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

Optimal. Leaf size=442 \[ \frac{4 B d i^2 (b c-a d) \text{PolyLog}\left (2,\frac{b (c+d x)}{d (a+b x)}\right ) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{b^3 g^2}+\frac{2 B^2 d i^2 (b c-a d) \text{PolyLog}\left (2,\frac{d (a+b x)}{b (c+d x)}\right )}{b^3 g^2}+\frac{4 B^2 d i^2 (b c-a d) \text{PolyLog}\left (3,\frac{b (c+d x)}{d (a+b x)}\right )}{b^3 g^2}+\frac{d^2 i^2 (a+b x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )^2}{b^3 g^2}-\frac{i^2 (c+d x) (b c-a d) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )^2}{b^2 g^2 (a+b x)}-\frac{2 B i^2 (c+d x) (b c-a d) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{b^2 g^2 (a+b x)}+\frac{2 B d i^2 (b c-a d) \log \left (\frac{b c-a d}{b (c+d x)}\right ) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{b^3 g^2}-\frac{2 d i^2 (b c-a d) \log \left (1-\frac{b (c+d x)}{d (a+b x)}\right ) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )^2}{b^3 g^2}-\frac{2 B^2 i^2 (c+d x) (b c-a d)}{b^2 g^2 (a+b x)} \]

[Out]

(-2*B^2*(b*c - a*d)*i^2*(c + d*x))/(b^2*g^2*(a + b*x)) - (2*B*(b*c - a*d)*i^2*(c + d*x)*(A + B*Log[(e*(a + b*x
))/(c + d*x)]))/(b^2*g^2*(a + b*x)) + (2*B*d*(b*c - a*d)*i^2*Log[(b*c - a*d)/(b*(c + d*x))]*(A + B*Log[(e*(a +
b*x))/(c + d*x)]))/(b^3*g^2) + (d^2*i^2*(a + b*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(b^3*g^2) - ((b*c -
a*d)*i^2*(c + d*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(b^2*g^2*(a + b*x)) - (2*d*(b*c - a*d)*i^2*(A + B*
Log[(e*(a + b*x))/(c + d*x)])^2*Log[1 - (b*(c + d*x))/(d*(a + b*x))])/(b^3*g^2) + (2*B^2*d*(b*c - a*d)*i^2*Pol
yLog[2, (d*(a + b*x))/(b*(c + d*x))])/(b^3*g^2) + (4*B*d*(b*c - a*d)*i^2*(A + B*Log[(e*(a + b*x))/(c + d*x)])*
PolyLog[2, (b*(c + d*x))/(d*(a + b*x))])/(b^3*g^2) + (4*B^2*d*(b*c - a*d)*i^2*PolyLog[3, (b*(c + d*x))/(d*(a +
b*x))])/(b^3*g^2)

________________________________________________________________________________________

Rubi [B]  time = 3.88662, antiderivative size = 1219, normalized size of antiderivative = 2.76, number of steps used = 65, number of rules used = 21, integrand size = 42, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {2528, 2523, 12, 2524, 2418, 2390, 2301, 2394, 2393, 2391, 2525, 44, 6688, 6742, 2411, 2344, 2317, 2507, 2488, 2506, 6610} \[ -\frac{a B^2 d^2 \log ^2(a+b x) i^2}{b^3 g^2}+\frac{B^2 d (b c-a d) \log ^2(a+b x) i^2}{b^3 g^2}-\frac{2 A B d (b c-a d) \log ^2(a+b x) i^2}{b^3 g^2}-\frac{2 B^2 d (b c-a d) \log \left (-\frac{b c-a d}{d (a+b x)}\right ) \log ^2\left (\frac{e (a+b x)}{c+d x}\right ) i^2}{b^3 g^2}-\frac{2 B^2 d (b c-a d) \log (a+b x) \log ^2\left (\frac{e (a+b x)}{c+d x}\right ) i^2}{b^3 g^2}+\frac{d^2 x \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2 i^2}{b^2 g^2}+\frac{2 d (b c-a d) \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2 i^2}{b^3 g^2}-\frac{(b c-a d)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2 i^2}{b^3 g^2 (a+b x)}-\frac{B^2 c d \log ^2(c+d x) i^2}{b^2 g^2}+\frac{B^2 d (b c-a d) \log ^2(c+d x) i^2}{b^3 g^2}-\frac{2 B^2 d (b c-a d) \log (a+b x) i^2}{b^3 g^2}+\frac{2 a B d^2 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) i^2}{b^3 g^2}-\frac{2 B d (b c-a d) \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) i^2}{b^3 g^2}-\frac{2 B (b c-a d)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) i^2}{b^3 g^2 (a+b x)}+\frac{2 B^2 c d \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x) i^2}{b^2 g^2}-\frac{2 B^2 d (b c-a d) \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x) i^2}{b^3 g^2}-\frac{2 B c d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x) i^2}{b^2 g^2}+\frac{2 B d (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x) i^2}{b^3 g^2}+\frac{2 B^2 d (b c-a d) \log (c+d x) i^2}{b^3 g^2}+\frac{2 a B^2 d^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right ) i^2}{b^3 g^2}-\frac{2 B^2 d (b c-a d) \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right ) i^2}{b^3 g^2}+\frac{4 A B d (b c-a d) \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right ) i^2}{b^3 g^2}+\frac{2 a B^2 d^2 \text{PolyLog}\left (2,-\frac{d (a+b x)}{b c-a d}\right ) i^2}{b^3 g^2}-\frac{2 B^2 d (b c-a d) \text{PolyLog}\left (2,-\frac{d (a+b x)}{b c-a d}\right ) i^2}{b^3 g^2}+\frac{4 A B d (b c-a d) \text{PolyLog}\left (2,-\frac{d (a+b x)}{b c-a d}\right ) i^2}{b^3 g^2}+\frac{2 B^2 c d \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right ) i^2}{b^2 g^2}-\frac{2 B^2 d (b c-a d) \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right ) i^2}{b^3 g^2}+\frac{4 B^2 d (b c-a d) \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 ) i^2}{b^3 g^2}+\frac{4 B^2 d (b c-a d) \text{PolyLog}\left (3,\frac{b c-a d}{d (a+b x)}+1\right ) i^2}{b^3 g^2}-\frac{2 B^2 (b c-a d)^2 i^2}{b^3 g^2 (a+b x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*B^2*(b*c - a*d)^2*i^2)/(b^3*g^2*(a + b*x)) - (2*B^2*d*(b*c - a*d)*i^2*Log[a + b*x])/(b^3*g^2) - (a*B^2*d^2
*i^2*Log[a + b*x]^2)/(b^3*g^2) - (2*A*B*d*(b*c - a*d)*i^2*Log[a + b*x]^2)/(b^3*g^2) + (B^2*d*(b*c - a*d)*i^2*L
og[a + b*x]^2)/(b^3*g^2) - (2*B^2*d*(b*c - a*d)*i^2*Log[-((b*c - a*d)/(d*(a + b*x)))]*Log[(e*(a + b*x))/(c + d
*x)]^2)/(b^3*g^2) - (2*B^2*d*(b*c - a*d)*i^2*Log[a + b*x]*Log[(e*(a + b*x))/(c + d*x)]^2)/(b^3*g^2) - (2*B*(b*
c - a*d)^2*i^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(b^3*g^2*(a + b*x)) + (2*a*B*d^2*i^2*Log[a + b*x]*(A + B*
Log[(e*(a + b*x))/(c + d*x)]))/(b^3*g^2) - (2*B*d*(b*c - a*d)*i^2*Log[a + b*x]*(A + B*Log[(e*(a + b*x))/(c + d
*x)]))/(b^3*g^2) + (d^2*i^2*x*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(b^2*g^2) - ((b*c - a*d)^2*i^2*(A + B*Lo
g[(e*(a + b*x))/(c + d*x)])^2)/(b^3*g^2*(a + b*x)) + (2*d*(b*c - a*d)*i^2*Log[a + b*x]*(A + B*Log[(e*(a + b*x)
)/(c + d*x)])^2)/(b^3*g^2) + (2*B^2*d*(b*c - a*d)*i^2*Log[c + d*x])/(b^3*g^2) + (2*B^2*c*d*i^2*Log[-((d*(a + b
*x))/(b*c - a*d))]*Log[c + d*x])/(b^2*g^2) - (2*B^2*d*(b*c - a*d)*i^2*Log[-((d*(a + b*x))/(b*c - a*d))]*Log[c
+ d*x])/(b^3*g^2) - (2*B*c*d*i^2*(A + B*Log[(e*(a + b*x))/(c + d*x)])*Log[c + d*x])/(b^2*g^2) + (2*B*d*(b*c -
a*d)*i^2*(A + B*Log[(e*(a + b*x))/(c + d*x)])*Log[c + d*x])/(b^3*g^2) - (B^2*c*d*i^2*Log[c + d*x]^2)/(b^2*g^2)
+ (B^2*d*(b*c - a*d)*i^2*Log[c + d*x]^2)/(b^3*g^2) + (2*a*B^2*d^2*i^2*Log[a + b*x]*Log[(b*(c + d*x))/(b*c - a
*d)])/(b^3*g^2) + (4*A*B*d*(b*c - a*d)*i^2*Log[a + b*x]*Log[(b*(c + d*x))/(b*c - a*d)])/(b^3*g^2) - (2*B^2*d*(
b*c - a*d)*i^2*Log[a + b*x]*Log[(b*(c + d*x))/(b*c - a*d)])/(b^3*g^2) + (2*a*B^2*d^2*i^2*PolyLog[2, -((d*(a +
b*x))/(b*c - a*d))])/(b^3*g^2) + (4*A*B*d*(b*c - a*d)*i^2*PolyLog[2, -((d*(a + b*x))/(b*c - a*d))])/(b^3*g^2)
- (2*B^2*d*(b*c - a*d)*i^2*PolyLog[2, -((d*(a + b*x))/(b*c - a*d))])/(b^3*g^2) + (2*B^2*c*d*i^2*PolyLog[2, (b*
(c + d*x))/(b*c - a*d)])/(b^2*g^2) - (2*B^2*d*(b*c - a*d)*i^2*PolyLog[2, (b*(c + d*x))/(b*c - a*d)])/(b^3*g^2)
+ (4*B^2*d*(b*c - a*d)*i^2*Log[(e*(a + b*x))/(c + d*x)]*PolyLog[2, 1 + (b*c - a*d)/(d*(a + b*x))])/(b^3*g^2)
+ (4*B^2*d*(b*c - a*d)*i^2*PolyLog[3, 1 + (b*c - a*d)/(d*(a + b*x))])/(b^3*g^2)

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 2523

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*Log[c*RFx^p])^n, x] - Dist[b*n*p
, Int[SimplifyIntegrand[(x*(a + b*Log[c*RFx^p])^(n - 1)*D[RFx, x])/RFx, x], x], x] /; FreeQ[{a, b, c, p}, x] &
& RationalFunctionQ[RFx, x] && IGtQ[n, 0]

Rule 12

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

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 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 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 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 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 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 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 2525

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)*((d_.) + (e_.)*(x_))^(m_.), x_Symbol] :> Simp[((d + e*x)^(m
+ 1)*(a + b*Log[c*RFx^p])^n)/(e*(m + 1)), x] - Dist[(b*n*p)/(e*(m + 1)), Int[SimplifyIntegrand[((d + e*x)^(m +
1)*(a + b*Log[c*RFx^p])^(n - 1)*D[RFx, x])/RFx, x], x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && RationalFunc
tionQ[RFx, x] && IGtQ[n, 0] && (EqQ[n, 1] || IntegerQ[m]) && NeQ[m, -1]

Rule 44

Int[((a_) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*
x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

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]]

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 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 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 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 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 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 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]

Rubi steps

\begin{align*} \int \frac{(69 c+69 d x)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^2} \, dx &=\int \left (\frac{4761 d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^2}+\frac{4761 (b c-a d)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^2 (a+b x)^2}+\frac{9522 d (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^2 (a+b x)}\right ) \, dx\\ &=\frac{\left (4761 d^2\right ) \int \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2 \, dx}{b^2 g^2}+\frac{(9522 d (b c-a d)) \int \frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{a+b x} \, dx}{b^2 g^2}+\frac{\left (4761 (b c-a d)^2\right ) \int \frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{(a+b x)^2} \, dx}{b^2 g^2}\\ &=\frac{4761 d^2 x \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^2}-\frac{4761 (b c-a d)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^2 (a+b x)}+\frac{9522 d (b c-a d) \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^2}-\frac{\left (9522 B d^2\right ) \int \frac{(b c-a d) x \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{(a+b x) (c+d x)} \, dx}{b^2 g^2}-\frac{(19044 B d (b c-a d)) \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}{b^3 g^2}+\frac{\left (9522 B (b c-a d)^2\right ) \int \frac{(b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{(a+b x)^2 (c+d x)} \, dx}{b^3 g^2}\\ &=\frac{4761 d^2 x \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^2}-\frac{4761 (b c-a d)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^2 (a+b x)}+\frac{9522 d (b c-a d) \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^2}-\frac{\left (9522 B d^2 (b c-a d)\right ) \int \frac{x \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{(a+b x) (c+d x)} \, dx}{b^2 g^2}+\frac{\left (9522 B (b c-a d)^3\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{(a+b x)^2 (c+d x)} \, dx}{b^3 g^2}-\frac{(19044 B d (b c-a d)) \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}{b^3 e g^2}\\ &=\frac{4761 d^2 x \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^2}-\frac{4761 (b c-a d)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^2 (a+b x)}+\frac{9522 d (b c-a d) \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^2}-\frac{\left (9522 B d^2 (b c-a d)\right ) \int \left (-\frac{a \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{(b c-a d) (a+b x)}+\frac{c \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{(b c-a d) (c+d x)}\right ) \, dx}{b^2 g^2}+\frac{\left (9522 B (b c-a d)^3\right ) \int \left (\frac{b \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{(b c-a d) (a+b x)^2}-\frac{b d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^2 (a+b x)}+\frac{d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^2 (c+d x)}\right ) \, dx}{b^3 g^2}-\frac{(19044 B d (b c-a d)) \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}{b^3 e g^2}\\ &=\frac{4761 d^2 x \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^2}-\frac{4761 (b c-a d)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^2 (a+b x)}+\frac{9522 d (b c-a d) \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^2}+\frac{\left (9522 a B d^2\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{b^2 g^2}-\frac{\left (9522 B c d^2\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{c+d x} \, dx}{b^2 g^2}-\frac{(9522 B d (b c-a d)) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{b^2 g^2}+\frac{\left (9522 B d^2 (b c-a d)\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{c+d x} \, dx}{b^3 g^2}+\frac{\left (9522 B (b c-a d)^2\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{(a+b x)^2} \, dx}{b^2 g^2}-\frac{\left (19044 B d (b c-a d)^2\right ) \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}{b^3 g^2}\\ &=-\frac{9522 B (b c-a d)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^3 g^2 (a+b x)}+\frac{9522 a B d^2 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^3 g^2}-\frac{9522 B d (b c-a d) \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^3 g^2}+\frac{4761 d^2 x \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^2}-\frac{4761 (b c-a d)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^2 (a+b x)}+\frac{9522 d (b c-a d) \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^2}-\frac{9522 B c d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{b^2 g^2}+\frac{9522 B d (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{b^3 g^2}+\frac{\left (9522 B^2 c d\right ) \int \frac{(c+d x) \left (-\frac{d e (a+b x)}{(c+d x)^2}+\frac{b e}{c+d x}\right ) \log (c+d x)}{e (a+b x)} \, dx}{b^2 g^2}-\frac{\left (9522 a B^2 d^2\right ) \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)}{e (a+b x)} \, dx}{b^3 g^2}+\frac{\left (9522 B^2 d (b c-a d)\right ) \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)}{e (a+b x)} \, dx}{b^3 g^2}-\frac{\left (9522 B^2 d (b c-a d)\right ) \int \frac{(c+d x) \left (-\frac{d e (a+b x)}{(c+d x)^2}+\frac{b e}{c+d x}\right ) \log (c+d x)}{e (a+b x)} \, dx}{b^3 g^2}+\frac{\left (9522 B^2 (b c-a d)^2\right ) \int \frac{b c-a d}{(a+b x)^2 (c+d x)} \, dx}{b^3 g^2}-\frac{\left (19044 B d (b c-a d)^2\right ) \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}{b^3 g^2}\\ &=-\frac{9522 B (b c-a d)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^3 g^2 (a+b x)}+\frac{9522 a B d^2 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^3 g^2}-\frac{9522 B d (b c-a d) \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^3 g^2}+\frac{4761 d^2 x \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^2}-\frac{4761 (b c-a d)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^2 (a+b x)}+\frac{9522 d (b c-a d) \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^2}-\frac{9522 B c d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{b^2 g^2}+\frac{9522 B d (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{b^3 g^2}-\frac{\left (19044 A B d (b c-a d)^2\right ) \int \frac{\log (a+b x)}{(a+b x) (c+d x)} \, dx}{b^3 g^2}-\frac{\left (19044 B^2 d (b c-a d)^2\right ) \int \frac{\log (a+b x) \log \left (\frac{e (a+b x)}{c+d x}\right )}{(a+b x) (c+d x)} \, dx}{b^3 g^2}+\frac{\left (9522 B^2 (b c-a d)^3\right ) \int \frac{1}{(a+b x)^2 (c+d x)} \, dx}{b^3 g^2}+\frac{\left (9522 B^2 c d\right ) \int \frac{(c+d x) \left (-\frac{d e (a+b x)}{(c+d x)^2}+\frac{b e}{c+d x}\right ) \log (c+d x)}{a+b x} \, dx}{b^2 e g^2}-\frac{\left (9522 a B^2 d^2\right ) \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)}{a+b x} \, dx}{b^3 e g^2}+\frac{\left (9522 B^2 d (b c-a d)\right ) \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)}{a+b x} \, dx}{b^3 e g^2}-\frac{\left (9522 B^2 d (b c-a d)\right ) \int \frac{(c+d x) \left (-\frac{d e (a+b x)}{(c+d x)^2}+\frac{b e}{c+d x}\right ) \log (c+d x)}{a+b x} \, dx}{b^3 e g^2}\\ &=-\frac{9522 B^2 d (b c-a d) \log (a+b x) \log ^2\left (\frac{e (a+b x)}{c+d x}\right )}{b^3 g^2}-\frac{9522 B (b c-a d)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^3 g^2 (a+b x)}+\frac{9522 a B d^2 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^3 g^2}-\frac{9522 B d (b c-a d) \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^3 g^2}+\frac{4761 d^2 x \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^2}-\frac{4761 (b c-a d)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^2 (a+b x)}+\frac{9522 d (b c-a d) \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^2}-\frac{9522 B c d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{b^2 g^2}+\frac{9522 B d (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{b^3 g^2}+\frac{\left (9522 B^2 d (b c-a d)\right ) \int \frac{\log ^2\left (\frac{e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{b^2 g^2}-\frac{\left (19044 A B d (b c-a d)^2\right ) \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 )}{b^4 g^2}+\frac{\left (9522 B^2 (b c-a d)^3\right ) \int \left (\frac{b}{(b c-a d) (a+b x)^2}-\frac{b d}{(b c-a d)^2 (a+b x)}+\frac{d^2}{(b c-a d)^2 (c+d x)}\right ) \, dx}{b^3 g^2}+\frac{\left (9522 B^2 c d\right ) \int \left (\frac{b e \log (c+d x)}{a+b x}-\frac{d e \log (c+d x)}{c+d x}\right ) \, dx}{b^2 e g^2}-\frac{\left (9522 a B^2 d^2\right ) \int \left (\frac{b e \log (a+b x)}{a+b x}-\frac{d e \log (a+b x)}{c+d x}\right ) \, dx}{b^3 e g^2}+\frac{\left (9522 B^2 d (b c-a d)\right ) \int \left (\frac{b e \log (a+b x)}{a+b x}-\frac{d e \log (a+b x)}{c+d x}\right ) \, dx}{b^3 e g^2}-\frac{\left (9522 B^2 d (b c-a d)\right ) \int \left (\frac{b e \log (c+d x)}{a+b x}-\frac{d e \log (c+d x)}{c+d x}\right ) \, dx}{b^3 e g^2}\\ &=-\frac{9522 B^2 (b c-a d)^2}{b^3 g^2 (a+b x)}-\frac{9522 B^2 d (b c-a d) \log (a+b x)}{b^3 g^2}-\frac{9522 B^2 d (b c-a d) \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^3 g^2}-\frac{9522 B^2 d (b c-a d) \log (a+b x) \log ^2\left (\frac{e (a+b x)}{c+d x}\right )}{b^3 g^2}-\frac{9522 B (b c-a d)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^3 g^2 (a+b x)}+\frac{9522 a B d^2 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^3 g^2}-\frac{9522 B d (b c-a d) \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^3 g^2}+\frac{4761 d^2 x \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^2}-\frac{4761 (b c-a d)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^2 (a+b x)}+\frac{9522 d (b c-a d) \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^2}+\frac{9522 B^2 d (b c-a d) \log (c+d x)}{b^3 g^2}-\frac{9522 B c d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{b^2 g^2}+\frac{9522 B d (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{b^3 g^2}+\frac{\left (9522 B^2 c d\right ) \int \frac{\log (c+d x)}{a+b x} \, dx}{b g^2}-\frac{\left (9522 a B^2 d^2\right ) \int \frac{\log (a+b x)}{a+b x} \, dx}{b^2 g^2}-\frac{\left (9522 B^2 c d^2\right ) \int \frac{\log (c+d x)}{c+d x} \, dx}{b^2 g^2}+\frac{\left (9522 a B^2 d^3\right ) \int \frac{\log (a+b x)}{c+d x} \, dx}{b^3 g^2}-\frac{(19044 A B d (b c-a d)) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a+b x\right )}{b^3 g^2}+\frac{\left (9522 B^2 d (b c-a d)\right ) \int \frac{\log (a+b x)}{a+b x} \, dx}{b^2 g^2}-\frac{\left (9522 B^2 d (b c-a d)\right ) \int \frac{\log (c+d x)}{a+b x} \, dx}{b^2 g^2}+\frac{\left (19044 A B d^2 (b c-a d)\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{\frac{b c-a d}{b}+\frac{d x}{b}} \, dx,x,a+b x\right )}{b^4 g^2}-\frac{\left (9522 B^2 d^2 (b c-a d)\right ) \int \frac{\log (a+b x)}{c+d x} \, dx}{b^3 g^2}+\frac{\left (9522 B^2 d^2 (b c-a d)\right ) \int \frac{\log (c+d x)}{c+d x} \, dx}{b^3 g^2}+\frac{\left (19044 B^2 d (b c-a d)^2\right ) \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}{b^3 g^2}\\ &=-\frac{9522 B^2 (b c-a d)^2}{b^3 g^2 (a+b x)}-\frac{9522 B^2 d (b c-a d) \log (a+b x)}{b^3 g^2}-\frac{9522 A B d (b c-a d) \log ^2(a+b x)}{b^3 g^2}-\frac{9522 B^2 d (b c-a d) \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^3 g^2}-\frac{9522 B^2 d (b c-a d) \log (a+b x) \log ^2\left (\frac{e (a+b x)}{c+d x}\right )}{b^3 g^2}-\frac{9522 B (b c-a d)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^3 g^2 (a+b x)}+\frac{9522 a B d^2 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^3 g^2}-\frac{9522 B d (b c-a d) \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^3 g^2}+\frac{4761 d^2 x \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^2}-\frac{4761 (b c-a d)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^2 (a+b x)}+\frac{9522 d (b c-a d) \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^2}+\frac{9522 B^2 d (b c-a d) \log (c+d x)}{b^3 g^2}+\frac{9522 B^2 c d \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b^2 g^2}-\frac{9522 B^2 d (b c-a d) \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b^3 g^2}-\frac{9522 B c d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{b^2 g^2}+\frac{9522 B d (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{b^3 g^2}+\frac{9522 a B^2 d^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^3 g^2}+\frac{19044 A B d (b c-a d) \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^3 g^2}-\frac{9522 B^2 d (b c-a d) \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^3 g^2}+\frac{19044 B^2 d (b c-a d) \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 )}{b^3 g^2}-\frac{\left (9522 B^2 c d\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,c+d x\right )}{b^2 g^2}-\frac{\left (9522 a B^2 d^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a+b x\right )}{b^3 g^2}-\frac{\left (9522 a B^2 d^2\right ) \int \frac{\log \left (\frac{b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{b^2 g^2}-\frac{\left (9522 B^2 c d^2\right ) \int \frac{\log \left (\frac{d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{b^2 g^2}-\frac{(19044 A B d (b c-a d)) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{b^3 g^2}+\frac{\left (9522 B^2 d (b c-a d)\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a+b x\right )}{b^3 g^2}+\frac{\left (9522 B^2 d (b c-a d)\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,c+d x\right )}{b^3 g^2}+\frac{\left (9522 B^2 d (b c-a d)\right ) \int \frac{\log \left (\frac{b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{b^2 g^2}+\frac{\left (9522 B^2 d^2 (b c-a d)\right ) \int \frac{\log \left (\frac{d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{b^3 g^2}-\frac{\left (19044 B^2 d (b c-a d)^2\right ) \int \frac{\text{Li}_2\left (1+\frac{b c-a d}{d (a+b x)}\right )}{(a+b x) (c+d x)} \, dx}{b^3 g^2}\\ &=-\frac{9522 B^2 (b c-a d)^2}{b^3 g^2 (a+b x)}-\frac{9522 B^2 d (b c-a d) \log (a+b x)}{b^3 g^2}-\frac{4761 a B^2 d^2 \log ^2(a+b x)}{b^3 g^2}-\frac{9522 A B d (b c-a d) \log ^2(a+b x)}{b^3 g^2}+\frac{4761 B^2 d (b c-a d) \log ^2(a+b x)}{b^3 g^2}-\frac{9522 B^2 d (b c-a d) \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^3 g^2}-\frac{9522 B^2 d (b c-a d) \log (a+b x) \log ^2\left (\frac{e (a+b x)}{c+d x}\right )}{b^3 g^2}-\frac{9522 B (b c-a d)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^3 g^2 (a+b x)}+\frac{9522 a B d^2 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^3 g^2}-\frac{9522 B d (b c-a d) \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^3 g^2}+\frac{4761 d^2 x \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^2}-\frac{4761 (b c-a d)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^2 (a+b x)}+\frac{9522 d (b c-a d) \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^2}+\frac{9522 B^2 d (b c-a d) \log (c+d x)}{b^3 g^2}+\frac{9522 B^2 c d \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b^2 g^2}-\frac{9522 B^2 d (b c-a d) \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b^3 g^2}-\frac{9522 B c d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{b^2 g^2}+\frac{9522 B d (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{b^3 g^2}-\frac{4761 B^2 c d \log ^2(c+d x)}{b^2 g^2}+\frac{4761 B^2 d (b c-a d) \log ^2(c+d x)}{b^3 g^2}+\frac{9522 a B^2 d^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^3 g^2}+\frac{19044 A B d (b c-a d) \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^3 g^2}-\frac{9522 B^2 d (b c-a d) \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^3 g^2}+\frac{19044 A B d (b c-a d) \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{b^3 g^2}+\frac{19044 B^2 d (b c-a d) \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 )}{b^3 g^2}+\frac{19044 B^2 d (b c-a d) \text{Li}_3\left (1+\frac{b c-a d}{d (a+b x)}\right )}{b^3 g^2}-\frac{\left (9522 B^2 c d\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{b^2 g^2}-\frac{\left (9522 a B^2 d^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{b^3 g^2}+\frac{\left (9522 B^2 d (b c-a d)\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{b^3 g^2}+\frac{\left (9522 B^2 d (b c-a d)\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{b^3 g^2}\\ &=-\frac{9522 B^2 (b c-a d)^2}{b^3 g^2 (a+b x)}-\frac{9522 B^2 d (b c-a d) \log (a+b x)}{b^3 g^2}-\frac{4761 a B^2 d^2 \log ^2(a+b x)}{b^3 g^2}-\frac{9522 A B d (b c-a d) \log ^2(a+b x)}{b^3 g^2}+\frac{4761 B^2 d (b c-a d) \log ^2(a+b x)}{b^3 g^2}-\frac{9522 B^2 d (b c-a d) \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^3 g^2}-\frac{9522 B^2 d (b c-a d) \log (a+b x) \log ^2\left (\frac{e (a+b x)}{c+d x}\right )}{b^3 g^2}-\frac{9522 B (b c-a d)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^3 g^2 (a+b x)}+\frac{9522 a B d^2 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^3 g^2}-\frac{9522 B d (b c-a d) \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^3 g^2}+\frac{4761 d^2 x \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^2}-\frac{4761 (b c-a d)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^2 (a+b x)}+\frac{9522 d (b c-a d) \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^2}+\frac{9522 B^2 d (b c-a d) \log (c+d x)}{b^3 g^2}+\frac{9522 B^2 c d \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b^2 g^2}-\frac{9522 B^2 d (b c-a d) \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b^3 g^2}-\frac{9522 B c d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{b^2 g^2}+\frac{9522 B d (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{b^3 g^2}-\frac{4761 B^2 c d \log ^2(c+d x)}{b^2 g^2}+\frac{4761 B^2 d (b c-a d) \log ^2(c+d x)}{b^3 g^2}+\frac{9522 a B^2 d^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^3 g^2}+\frac{19044 A B d (b c-a d) \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^3 g^2}-\frac{9522 B^2 d (b c-a d) \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^3 g^2}+\frac{9522 a B^2 d^2 \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{b^3 g^2}+\frac{19044 A B d (b c-a d) \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{b^3 g^2}-\frac{9522 B^2 d (b c-a d) \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{b^3 g^2}+\frac{9522 B^2 c d \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{b^2 g^2}-\frac{9522 B^2 d (b c-a d) \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{b^3 g^2}+\frac{19044 B^2 d (b c-a d) \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 )}{b^3 g^2}+\frac{19044 B^2 d (b c-a d) \text{Li}_3\left (1+\frac{b c-a d}{d (a+b x)}\right )}{b^3 g^2}\\ \end{align*}

Mathematica [B]  time = 8.36429, size = 2652, normalized size = 6. \[ \text{Result too large to show} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(i^2*(3*A^2*b*d^2*x - (3*A^2*(b*c - a*d)^2)/(a + b*x) + 6*A^2*d*(b*c - a*d)*Log[a + b*x] - (6*A*b^2*B*c^2*(-(d
*(a + b*x)*Log[c/d + x]) + d*(a + b*x)*Log[(d*(a + b*x))/(-(b*c) + a*d)] + (b*c - a*d)*(1 + Log[(e*(a + b*x))/
(c + d*x)])))/((b*c - a*d)*(a + b*x)) + (3*b^2*B^2*c^2*(-2*b*c + 2*a*d - 2*d*(a + b*x)*Log[a + b*x] - 2*(b*c -
a*d)*Log[(e*(a + b*x))/(c + d*x)] - 2*d*(a + b*x)*Log[a + b*x]*Log[(e*(a + b*x))/(c + d*x)] - (b*c - a*d)*Log
[(e*(a + b*x))/(c + d*x)]^2 + 2*d*(a + b*x)*Log[c + d*x] - 2*d*(a + b*x)*Log[(e*(a + b*x))/(c + d*x)]*Log[(b*c
- a*d)/(b*c + b*d*x)] + d*(a + b*x)*(Log[a + b*x]*(Log[a + b*x] - 2*Log[(b*(c + d*x))/(b*c - a*d)]) - 2*PolyL
og[2, (d*(a + b*x))/(-(b*c) + a*d)]) + d*(a + b*x)*(Log[(b*c - a*d)/(b*c + b*d*x)]*(2*Log[(d*(a + b*x))/(-(b*c
) + a*d)] + Log[(b*c - a*d)/(b*c + b*d*x)]) - 2*PolyLog[2, (b*(c + d*x))/(b*c - a*d)])))/((b*c - a*d)*(a + b*x
)) + 6*A*b*B*c*d*(Log[a/b + x]^2 - 2*Log[a/b + x]*Log[a + b*x] - 2*Log[c/d + x]*Log[(d*(a + b*x))/(-(b*c) + a*
d)] + 2*Log[a + b*x]*((a*d)/(b*c - a*d) + Log[c/d + x] + Log[(e*(a + b*x))/(c + d*x)]) + 2*a*((a + b*x)^(-1) +
Log[(e*(a + b*x))/(c + d*x)]/(a + b*x) + (d*Log[c + d*x])/(-(b*c) + a*d)) - 2*PolyLog[2, (b*(c + d*x))/(b*c -
a*d)]) + 6*A*B*d^2*((a + b*x)*(-1 + Log[a/b + x]) - a*Log[a/b + x]^2 - (a^2*(1 + Log[a/b + x]))/(a + b*x) - b
*(c/d + x)*(-1 + Log[c/d + x]) + (a^2*Log[c/d + x])/(a + b*x) + (b*x - a^2/(a + b*x) - 2*a*Log[a + b*x])*(-Log
[a/b + x] + Log[c/d + x] + Log[(e*(a + b*x))/(c + d*x)]) + (a^2*d*(Log[a + b*x] - Log[c + d*x]))/(-(b*c) + a*d
) + 2*a*(Log[c/d + x]*Log[(d*(a + b*x))/(-(b*c) + a*d)] + PolyLog[2, (b*(c + d*x))/(b*c - a*d)])) + B^2*d^2*(6
*b*x - 6*(a + b*x)*Log[a/b + x] + 3*(a + b*x)*Log[a/b + x]^2 - 2*a*Log[a/b + x]^3 - (3*a^2*(2 + 2*Log[a/b + x]
+ Log[a/b + x]^2))/(a + b*x) + (3*b*(2*d*x - 2*(c + d*x)*Log[c/d + x] + (c + d*x)*Log[c/d + x]^2))/d + 3*(b*x
- a^2/(a + b*x) - 2*a*Log[a + b*x])*(-Log[a/b + x] + Log[c/d + x] + Log[(e*(a + b*x))/(c + d*x)])^2 - (6*(a*d
+ 2*b*d*x - b*d*x*Log[c/d + x] - b*c*Log[c + d*x] + Log[a/b + x]*(-(d*(a + b*x)) + d*(a + b*x)*Log[c/d + x] +
(b*c - a*d)*Log[(b*(c + d*x))/(b*c - a*d)]) + (b*c - a*d)*PolyLog[2, (d*(a + b*x))/(-(b*c) + a*d)]))/d + (3*a
^2*(d*(a + b*x)*Log[a/b + x]^2 + 2*((-(b*c) + a*d)*Log[c/d + x] + d*(a + b*x)*(Log[a + b*x] - Log[c + d*x])) -
2*Log[a/b + x]*((b*c - a*d)*Log[c/d + x] + d*(a + b*x)*Log[(b*(c + d*x))/(b*c - a*d)]) - 2*d*(a + b*x)*PolyLo
g[2, (d*(a + b*x))/(-(b*c) + a*d)]))/((-(b*c) + a*d)*(a + b*x)) + (3*a^2*(-(b*(c + d*x)*Log[c/d + x]^2) + 2*d*
(a + b*x)*Log[c/d + x]*Log[(d*(a + b*x))/(-(b*c) + a*d)] + 2*d*(a + b*x)*PolyLog[2, (b*(c + d*x))/(b*c - a*d)]
))/((b*c - a*d)*(a + b*x)) + 6*(-Log[a/b + x] + Log[c/d + x] + Log[(e*(a + b*x))/(c + d*x)])*((a + b*x)*(-1 +
Log[a/b + x]) - a*Log[a/b + x]^2 - (a^2*(1 + Log[a/b + x]))/(a + b*x) - b*(c/d + x)*(-1 + Log[c/d + x]) + (a^2
*Log[c/d + x])/(a + b*x) + (a^2*d*(Log[a + b*x] - Log[c + d*x]))/(-(b*c) + a*d) + 2*a*(Log[c/d + x]*Log[(d*(a
+ b*x))/(-(b*c) + a*d)] + PolyLog[2, (b*(c + d*x))/(b*c - a*d)])) + 6*a*(Log[a/b + x]^2*(Log[c/d + x] - Log[(b
*(c + d*x))/(b*c - a*d)]) - 2*Log[a/b + x]*PolyLog[2, (d*(a + b*x))/(-(b*c) + a*d)] + 2*PolyLog[3, (d*(a + b*x
))/(-(b*c) + a*d)]) - 6*a*(Log[c/d + x]^2*Log[(d*(a + b*x))/(-(b*c) + a*d)] + 2*Log[c/d + x]*PolyLog[2, (b*(c
+ d*x))/(b*c - a*d)] - 2*PolyLog[3, (b*(c + d*x))/(b*c - a*d)])) + (2*b*B^2*c*d*(6*b*c - 6*a*d - (6*b^2*c*x)/(
a + b*x) + (6*a*b*d*x)/(a + b*x) + 6*a*d*Log[a/b + x] + 3*b*c*Log[a/b + x]^2 - 3*a*d*Log[a/b + x]^2 - 6*b*c*Lo
g[c/d + x] + 6*b*c*Log[a + b*x] - 6*a*d*Log[a + b*x] - 6*b*c*Log[a/b + x]*Log[a + b*x] + 6*a*d*Log[a/b + x]*Lo
g[a + b*x] + 6*b*c*Log[c/d + x]*Log[a + b*x] - 6*a*d*Log[c/d + x]*Log[a + b*x] - 6*b*c*Log[c/d + x]*Log[(d*(a
+ b*x))/(-(b*c) + a*d)] + 6*a*d*Log[c/d + x]*Log[(d*(a + b*x))/(-(b*c) + a*d)] - (6*b*(b*c - a*d)*x*Log[(e*(a
+ b*x))/(c + d*x)])/(a + b*x) + 6*b*c*Log[a + b*x]*Log[(e*(a + b*x))/(c + d*x)] - 6*a*d*Log[a + b*x]*Log[(e*(a
+ b*x))/(c + d*x)] + 3*a*d*Log[(e*(a + b*x))/(c + d*x)]^2 + 3*b*d*x*Log[(e*(a + b*x))/(c + d*x)]^2 - (3*b^2*x
*(c + d*x)*Log[(e*(a + b*x))/(c + d*x)]^2)/(a + b*x) - 3*b*c*Log[(-(b*c) + a*d)/(d*(a + b*x))]*Log[(e*(a + b*x
))/(c + d*x)]^2 - a*d*Log[(e*(a + b*x))/(c + d*x)]^3 + 6*b*c*Log[(e*(a + b*x))/(c + d*x)]*Log[(b*c - a*d)/(b*c
+ b*d*x)] - 6*a*d*Log[(e*(a + b*x))/(c + d*x)]*Log[(b*c - a*d)/(b*c + b*d*x)] + 3*a*d*Log[(e*(a + b*x))/(c +
d*x)]^2*Log[(b*c - a*d)/(b*c + b*d*x)] + 6*(b*c - a*d + a*d*Log[(e*(a + b*x))/(c + d*x)])*PolyLog[2, (d*(a + b
*x))/(b*(c + d*x))] - 6*(b*c - a*d)*PolyLog[2, (b*(c + d*x))/(b*c - a*d)] + 6*b*c*Log[(e*(a + b*x))/(c + d*x)]
*PolyLog[2, (b*(c + d*x))/(d*(a + b*x))] - 6*a*d*PolyLog[3, (d*(a + b*x))/(b*(c + d*x))] + 6*b*c*PolyLog[3, (b
*(c + d*x))/(d*(a + b*x))]))/(b*c - a*d)))/(3*b^3*g^2)

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Maple [F]  time = 3.704, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( dix+ci \right ) ^{2}}{ \left ( bgx+ag \right ) ^{2}} \left ( A+B\ln \left ({\frac{e \left ( bx+a \right ) }{dx+c}} \right ) \right ) ^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-A^2*(a^2/(b^4*g^2*x + a*b^3*g^2) - x/(b^2*g^2) + 2*a*log(b*x + a)/(b^3*g^2))*d^2*i^2 + 2*A^2*c*d*i^2*(a/(b^3*
g^2*x + a*b^2*g^2) + log(b*x + a)/(b^2*g^2)) - 2*A*B*c^2*i^2*(log(b*e*x/(d*x + c) + a*e/(d*x + c))/(b^2*g^2*x
+ a*b*g^2) + 1/(b^2*g^2*x + a*b*g^2) + d*log(b*x + a)/((b^2*c - a*b*d)*g^2) - d*log(d*x + c)/((b^2*c - a*b*d)*
g^2)) - A^2*c^2*i^2/(b^2*g^2*x + a*b*g^2) + (B^2*b^2*d^2*i^2*x^2 + B^2*a*b*d^2*i^2*x - (b^2*c^2*i^2 - 2*a*b*c*
d*i^2 + a^2*d^2*i^2)*B^2 + 2*((b^2*c*d*i^2 - a*b*d^2*i^2)*B^2*x + (a*b*c*d*i^2 - a^2*d^2*i^2)*B^2)*log(b*x + a
))*log(d*x + c)^2/(b^4*g^2*x + a*b^3*g^2) - integrate(-(B^2*b^3*c^3*i^2*log(e)^2 + (B^2*b^3*d^3*i^2*log(e)^2 +
2*A*B*b^3*d^3*i^2*log(e))*x^3 + 3*(B^2*b^3*c*d^2*i^2*log(e)^2 + 2*A*B*b^3*c*d^2*i^2*log(e))*x^2 + (B^2*b^3*d^
3*i^2*x^3 + 3*B^2*b^3*c*d^2*i^2*x^2 + 3*B^2*b^3*c^2*d*i^2*x + B^2*b^3*c^3*i^2)*log(b*x + a)^2 + (3*B^2*b^3*c^2
*d*i^2*log(e)^2 + 4*A*B*b^3*c^2*d*i^2*log(e))*x + 2*(B^2*b^3*c^3*i^2*log(e) + (B^2*b^3*d^3*i^2*log(e) + A*B*b^
3*d^3*i^2)*x^3 + 3*(B^2*b^3*c*d^2*i^2*log(e) + A*B*b^3*c*d^2*i^2)*x^2 + (3*B^2*b^3*c^2*d*i^2*log(e) + 2*A*B*b^
3*c^2*d*i^2)*x)*log(b*x + a) - 2*((A*B*b^3*d^3*i^2 + (i^2*log(e) + i^2)*B^2*b^3*d^3)*x^3 + (b^3*c^3*i^2*log(e)
- a*b^2*c^2*d*i^2 + 2*a^2*b*c*d^2*i^2 - a^3*d^3*i^2)*B^2 + (3*A*B*b^3*c*d^2*i^2 + (3*b^3*c*d^2*i^2*log(e) + 2
*a*b^2*d^3*i^2)*B^2)*x^2 + (2*A*B*b^3*c^2*d*i^2 + (2*a*b^2*c*d^2*i^2 + (3*i^2*log(e) - i^2)*b^3*c^2*d)*B^2)*x
+ (B^2*b^3*d^3*i^2*x^3 + (5*b^3*c*d^2*i^2 - 2*a*b^2*d^3*i^2)*B^2*x^2 + (3*b^3*c^2*d*i^2 + 4*a*b^2*c*d^2*i^2 -
4*a^2*b*d^3*i^2)*B^2*x + (b^3*c^3*i^2 + 2*a^2*b*c*d^2*i^2 - 2*a^3*d^3*i^2)*B^2)*log(b*x + a))*log(d*x + c))/(b
^5*d*g^2*x^3 + a^2*b^3*c*g^2 + (b^5*c*g^2 + 2*a*b^4*d*g^2)*x^2 + (2*a*b^4*c*g^2 + a^2*b^3*d*g^2)*x), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{A^{2} d^{2} i^{2} x^{2} + 2 \, A^{2} c d i^{2} x + A^{2} c^{2} i^{2} +{\left (B^{2} d^{2} i^{2} x^{2} + 2 \, B^{2} c d i^{2} x + B^{2} c^{2} i^{2}\right )} \log \left (\frac{b e x + a e}{d x + c}\right )^{2} + 2 \,{\left (A B d^{2} i^{2} x^{2} + 2 \, A B c d i^{2} x + A B c^{2} i^{2}\right )} \log \left (\frac{b e x + a e}{d x + c}\right )}{b^{2} g^{2} x^{2} + 2 \, a b g^{2} x + a^{2} g^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (d i x + c i\right )}^{2}{\left (B \log \left (\frac{{\left (b x + a\right )} e}{d x + c}\right ) + A\right )}^{2}}{{\left (b g x + a g\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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