*g) - (2*B^2*PolyLog[3, (b*(c + d*x))/(b*c - a*d)])/(b*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 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

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 2302

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/(x_), x_Symbol] :> Dist[1/(b*n), Subst[Int[x^p, x], x, a + b*L
og[c*x^n]], x] /; FreeQ[{a, b, c, n, p}, x]

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 2396

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

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 2434

Int[(((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))*((f_.) + Log[(h_.)*((i_.) + (j_.)*(x_))^(m_.)]*(g_.)
))/(x_), x_Symbol] :> Simp[Log[x]*(a + b*Log[c*(d + e*x)^n])*(f + g*Log[h*(i + j*x)^m]), x] + (-Dist[e*g*m, In
t[(Log[x]*(a + b*Log[c*(d + e*x)^n]))/(d + e*x), x], x] - Dist[b*j*n, Int[(Log[x]*(f + g*Log[h*(i + j*x)^m]))/
(i + j*x), x], x]) /; FreeQ[{a, b, c, d, e, f, g, h, i, j, m, n}, x] && EqQ[e*i - d*j, 0]

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 2499

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

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}{a g+b g x} \, dx &=\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (a g+b g x)}{b 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 (a g+b g x)}{e (a+b x)} \, dx}{b g}\\ &=\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (a g+b g x)}{b 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 (a g+b g x)}{a+b x} \, dx}{b e g}\\ &=\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (a g+b g x)}{b 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 (a g+b g x)}{(a+b x) (c+d x)} \, dx}{b e g}\\ &=\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (a g+b g x)}{b 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 (a g+b g x)}{(a+b x) (c+d x)} \, dx}{b g}\\ &=\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (a g+b g x)}{b g}-\frac {(2 B (b c-a d)) \int \left (\frac {d \left (-A-B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (a g+b g x)}{(b c-a d) (c+d x)}+\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (a g+b g x)}{(b c-a d) (a+b x)}\right ) \, dx}{b g}\\ &=\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (a g+b g x)}{b g}-\frac {(2 B) \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (a g+b 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 (a g+b g x)}{c+d x} \, dx}{b g}\\ &=\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (a g+b g x)}{b g}-\frac {(2 B) \int \left (\frac {A \log (a g+b g x)}{a+b x}+\frac {B \log \left (\frac {e (a+b x)}{c+d x}\right ) \log (a g+b g x)}{a+b x}\right ) \, dx}{g}-\frac {(2 B d) \int \left (\frac {A \log (a g+b g x)}{-c-d x}+\frac {B \log \left (\frac {e (a+b x)}{c+d x}\right ) \log (a g+b g x)}{-c-d x}\right ) \, dx}{b g}\\ &=\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (a g+b g x)}{b g}-\frac {(2 A B) \int \frac {\log (a g+b g x)}{a+b x} \, dx}{g}-\frac {\left (2 B^2\right ) \int \frac {\log \left (\frac {e (a+b x)}{c+d x}\right ) \log (a g+b g x)}{a+b x} \, dx}{g}-\frac {(2 A B d) \int \frac {\log (a g+b g x)}{-c-d x} \, dx}{b g}-\frac {\left (2 B^2 d\right ) \int \frac {\log \left (\frac {e (a+b x)}{c+d x}\right ) \log (a g+b g x)}{-c-d x} \, dx}{b g}\\ &=\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (a g+b g x)}{b g}+\frac {2 A B \log \left (\frac {b (c+d x)}{b c-a d}\right ) \log (a g+b g x)}{b g}-\frac {B^2 \log \left (\frac {e (a+b x)}{c+d x}\right ) \log ^2(a g+b g x)}{b g}-(2 A B) \int \frac {\log \left (\frac {b g (-c-d x)}{-b c g+a d g}\right )}{a g+b g x} \, dx-\frac {(2 A B) \operatorname {Subst}\left (\int \frac {g \log (x)}{x} \, dx,x,a g+b g x\right )}{b g^2}+\frac {B^2 \int \frac {\log ^2(a g+b g x)}{a+b x} \, dx}{g}-\frac {\left (B^2 d\right ) \int \frac {\log ^2(a g+b g x)}{c+d x} \, dx}{b g}-\frac {\left (2 B^2 d\right ) \int \frac {\log (a+b x) \log (a g+b g x)}{-c-d x} \, dx}{b g}-\frac {\left (2 B^2 d\right ) \int \frac {\log \left (\frac {1}{c+d x}\right ) \log (a g+b g x)}{-c-d x} \, dx}{b 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 (a g+b g x)}{-c-d x} \, dx}{b g}\\ &=\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (a g+b g x)}{b g}+\frac {2 A B \log \left (\frac {b (c+d x)}{b c-a d}\right ) \log (a g+b g x)}{b 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 {b (c+d x)}{b c-a d}\right ) \log (a g+b g x)}{b g}-\frac {B^2 \log \left (\frac {e (a+b x)}{c+d x}\right ) \log ^2(a g+b g x)}{b g}-\frac {B^2 \log \left (\frac {b (c+d x)}{b c-a d}\right ) \log ^2(a g+b g x)}{b g}+\left (2 B^2\right ) \int \frac {\log \left (\frac {b g (c+d x)}{b c g-a d g}\right ) \log (a g+b g x)}{a g+b g x} \, dx+\frac {B^2 \operatorname {Subst}\left (\int \frac {g \log ^2(x)}{x} \, dx,x,a g+b g x\right )}{b g^2}-\frac {(2 A B) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a g+b g x\right )}{b g}-\frac {(2 A B) \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {d x}{-b c g+a d g}\right )}{x} \, dx,x,a g+b g x\right )}{b 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 {-b c g+a d g}{d}-\frac {b g x}{d}\right )}{x} \, dx,x,-c-d x\right )}{b g}+\frac {\left (2 B^2\right ) \operatorname {Subst}\left (\int \frac {\log \left (\frac {1}{x}\right ) \log \left (\frac {-b c g+a d g}{d}+\frac {b g x}{d}\right )}{x} \, dx,x,c+d x\right )}{b 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 {b g (-c-d x)}{-b c g+a d g}\right )}{a g+b g x} \, dx\\ &=-\frac {A B \log ^2(g (a+b x))}{b g}+\frac {2 B^2 \log (a+b x) \log (g (a+b x)) \log (-c-d x)}{b g}-\frac {B^2 \log (g (a+b x)) \log ^2\left (\frac {1}{c+d x}\right )}{b g}+\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (a g+b g x)}{b g}+\frac {2 A B \log \left (\frac {b (c+d x)}{b c-a d}\right ) \log (a g+b g x)}{b 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 {b (c+d x)}{b c-a d}\right ) \log (a g+b g x)}{b g}-\frac {B^2 \log \left (\frac {e (a+b x)}{c+d x}\right ) \log ^2(a g+b g x)}{b g}-\frac {B^2 \log \left (\frac {b (c+d x)}{b c-a d}\right ) \log ^2(a g+b g x)}{b g}+\frac {2 A B \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{b g}+\frac {B^2 \operatorname {Subst}\left (\int \frac {\log ^2\left (\frac {1}{x}\right )}{\frac {-b c g+a d g}{d}+\frac {b g x}{d}} \, dx,x,c+d x\right )}{d}+\frac {\left (2 B^2\right ) \operatorname {Subst}\left (\int \frac {\log (x) \log \left (\frac {-b c g+a d g}{d}-\frac {b g x}{d}\right )}{\frac {-b c g+a d g}{d}-\frac {b g x}{d}} \, dx,x,-c-d x\right )}{d}+\frac {B^2 \operatorname {Subst}\left (\int \frac {\log ^2(x)}{x} \, dx,x,a g+b g x\right )}{b g}+\frac {\left (2 B^2\right ) \operatorname {Subst}\left (\int \frac {\log (x) \log \left (\frac {b g \left (\frac {b c g-a d g}{b g}+\frac {d x}{b g}\right )}{b c g-a d g}\right )}{x} \, dx,x,a g+b g x\right )}{b g}+\frac {\left (2 B^2\right ) \operatorname {Subst}\left (\int \frac {\log (x) \log \left (\frac {-b c+a d}{d}-\frac {b x}{d}\right )}{\frac {-b c+a d}{d}-\frac {b x}{d}} \, dx,x,-c-d x\right )}{d 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}{-b c g+a d g}\right )}{x} \, dx,x,a g+b g x\right )}{b g}\\ &=-\frac {A B \log ^2(g (a+b x))}{b g}+\frac {2 B^2 \log (a+b x) \log (g (a+b x)) \log (-c-d x)}{b g}+\frac {B^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2\left (\frac {1}{c+d x}\right )}{b g}-\frac {B^2 \log (g (a+b x)) \log ^2\left (\frac {1}{c+d x}\right )}{b g}+\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (a g+b g x)}{b g}+\frac {2 A B \log \left (\frac {b (c+d x)}{b c-a d}\right ) \log (a g+b g x)}{b 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 {b (c+d x)}{b c-a d}\right ) \log (a g+b g x)}{b g}-\frac {B^2 \log \left (\frac {e (a+b x)}{c+d x}\right ) \log ^2(a g+b g x)}{b g}-\frac {B^2 \log \left (\frac {b (c+d x)}{b c-a d}\right ) \log ^2(a g+b g x)}{b g}+\frac {2 A B \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{b g}-\frac {2 B^2 \log (g (a+b x)) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{b 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 (a+b x)}{b c-a d}\right )}{b g}+\frac {B^2 \operatorname {Subst}\left (\int x^2 \, dx,x,\log (g (a+b x))\right )}{b g}-\frac {\left (2 B^2\right ) \operatorname {Subst}\left (\int \frac {\log (x) \log \left (\frac {-b c+a d}{b}-\frac {d x}{b}\right )}{x} \, dx,x,a+b x\right )}{b g}-\frac {\left (2 B^2\right ) \operatorname {Subst}\left (\int \frac {\log (x) \log \left (\frac {-b c g+a d g}{b g}-\frac {d x}{b g}\right )}{x} \, dx,x,a g+b g x\right )}{b g}+\frac {\left (2 B^2\right ) \operatorname {Subst}\left (\int \frac {\log \left (\frac {1}{x}\right ) \log \left (1+\frac {b g x}{-b c g+a d g}\right )}{x} \, dx,x,c+d x\right )}{b g}+\frac {\left (2 B^2\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {d x}{b c g-a d g}\right )}{x} \, dx,x,a g+b g x\right )}{b g}\\ &=-\frac {A B \log ^2(g (a+b x))}{b g}+\frac {B^2 \log ^3(g (a+b x))}{3 b g}-\frac {B^2 \log ^2(a+b x) \log (-c-d x)}{b g}+\frac {2 B^2 \log (a+b x) \log (g (a+b x)) \log (-c-d x)}{b g}-\frac {B^2 \log ^2(g (a+b x)) \log (-c-d x)}{b g}+\frac {B^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2\left (\frac {1}{c+d x}\right )}{b g}-\frac {B^2 \log (g (a+b x)) \log ^2\left (\frac {1}{c+d x}\right )}{b g}+\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (a g+b g x)}{b g}+\frac {2 A B \log \left (\frac {b (c+d x)}{b c-a d}\right ) \log (a g+b g x)}{b 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 {b (c+d x)}{b c-a d}\right ) \log (a g+b g x)}{b g}-\frac {B^2 \log \left (\frac {e (a+b x)}{c+d x}\right ) \log ^2(a g+b g x)}{b g}-\frac {B^2 \log \left (\frac {b (c+d x)}{b c-a d}\right ) \log ^2(a g+b g x)}{b g}+\frac {2 A B \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{b g}-\frac {2 B^2 \log (g (a+b x)) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{b 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 (a+b x)}{b c-a d}\right )}{b g}-\frac {2 B^2 \log \left (\frac {1}{c+d x}\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{b g}+\frac {2 B^2 \text {Li}_3\left (-\frac {d (a+b x)}{b c-a d}\right )}{b g}-\frac {\left (B^2 d\right ) \operatorname {Subst}\left (\int \frac {\log ^2(x)}{\frac {-b c g+a d g}{b g}-\frac {d x}{b g}} \, dx,x,a g+b g x\right )}{b^2 g^2}-\frac {\left (2 B^2\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {b g x}{-b c g+a d g}\right )}{x} \, dx,x,c+d x\right )}{b g}-\frac {\left (B^2 d\right ) \operatorname {Subst}\left (\int \frac {\log ^2(x)}{\frac {-b c+a d}{b}-\frac {d x}{b}} \, dx,x,a+b x\right )}{b^2 g}\\ &=-\frac {A B \log ^2(g (a+b x))}{b g}+\frac {B^2 \log ^3(g (a+b x))}{3 b g}-\frac {B^2 \log ^2(a+b x) \log (-c-d x)}{b g}+\frac {2 B^2 \log (a+b x) \log (g (a+b x)) \log (-c-d x)}{b g}-\frac {B^2 \log ^2(g (a+b x)) \log (-c-d x)}{b g}+\frac {B^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2\left (\frac {1}{c+d x}\right )}{b g}-\frac {B^2 \log (g (a+b x)) \log ^2\left (\frac {1}{c+d x}\right )}{b g}+\frac {B^2 \log ^2(a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b g}+\frac {B^2 \log ^2(g (a+b x)) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b g}+\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (a g+b g x)}{b g}+\frac {2 A B \log \left (\frac {b (c+d x)}{b c-a d}\right ) \log (a g+b g x)}{b 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 {b (c+d x)}{b c-a d}\right ) \log (a g+b g x)}{b g}-\frac {B^2 \log \left (\frac {e (a+b x)}{c+d x}\right ) \log ^2(a g+b g x)}{b g}-\frac {B^2 \log \left (\frac {b (c+d x)}{b c-a d}\right ) \log ^2(a g+b g x)}{b g}+\frac {2 A B \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{b g}-\frac {2 B^2 \log (g (a+b x)) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{b 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 (a+b x)}{b c-a d}\right )}{b g}-\frac {2 B^2 \log \left (\frac {1}{c+d x}\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{b g}+\frac {2 B^2 \text {Li}_3\left (-\frac {d (a+b x)}{b c-a d}\right )}{b g}-\frac {2 B^2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{b g}-\frac {\left (2 B^2\right ) \operatorname {Subst}\left (\int \frac {\log (x) \log \left (1-\frac {d x}{-b c+a d}\right )}{x} \, dx,x,a+b x\right )}{b g}-\frac {\left (2 B^2\right ) \operatorname {Subst}\left (\int \frac {\log (x) \log \left (1-\frac {d x}{-b c g+a d g}\right )}{x} \, dx,x,a g+b g x\right )}{b g}\\ &=-\frac {A B \log ^2(g (a+b x))}{b g}+\frac {B^2 \log ^3(g (a+b x))}{3 b g}-\frac {B^2 \log ^2(a+b x) \log (-c-d x)}{b g}+\frac {2 B^2 \log (a+b x) \log (g (a+b x)) \log (-c-d x)}{b g}-\frac {B^2 \log ^2(g (a+b x)) \log (-c-d x)}{b g}+\frac {B^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2\left (\frac {1}{c+d x}\right )}{b g}-\frac {B^2 \log (g (a+b x)) \log ^2\left (\frac {1}{c+d x}\right )}{b g}+\frac {B^2 \log ^2(a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b g}+\frac {B^2 \log ^2(g (a+b x)) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b g}+\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (a g+b g x)}{b g}+\frac {2 A B \log \left (\frac {b (c+d x)}{b c-a d}\right ) \log (a g+b g x)}{b 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 {b (c+d x)}{b c-a d}\right ) \log (a g+b g x)}{b g}-\frac {B^2 \log \left (\frac {e (a+b x)}{c+d x}\right ) \log ^2(a g+b g x)}{b g}-\frac {B^2 \log \left (\frac {b (c+d x)}{b c-a d}\right ) \log ^2(a g+b g x)}{b g}+\frac {2 A B \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{b g}+\frac {2 B^2 \log (a+b x) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{b 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 (a+b x)}{b c-a d}\right )}{b g}-\frac {2 B^2 \log \left (\frac {1}{c+d x}\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{b g}+\frac {2 B^2 \text {Li}_3\left (-\frac {d (a+b x)}{b c-a d}\right )}{b g}-\frac {2 B^2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{b g}-\frac {\left (2 B^2\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (\frac {d x}{-b c+a d}\right )}{x} \, dx,x,a+b x\right )}{b g}-\frac {\left (2 B^2\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (\frac {d x}{-b c g+a d g}\right )}{x} \, dx,x,a g+b g x\right )}{b g}\\ &=-\frac {A B \log ^2(g (a+b x))}{b g}+\frac {B^2 \log ^3(g (a+b x))}{3 b g}-\frac {B^2 \log ^2(a+b x) \log (-c-d x)}{b g}+\frac {2 B^2 \log (a+b x) \log (g (a+b x)) \log (-c-d x)}{b g}-\frac {B^2 \log ^2(g (a+b x)) \log (-c-d x)}{b g}+\frac {B^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2\left (\frac {1}{c+d x}\right )}{b g}-\frac {B^2 \log (g (a+b x)) \log ^2\left (\frac {1}{c+d x}\right )}{b g}+\frac {B^2 \log ^2(a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b g}+\frac {B^2 \log ^2(g (a+b x)) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b g}+\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (a g+b g x)}{b g}+\frac {2 A B \log \left (\frac {b (c+d x)}{b c-a d}\right ) \log (a g+b g x)}{b 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 {b (c+d x)}{b c-a d}\right ) \log (a g+b g x)}{b g}-\frac {B^2 \log \left (\frac {e (a+b x)}{c+d x}\right ) \log ^2(a g+b g x)}{b g}-\frac {B^2 \log \left (\frac {b (c+d x)}{b c-a d}\right ) \log ^2(a g+b g x)}{b g}+\frac {2 A B \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{b g}+\frac {2 B^2 \log (a+b x) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{b 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 (a+b x)}{b c-a d}\right )}{b g}-\frac {2 B^2 \log \left (\frac {1}{c+d x}\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{b g}-\frac {2 B^2 \text {Li}_3\left (-\frac {d (a+b x)}{b c-a d}\right )}{b g}-\frac {2 B^2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{b g}\\ \end {align*}

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Mathematica [A]  time = 0.61, size = 250, normalized size = 1.95 \[ \frac {A^2 \log (a+b x)+2 A B \log (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )-2 A B \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )+2 A B \log (a+b x) \log \left (\frac {c}{d}+x\right )-2 A B \log \left (\frac {c}{d}+x\right ) \log \left (\frac {d (a+b x)}{a d-b c}\right )+A B \log ^2\left (\frac {a}{b}+x\right )-2 A B \log \left (\frac {a}{b}+x\right ) \log (a+b x)+2 B^2 \text {Li}_2\left (\frac {b (c+d x)}{d (a+b x)}\right ) \log \left (\frac {e (a+b x)}{c+d x}\right )-B^2 \log \left (\frac {a d-b c}{d (a+b x)}\right ) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )+2 B^2 \text {Li}_3\left (\frac {b (c+d x)}{d (a+b x)}\right )}{b g} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [F]  time = 1.64, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {B^{2} \log \left (\frac {b e x + a e}{d x + c}\right )^{2} + 2 \, A B \log \left (\frac {b e x + a e}{d x + c}\right ) + A^{2}}{b g x + a g}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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maple [B]  time = 0.06, size = 1186, normalized size = 9.27 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (A+B\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )\right )}^2}{a\,g+b\,g\,x} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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