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

Optimal. Leaf size=44 \[ \frac {\left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^3}{3 B g i (b c-a d)} \]

[Out]

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

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Rubi [C]  time = 5.53, antiderivative size = 1163, normalized size of antiderivative = 26.43, number of steps used = 61, number of rules used = 29, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.690, Rules used = {2528, 2524, 12, 2418, 2390, 2301, 2394, 2393, 2391, 6688, 6742, 2411, 2344, 2317, 2507, 2488, 2506, 6610, 2500, 2433, 2375, 2374, 6589, 2440, 2434, 2499, 2396, 2302, 30} \[ -\frac {B^2 \log ^3(c+d x)}{3 (b c-a d) g i}+\frac {B^2 \log (a+b x) \log ^2(c+d x)}{(b c-a d) g i}-\frac {B^2 \log \left (\frac {e (a+b x)}{c+d x}\right ) \log ^2(c+d x)}{(b c-a d) g i}-\frac {A B \log ^2(c+d x)}{(b c-a d) g i}+\frac {B^2 \log ^2(a+b x) \log (c+d x)}{(b c-a d) g i}-\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{(b c-a d) g i}+\frac {2 A B \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{(b c-a d) g i}+\frac {2 B^2 \log (a+b x) \log \left (\frac {1}{c+d x}\right ) \log (c+d x)}{(b c-a d) g i}-\frac {2 B^2 \log \left (-\frac {d (a+b x)}{b c-a d}\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 (c+d x)}{(b c-a d) g i}-\frac {A B \log ^2(a+b x)}{(b c-a d) g i}+\frac {B^2 \log (a+b x) \log ^2\left (\frac {1}{c+d x}\right )}{(b c-a d) g i}-\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 c-a d) g i}-\frac {B^2 \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{(b c-a d) g i}-\frac {B^2 \log (a+b x) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{(b c-a d) g i}+\frac {\log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(b c-a d) g i}-\frac {B^2 \log ^2(a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{(b c-a d) g i}+\frac {2 A B \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{(b c-a d) g i}-\frac {2 B^2 \log (a+b x) \text {PolyLog}\left (2,-\frac {d (a+b x)}{b c-a d}\right )}{(b c-a d) g i}+\frac {2 A B \text {PolyLog}\left (2,-\frac {d (a+b x)}{b c-a d}\right )}{(b c-a d) g i}+\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 c-a d) g i}-\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 {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )}{(b c-a d) g i}+\frac {2 A B \text {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )}{(b c-a d) g i}+\frac {2 B^2 \log \left (\frac {e (a+b x)}{c+d x}\right ) \text {PolyLog}\left (2,\frac {b c-a d}{d (a+b x)}+1\right )}{(b c-a d) g i}+\frac {2 B^2 \text {PolyLog}\left (3,-\frac {d (a+b x)}{b c-a d}\right )}{(b c-a d) g i}+\frac {2 B^2 \text {PolyLog}\left (3,\frac {b (c+d x)}{b c-a d}\right )}{(b c-a d) g i}+\frac {2 B^2 \text {PolyLog}\left (3,\frac {b c-a d}{d (a+b x)}+1\right )}{(b c-a d) g i} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 2344

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_))), x_Symbol] :> Dist[1/d, Int[(a + b*
Log[c*x^n])^p/x, x], x] - Dist[e/d, Int[(a + b*Log[c*x^n])^p/(d + e*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 2411

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

Rule 2418

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*(RFx_), x_Symbol] :> With[{u = ExpandIntegrand[
(a + b*Log[c*(d + e*x)^n])^p, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, n}, x] && RationalFunct
ionQ[RFx, x] && IntegerQ[p]

Rule 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 2488

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

Rule 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 2506

Int[Log[v_]*Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.))^(r_.)]^(s_.)*(u_), x_Symbo
l] :> With[{g = Simplify[((v - 1)*(c + d*x))/(a + b*x)], h = Simplify[u*(a + b*x)*(c + d*x)]}, -Simp[(h*PolyLo
g[2, 1 - v]*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^s)/(b*c - a*d), x] + Dist[h*p*r*s, Int[(PolyLog[2, 1 - v]*Log
[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^(s - 1))/((a + b*x)*(c + d*x)), x], x] /; FreeQ[{g, h}, x]] /; FreeQ[{a, b,
c, d, e, f, p, q, r, s}, x] && NeQ[b*c - a*d, 0] && IGtQ[s, 0] && EqQ[p + q, 0]

Rule 2507

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

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 6610

Int[(u_)*PolyLog[n_, v_], x_Symbol] :> With[{w = DerivativeDivides[v, u*v, x]}, Simp[w*PolyLog[n + 1, v], x] /
;  !FalseQ[w]] /; FreeQ[n, x]

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}{(88 c+88 d x) (a g+b g x)} \, dx &=\int \left (\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{88 (b c-a d) g (a+b x)}-\frac {d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{88 (b c-a d) g (c+d x)}\right ) \, dx\\ &=\frac {b \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{a+b x} \, dx}{88 (b c-a d) g}-\frac {d \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{c+d x} \, dx}{88 (b c-a d) g}\\ &=\frac {\log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{88 (b c-a d) g}-\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{88 (b c-a d) g}-\frac {B \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{e (a+b x)} \, dx}{44 (b c-a d) g}+\frac {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 (c+d x)}{e (a+b x)} \, dx}{44 (b c-a d) g}\\ &=\frac {\log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{88 (b c-a d) g}-\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{88 (b c-a d) g}-\frac {B \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{a+b x} \, dx}{44 (b c-a d) e g}+\frac {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 (c+d x)}{a+b x} \, dx}{44 (b c-a d) e g}\\ &=\frac {\log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{88 (b c-a d) g}-\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{88 (b c-a d) g}-\frac {B \int \frac {(b c-a d) e \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a+b x) (c+d x)} \, dx}{44 (b c-a d) e g}+\frac {B \int \frac {(b c-a d) e \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{(a+b x) (c+d x)} \, dx}{44 (b c-a d) e g}\\ &=\frac {\log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{88 (b c-a d) g}-\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{88 (b c-a d) g}-\frac {B \int \frac {\log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a+b x) (c+d x)} \, dx}{44 g}+\frac {B \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{(a+b x) (c+d x)} \, dx}{44 g}\\ &=\frac {\log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{88 (b c-a d) g}-\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{88 (b c-a d) g}-\frac {B \int \left (\frac {A \log (a+b x)}{(a+b x) (c+d x)}+\frac {B \log (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x) (c+d x)}\right ) \, dx}{44 g}+\frac {B \int \left (\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d 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 (c+d x)}{(b c-a d) (c+d x)}\right ) \, dx}{44 g}\\ &=\frac {\log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{88 (b c-a d) g}-\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{88 (b c-a d) g}-\frac {(A B) \int \frac {\log (a+b x)}{(a+b x) (c+d x)} \, dx}{44 g}-\frac {B^2 \int \frac {\log (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x) (c+d x)} \, dx}{44 g}+\frac {(b B) \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{a+b x} \, dx}{44 (b c-a d) g}-\frac {(B d) \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{c+d x} \, dx}{44 (b c-a d) g}\\ &=-\frac {B^2 \log (a+b x) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{88 (b c-a d) g}+\frac {\log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{88 (b c-a d) g}-\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{88 (b c-a d) g}-\frac {(A B) \operatorname {Subst}\left (\int \frac {\log (x)}{x \left (\frac {b c-a d}{b}+\frac {d x}{b}\right )} \, dx,x,a+b x\right )}{44 b g}+\frac {(b B) \int \left (\frac {A \log (c+d x)}{a+b x}+\frac {B \log \left (\frac {e (a+b x)}{c+d x}\right ) \log (c+d x)}{a+b x}\right ) \, dx}{44 (b c-a d) g}+\frac {\left (b B^2\right ) \int \frac {\log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{88 (b c-a d) g}-\frac {(B d) \int \left (\frac {A \log (c+d x)}{c+d x}+\frac {B \log \left (\frac {e (a+b x)}{c+d x}\right ) \log (c+d x)}{c+d x}\right ) \, dx}{44 (b c-a d) g}\\ &=-\frac {B^2 \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{88 (b c-a d) g}-\frac {B^2 \log (a+b x) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{88 (b c-a d) g}+\frac {\log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{88 (b c-a d) g}-\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{88 (b c-a d) g}+\frac {B^2 \int \frac {\log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x) (c+d x)} \, dx}{44 g}-\frac {(A B) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{44 (b c-a d) g}+\frac {(A b B) \int \frac {\log (c+d x)}{a+b x} \, dx}{44 (b c-a d) g}+\frac {\left (b B^2\right ) \int \frac {\log \left (\frac {e (a+b x)}{c+d x}\right ) \log (c+d x)}{a+b x} \, dx}{44 (b c-a d) g}-\frac {(A B d) \int \frac {\log (c+d x)}{c+d x} \, dx}{44 (b c-a d) g}+\frac {(A B d) \operatorname {Subst}\left (\int \frac {\log (x)}{\frac {b c-a d}{b}+\frac {d x}{b}} \, dx,x,a+b x\right )}{44 b (b c-a d) g}-\frac {\left (B^2 d\right ) \int \frac {\log \left (\frac {e (a+b x)}{c+d x}\right ) \log (c+d x)}{c+d x} \, dx}{44 (b c-a d) g}\\ &=-\frac {A B \log ^2(a+b x)}{88 (b c-a d) g}-\frac {B^2 \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{88 (b c-a d) g}-\frac {B^2 \log (a+b x) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{88 (b c-a d) g}+\frac {\log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{88 (b c-a d) g}+\frac {A B \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{44 (b c-a d) g}-\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{88 (b c-a d) g}-\frac {B^2 \log \left (\frac {e (a+b x)}{c+d x}\right ) \log ^2(c+d x)}{88 (b c-a d) g}+\frac {A B \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{44 (b c-a d) g}+\frac {B^2 \log \left (\frac {e (a+b x)}{c+d x}\right ) \text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{44 (b c-a d) g}-\frac {B^2 \int \frac {\text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{(a+b x) (c+d x)} \, dx}{44 g}-\frac {(A B) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{44 (b c-a d) g}-\frac {(A B) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{44 (b c-a d) g}+\frac {\left (b B^2\right ) \int \frac {\log ^2(c+d x)}{a+b x} \, dx}{88 (b c-a d) g}+\frac {\left (b B^2\right ) \int \frac {\log (a+b x) \log (c+d x)}{a+b x} \, dx}{44 (b c-a d) g}+\frac {\left (b B^2\right ) \int \frac {\log \left (\frac {1}{c+d x}\right ) \log (c+d x)}{a+b x} \, dx}{44 (b c-a d) g}-\frac {(A B d) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{44 (b c-a d) g}-\frac {\left (B^2 d\right ) \int \frac {\log ^2(c+d x)}{c+d x} \, dx}{88 (b c-a d) g}+\frac {\left (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 (c+d x)}{a+b x} \, dx}{44 (b c-a d) g}\\ &=-\frac {A B \log ^2(a+b x)}{88 (b c-a d) g}-\frac {B^2 \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{88 (b c-a d) g}-\frac {B^2 \log (a+b x) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{88 (b c-a d) g}+\frac {\log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{88 (b c-a d) g}+\frac {A B \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{44 (b c-a d) g}-\frac {B^2 \log \left (-\frac {d (a+b x)}{b c-a d}\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 (c+d x)}{44 (b c-a d) g}-\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{88 (b c-a d) g}-\frac {A B \log ^2(c+d x)}{88 (b c-a d) g}+\frac {B^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(c+d x)}{88 (b c-a d) g}-\frac {B^2 \log \left (\frac {e (a+b x)}{c+d x}\right ) \log ^2(c+d x)}{88 (b c-a d) g}+\frac {A B \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{44 (b c-a d) g}+\frac {A B \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{44 (b c-a d) g}+\frac {B^2 \log \left (\frac {e (a+b x)}{c+d x}\right ) \text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{44 (b c-a d) g}+\frac {B^2 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{44 (b c-a d) g}-\frac {(A B) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{44 (b c-a d) g}-\frac {B^2 \operatorname {Subst}\left (\int \frac {\log ^2(x)}{x} \, dx,x,c+d x\right )}{88 (b c-a d) g}+\frac {B^2 \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 )}{44 (b c-a d) g}+\frac {B^2 \operatorname {Subst}\left (\int \frac {\log \left (\frac {1}{-\frac {-b c+a d}{b}+\frac {d x}{b}}\right ) \log \left (-\frac {-b c+a d}{b}+\frac {d x}{b}\right )}{x} \, dx,x,a+b x\right )}{44 (b c-a d) g}-\frac {\left (B^2 d\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right ) \log (c+d x)}{c+d x} \, dx}{44 (b c-a d) g}-\frac {\left (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 \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{44 (b c-a d) g}\\ &=-\frac {A B \log ^2(a+b x)}{88 (b c-a d) g}-\frac {B^2 \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{88 (b c-a d) g}-\frac {B^2 \log (a+b x) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{88 (b c-a d) g}+\frac {\log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{88 (b c-a d) g}+\frac {B^2 \log ^2(a+b x) \log (c+d x)}{88 (b c-a d) g}+\frac {A B \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{44 (b c-a d) g}+\frac {B^2 \log (a+b x) \log \left (\frac {1}{c+d x}\right ) \log (c+d x)}{44 (b c-a d) g}-\frac {B^2 \log \left (-\frac {d (a+b x)}{b c-a d}\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 (c+d x)}{44 (b c-a d) g}-\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{88 (b c-a d) g}-\frac {A B \log ^2(c+d x)}{88 (b c-a d) g}+\frac {B^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(c+d x)}{88 (b c-a d) g}-\frac {B^2 \log \left (\frac {e (a+b x)}{c+d x}\right ) \log ^2(c+d x)}{88 (b c-a d) g}+\frac {A B \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{44 (b c-a d) g}+\frac {A B \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{44 (b c-a d) g}+\frac {A B \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{44 (b c-a d) g}+\frac {B^2 \log \left (\frac {e (a+b x)}{c+d x}\right ) \text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{44 (b c-a d) g}+\frac {B^2 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{44 (b c-a d) g}-\frac {B^2 \operatorname {Subst}\left (\int x^2 \, dx,x,\log (c+d x)\right )}{88 (b c-a d) g}-\frac {B^2 \operatorname {Subst}\left (\int \frac {\log (x) \log \left (\frac {d \left (\frac {-b c+a d}{d}+\frac {b x}{d}\right )}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{44 (b c-a d) 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 )}{88 b (b c-a d) g}-\frac {\left (B^2 d\right ) \operatorname {Subst}\left (\int \frac {\log (x) \log \left (\frac {1}{-\frac {-b c+a d}{b}+\frac {d x}{b}}\right )}{-\frac {-b c+a d}{b}+\frac {d x}{b}} \, dx,x,a+b x\right )}{44 b (b c-a d) g}+\frac {\left (B^2 d\right ) \operatorname {Subst}\left (\int \frac {\log (x) \log \left (-\frac {-b c+a d}{b}+\frac {d x}{b}\right )}{-\frac {-b c+a d}{b}+\frac {d x}{b}} \, dx,x,a+b x\right )}{44 b (b c-a d) g}-\frac {\left (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 c+a d}\right )}{x} \, dx,x,c+d x\right )}{44 (b c-a d) g}\\ &=-\frac {A B \log ^2(a+b x)}{88 (b c-a d) g}-\frac {B^2 \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{88 (b c-a d) g}-\frac {B^2 \log (a+b x) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{88 (b c-a d) g}+\frac {\log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{88 (b c-a d) g}+\frac {B^2 \log ^2(a+b x) \log (c+d x)}{88 (b c-a d) g}+\frac {A B \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{44 (b c-a d) g}+\frac {B^2 \log (a+b x) \log \left (\frac {1}{c+d x}\right ) \log (c+d x)}{44 (b c-a d) g}-\frac {B^2 \log \left (-\frac {d (a+b x)}{b c-a d}\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 (c+d x)}{44 (b c-a d) g}-\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{88 (b c-a d) g}-\frac {A B \log ^2(c+d x)}{88 (b c-a d) g}+\frac {B^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(c+d x)}{88 (b c-a d) g}-\frac {B^2 \log \left (\frac {e (a+b x)}{c+d x}\right ) \log ^2(c+d x)}{88 (b c-a d) g}-\frac {B^2 \log ^3(c+d x)}{264 (b c-a d) g}+\frac {A B \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{44 (b c-a d) g}-\frac {B^2 \log ^2(a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{88 (b c-a d) g}+\frac {A B \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{44 (b c-a d) g}+\frac {A B \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{44 (b c-a d) g}-\frac {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 (c+d x)}{b c-a d}\right )}{44 (b c-a d) g}+\frac {B^2 \log (c+d x) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{44 (b c-a d) g}+\frac {B^2 \log \left (\frac {e (a+b x)}{c+d x}\right ) \text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{44 (b c-a d) g}+\frac {B^2 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{44 (b c-a d) g}-\frac {B^2 \operatorname {Subst}\left (\int \frac {\log \left (\frac {1}{x}\right ) \log \left (\frac {-b c+a d}{d}+\frac {b x}{d}\right )}{x} \, dx,x,c+d x\right )}{44 (b c-a d) g}+\frac {B^2 \operatorname {Subst}\left (\int \frac {\log (x) \log \left (\frac {-b c+a d}{d}+\frac {b x}{d}\right )}{x} \, dx,x,c+d x\right )}{44 (b c-a d) g}+\frac {B^2 \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 )}{44 (b c-a d) g}-\frac {B^2 \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{44 (b c-a d) g}\\ &=-\frac {A B \log ^2(a+b x)}{88 (b c-a d) g}+\frac {B^2 \log (a+b x) \log ^2\left (\frac {1}{c+d x}\right )}{88 (b c-a d) g}-\frac {B^2 \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{88 (b c-a d) g}-\frac {B^2 \log (a+b x) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{88 (b c-a d) g}+\frac {\log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{88 (b c-a d) g}+\frac {B^2 \log ^2(a+b x) \log (c+d x)}{88 (b c-a d) g}+\frac {A B \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{44 (b c-a d) g}+\frac {B^2 \log (a+b x) \log \left (\frac {1}{c+d x}\right ) \log (c+d x)}{44 (b c-a d) g}-\frac {B^2 \log \left (-\frac {d (a+b x)}{b c-a d}\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 (c+d x)}{44 (b c-a d) g}-\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{88 (b c-a d) g}-\frac {A B \log ^2(c+d x)}{88 (b c-a d) g}+\frac {B^2 \log (a+b x) \log ^2(c+d x)}{88 (b c-a d) g}+\frac {B^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(c+d x)}{88 (b c-a d) g}-\frac {B^2 \log \left (\frac {e (a+b x)}{c+d x}\right ) \log ^2(c+d x)}{88 (b c-a d) g}-\frac {B^2 \log ^3(c+d x)}{264 (b c-a d) g}+\frac {A B \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{44 (b c-a d) g}-\frac {B^2 \log ^2(a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{88 (b c-a d) g}+\frac {A B \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{44 (b c-a d) g}-\frac {B^2 \log (a+b x) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{44 (b c-a d) g}+\frac {A B \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{44 (b c-a d) g}-\frac {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 (c+d x)}{b c-a d}\right )}{44 (b c-a d) g}+\frac {B^2 \log (c+d x) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{44 (b c-a d) g}+\frac {B^2 \log \left (\frac {e (a+b x)}{c+d x}\right ) \text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{44 (b c-a d) g}-\frac {B^2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{44 (b c-a d) g}+\frac {B^2 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{44 (b c-a d) g}+\frac {B^2 \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{44 (b c-a d) g}-\frac {\left (b B^2\right ) \operatorname {Subst}\left (\int \frac {\log ^2\left (\frac {1}{x}\right )}{\frac {-b c+a d}{d}+\frac {b x}{d}} \, dx,x,c+d x\right )}{88 d (b c-a d) g}-\frac {\left (b B^2\right ) \operatorname {Subst}\left (\int \frac {\log ^2(x)}{\frac {-b c+a d}{d}+\frac {b x}{d}} \, dx,x,c+d x\right )}{88 d (b c-a d) g}\\ &=-\frac {A B \log ^2(a+b x)}{88 (b c-a d) g}+\frac {B^2 \log (a+b x) \log ^2\left (\frac {1}{c+d x}\right )}{88 (b c-a d) 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 )}{88 (b c-a d) g}-\frac {B^2 \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{88 (b c-a d) g}-\frac {B^2 \log (a+b x) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{88 (b c-a d) g}+\frac {\log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{88 (b c-a d) g}+\frac {B^2 \log ^2(a+b x) \log (c+d x)}{88 (b c-a d) g}+\frac {A B \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{44 (b c-a d) g}+\frac {B^2 \log (a+b x) \log \left (\frac {1}{c+d x}\right ) \log (c+d x)}{44 (b c-a d) g}-\frac {B^2 \log \left (-\frac {d (a+b x)}{b c-a d}\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 (c+d x)}{44 (b c-a d) g}-\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{88 (b c-a d) g}-\frac {A B \log ^2(c+d x)}{88 (b c-a d) g}+\frac {B^2 \log (a+b x) \log ^2(c+d x)}{88 (b c-a d) g}-\frac {B^2 \log \left (\frac {e (a+b x)}{c+d x}\right ) \log ^2(c+d x)}{88 (b c-a d) g}-\frac {B^2 \log ^3(c+d x)}{264 (b c-a d) g}+\frac {A B \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{44 (b c-a d) g}-\frac {B^2 \log ^2(a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{88 (b c-a d) g}+\frac {A B \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{44 (b c-a d) g}-\frac {B^2 \log (a+b x) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{44 (b c-a d) g}+\frac {A B \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{44 (b c-a d) g}-\frac {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 (c+d x)}{b c-a d}\right )}{44 (b c-a d) g}+\frac {B^2 \log (c+d x) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{44 (b c-a d) g}+\frac {B^2 \log \left (\frac {e (a+b x)}{c+d x}\right ) \text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{44 (b c-a d) g}+\frac {B^2 \text {Li}_3\left (-\frac {d (a+b x)}{b c-a d}\right )}{44 (b c-a d) g}-\frac {B^2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{44 (b c-a d) g}+\frac {B^2 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{44 (b c-a d) g}-\frac {B^2 \operatorname {Subst}\left (\int \frac {\log \left (\frac {1}{x}\right ) \log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{44 (b c-a d) g}+\frac {B^2 \operatorname {Subst}\left (\int \frac {\log (x) \log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{44 (b c-a d) g}\\ &=-\frac {A B \log ^2(a+b x)}{88 (b c-a d) g}+\frac {B^2 \log (a+b x) \log ^2\left (\frac {1}{c+d x}\right )}{88 (b c-a d) 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 )}{88 (b c-a d) g}-\frac {B^2 \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{88 (b c-a d) g}-\frac {B^2 \log (a+b x) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{88 (b c-a d) g}+\frac {\log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{88 (b c-a d) g}+\frac {B^2 \log ^2(a+b x) \log (c+d x)}{88 (b c-a d) g}+\frac {A B \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{44 (b c-a d) g}+\frac {B^2 \log (a+b x) \log \left (\frac {1}{c+d x}\right ) \log (c+d x)}{44 (b c-a d) g}-\frac {B^2 \log \left (-\frac {d (a+b x)}{b c-a d}\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 (c+d x)}{44 (b c-a d) g}-\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{88 (b c-a d) g}-\frac {A B \log ^2(c+d x)}{88 (b c-a d) g}+\frac {B^2 \log (a+b x) \log ^2(c+d x)}{88 (b c-a d) g}-\frac {B^2 \log \left (\frac {e (a+b x)}{c+d x}\right ) \log ^2(c+d x)}{88 (b c-a d) g}-\frac {B^2 \log ^3(c+d x)}{264 (b c-a d) g}+\frac {A B \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{44 (b c-a d) g}-\frac {B^2 \log ^2(a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{88 (b c-a d) g}+\frac {A B \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{44 (b c-a d) g}-\frac {B^2 \log (a+b x) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{44 (b c-a d) g}+\frac {A B \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{44 (b c-a d) g}+\frac {B^2 \log \left (\frac {1}{c+d x}\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{44 (b c-a d) g}-\frac {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 (c+d x)}{b c-a d}\right )}{44 (b c-a d) g}+\frac {B^2 \log \left (\frac {e (a+b x)}{c+d x}\right ) \text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{44 (b c-a d) g}+\frac {B^2 \text {Li}_3\left (-\frac {d (a+b x)}{b c-a d}\right )}{44 (b c-a d) g}-\frac {B^2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{44 (b c-a d) g}+\frac {B^2 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{44 (b c-a d) g}+2 \frac {B^2 \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{44 (b c-a d) g}\\ &=-\frac {A B \log ^2(a+b x)}{88 (b c-a d) g}+\frac {B^2 \log (a+b x) \log ^2\left (\frac {1}{c+d x}\right )}{88 (b c-a d) 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 )}{88 (b c-a d) g}-\frac {B^2 \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{88 (b c-a d) g}-\frac {B^2 \log (a+b x) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{88 (b c-a d) g}+\frac {\log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{88 (b c-a d) g}+\frac {B^2 \log ^2(a+b x) \log (c+d x)}{88 (b c-a d) g}+\frac {A B \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{44 (b c-a d) g}+\frac {B^2 \log (a+b x) \log \left (\frac {1}{c+d x}\right ) \log (c+d x)}{44 (b c-a d) g}-\frac {B^2 \log \left (-\frac {d (a+b x)}{b c-a d}\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 (c+d x)}{44 (b c-a d) g}-\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{88 (b c-a d) g}-\frac {A B \log ^2(c+d x)}{88 (b c-a d) g}+\frac {B^2 \log (a+b x) \log ^2(c+d x)}{88 (b c-a d) g}-\frac {B^2 \log \left (\frac {e (a+b x)}{c+d x}\right ) \log ^2(c+d x)}{88 (b c-a d) g}-\frac {B^2 \log ^3(c+d x)}{264 (b c-a d) g}+\frac {A B \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{44 (b c-a d) g}-\frac {B^2 \log ^2(a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{88 (b c-a d) g}+\frac {A B \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{44 (b c-a d) g}-\frac {B^2 \log (a+b x) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{44 (b c-a d) g}+\frac {A B \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{44 (b c-a d) g}+\frac {B^2 \log \left (\frac {1}{c+d x}\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{44 (b c-a d) g}-\frac {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 (c+d x)}{b c-a d}\right )}{44 (b c-a d) g}+\frac {B^2 \log \left (\frac {e (a+b x)}{c+d x}\right ) \text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{44 (b c-a d) g}+\frac {B^2 \text {Li}_3\left (-\frac {d (a+b x)}{b c-a d}\right )}{44 (b c-a d) g}+\frac {B^2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{44 (b c-a d) g}+\frac {B^2 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{44 (b c-a d) g}\\ \end {align*}

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Mathematica [A]  time = 0.37, size = 79, normalized size = 1.80 \[ \frac {3 A^2 \log \left (\frac {e (a+b x)}{c+d x}\right )+3 A B \log ^2\left (\frac {e (a+b x)}{c+d x}\right )+B^2 \log ^3\left (\frac {e (a+b x)}{c+d x}\right )}{3 b c g i-3 a d g i} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 0.88, size = 87, normalized size = 1.98 \[ \frac {B^{2} \log \left (\frac {b e x + a e}{d x + c}\right )^{3} + 3 \, A B \log \left (\frac {b e x + a e}{d x + c}\right )^{2} + 3 \, A^{2} \log \left (\frac {b e x + a e}{d x + c}\right )}{3 \, {\left (b c - a d\right )} g i} \]

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)/(d*i*x+c*i),x, algorithm="fricas")

[Out]

1/3*(B^2*log((b*e*x + a*e)/(d*x + c))^3 + 3*A*B*log((b*e*x + a*e)/(d*x + c))^2 + 3*A^2*log((b*e*x + a*e)/(d*x
+ c)))/((b*c - a*d)*g*i)

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giac [B]  time = 0.68, size = 145, normalized size = 3.30 \[ -\frac {{\left (B^{2} i e \log \left (\frac {b x e + a e}{d x + c}\right )^{3} + 3 \, A B i e \log \left (\frac {b x e + a e}{d x + c}\right )^{2} + 3 \, A^{2} i e \log \left (\frac {b x e + a e}{d x + c}\right )\right )} {\left (\frac {b c}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}} - \frac {a d}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}}\right )}}{3 \, g} \]

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)/(d*i*x+c*i),x, algorithm="giac")

[Out]

-1/3*(B^2*i*e*log((b*x*e + a*e)/(d*x + c))^3 + 3*A*B*i*e*log((b*x*e + a*e)/(d*x + c))^2 + 3*A^2*i*e*log((b*x*e
 + a*e)/(d*x + c)))*(b*c/((b*c*e - a*d*e)*(b*c - a*d)) - a*d/((b*c*e - a*d*e)*(b*c - a*d)))/g

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maple [B]  time = 0.05, size = 312, normalized size = 7.09 \[ -\frac {B^{2} a d \ln \left (\frac {b e}{d}+\frac {\left (a d -b c \right ) e}{\left (d x +c \right ) d}\right )^{3}}{3 \left (a d -b c \right )^{2} g i}+\frac {B^{2} b c \ln \left (\frac {b e}{d}+\frac {\left (a d -b c \right ) e}{\left (d x +c \right ) d}\right )^{3}}{3 \left (a d -b c \right )^{2} g i}-\frac {A B a d \ln \left (\frac {b e}{d}+\frac {\left (a d -b c \right ) e}{\left (d x +c \right ) d}\right )^{2}}{\left (a d -b c \right )^{2} g i}+\frac {A B b c \ln \left (\frac {b e}{d}+\frac {\left (a d -b c \right ) e}{\left (d x +c \right ) d}\right )^{2}}{\left (a d -b c \right )^{2} g i}-\frac {A^{2} a d \ln \left (\frac {b e}{d}+\frac {\left (a d -b c \right ) e}{\left (d x +c \right ) d}\right )}{\left (a d -b c \right )^{2} g i}+\frac {A^{2} b c \ln \left (\frac {b e}{d}+\frac {\left (a d -b c \right ) e}{\left (d x +c \right ) d}\right )}{\left (a d -b c \right )^{2} g i} \]

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)/(d*i*x+c*i),x)

[Out]

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

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maxima [B]  time = 1.35, size = 397, normalized size = 9.02 \[ B^{2} {\left (\frac {\log \left (b x + a\right )}{{\left (b c - a d\right )} g i} - \frac {\log \left (d x + c\right )}{{\left (b c - a d\right )} g i}\right )} \log \left (\frac {b e x}{d x + c} + \frac {a e}{d x + c}\right )^{2} + 2 \, A B {\left (\frac {\log \left (b x + a\right )}{{\left (b c - a d\right )} g i} - \frac {\log \left (d x + c\right )}{{\left (b c - a d\right )} g i}\right )} \log \left (\frac {b e x}{d x + c} + \frac {a e}{d x + c}\right ) - \frac {1}{3} \, B^{2} {\left (\frac {3 \, {\left (\log \left (b x + a\right )^{2} - 2 \, \log \left (b x + a\right ) \log \left (d x + c\right ) + \log \left (d x + c\right )^{2}\right )} \log \left (\frac {b e x}{d x + c} + \frac {a e}{d x + c}\right )}{b c g i - a d g i} - \frac {\log \left (b x + a\right )^{3} - 3 \, \log \left (b x + a\right )^{2} \log \left (d x + c\right ) + 3 \, \log \left (b x + a\right ) \log \left (d x + c\right )^{2} - \log \left (d x + c\right )^{3}}{b c g i - a d g i}\right )} + A^{2} {\left (\frac {\log \left (b x + a\right )}{{\left (b c - a d\right )} g i} - \frac {\log \left (d x + c\right )}{{\left (b c - a d\right )} g i}\right )} - \frac {{\left (\log \left (b x + a\right )^{2} - 2 \, \log \left (b x + a\right ) \log \left (d x + c\right ) + \log \left (d x + c\right )^{2}\right )} A B}{b c g i - a d g i} \]

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)/(d*i*x+c*i),x, algorithm="maxima")

[Out]

B^2*(log(b*x + a)/((b*c - a*d)*g*i) - log(d*x + c)/((b*c - a*d)*g*i))*log(b*e*x/(d*x + c) + a*e/(d*x + c))^2 +
 2*A*B*(log(b*x + a)/((b*c - a*d)*g*i) - log(d*x + c)/((b*c - a*d)*g*i))*log(b*e*x/(d*x + c) + a*e/(d*x + c))
- 1/3*B^2*(3*(log(b*x + a)^2 - 2*log(b*x + a)*log(d*x + c) + log(d*x + c)^2)*log(b*e*x/(d*x + c) + a*e/(d*x +
c))/(b*c*g*i - a*d*g*i) - (log(b*x + a)^3 - 3*log(b*x + a)^2*log(d*x + c) + 3*log(b*x + a)*log(d*x + c)^2 - lo
g(d*x + c)^3)/(b*c*g*i - a*d*g*i)) + A^2*(log(b*x + a)/((b*c - a*d)*g*i) - log(d*x + c)/((b*c - a*d)*g*i)) - (
log(b*x + a)^2 - 2*log(b*x + a)*log(d*x + c) + log(d*x + c)^2)*A*B/(b*c*g*i - a*d*g*i)

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mupad [B]  time = 5.76, size = 96, normalized size = 2.18 \[ -\frac {-6{}\mathrm {i}\,\mathrm {atan}\left (\frac {a\,d\,1{}\mathrm {i}+b\,c\,1{}\mathrm {i}+b\,d\,x\,2{}\mathrm {i}}{a\,d-b\,c}\right )\,A^2+3\,A\,B\,{\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}^2+B^2\,{\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}^3}{3\,g\,i\,\left (a\,d-b\,c\right )} \]

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)*(c*i + d*i*x)),x)

[Out]

-(B^2*log((e*(a + b*x))/(c + d*x))^3 - A^2*atan((a*d*1i + b*c*1i + b*d*x*2i)/(a*d - b*c))*6i + 3*A*B*log((e*(a
 + b*x))/(c + d*x))^2)/(3*g*i*(a*d - b*c))

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sympy [B]  time = 1.63, size = 206, normalized size = 4.68 \[ A^{2} \left (\frac {\log {\left (x + \frac {- \frac {a^{2} d^{2}}{a d - b c} + \frac {2 a b c d}{a d - b c} + a d - \frac {b^{2} c^{2}}{a d - b c} + b c}{2 b d} \right )}}{g i \left (a d - b c\right )} - \frac {\log {\left (x + \frac {\frac {a^{2} d^{2}}{a d - b c} - \frac {2 a b c d}{a d - b c} + a d + \frac {b^{2} c^{2}}{a d - b c} + b c}{2 b d} \right )}}{g i \left (a d - b c\right )}\right ) - \frac {A B \log {\left (\frac {e \left (a + b x\right )}{c + d x} \right )}^{2}}{a d g i - b c g i} - \frac {B^{2} \log {\left (\frac {e \left (a + b x\right )}{c + d x} \right )}^{3}}{3 a d g i - 3 b c g i} \]

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)/(d*i*x+c*i),x)

[Out]

A**2*(log(x + (-a**2*d**2/(a*d - b*c) + 2*a*b*c*d/(a*d - b*c) + a*d - b**2*c**2/(a*d - b*c) + b*c)/(2*b*d))/(g
*i*(a*d - b*c)) - log(x + (a**2*d**2/(a*d - b*c) - 2*a*b*c*d/(a*d - b*c) + a*d + b**2*c**2/(a*d - b*c) + b*c)/
(2*b*d))/(g*i*(a*d - b*c))) - A*B*log(e*(a + b*x)/(c + d*x))**2/(a*d*g*i - b*c*g*i) - B**2*log(e*(a + b*x)/(c
+ d*x))**3/(3*a*d*g*i - 3*b*c*g*i)

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