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

Optimal. Leaf size=536 \[ \frac {b^2 g^2 (c+d x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{2 d^3 i}-\frac {2 B g^2 (b c-a d)^2 \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right ) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{d^3 i}-\frac {g^2 (b c-a d)^2 \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{d^3 i}-\frac {4 B g^2 (b c-a d)^2 \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{d^3 i}+\frac {B g^2 (b c-a d)^2 \log \left (1-\frac {b (c+d x)}{d (a+b x)}\right ) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{d^3 i}-\frac {2 g^2 (a+b x) (b c-a d) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{d^2 i}-\frac {B g^2 (a+b x) (b c-a d) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{d^2 i}-\frac {4 B^2 g^2 (b c-a d)^2 \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right )}{d^3 i}-\frac {B^2 g^2 (b c-a d)^2 \text {Li}_2\left (\frac {b (c+d x)}{d (a+b x)}\right )}{d^3 i}+\frac {2 B^2 g^2 (b c-a d)^2 \text {Li}_3\left (\frac {d (a+b x)}{b (c+d x)}\right )}{d^3 i}+\frac {B^2 g^2 (b c-a d)^2 \log (c+d x)}{d^3 i} \]

[Out]

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

________________________________________________________________________________________

Rubi [B]  time = 4.82, antiderivative size = 1666, normalized size of antiderivative = 3.11, number of steps used = 86, number of rules used = 27, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.643, Rules used = {2528, 2523, 12, 2524, 2418, 2390, 2301, 2394, 2393, 2391, 2525, 2486, 31, 6688, 6742, 2499, 2396, 2433, 2374, 6589, 2302, 30, 2500, 2375, 2317, 2440, 2434} \[ \text {result too large to display} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((A*b*B*(b*c - a*d)*g^2*x)/(d^2*i)) + (a*B^2*(b*c - a*d)*g^2*Log[a + b*x]^2)/(d^2*i) - (B^2*(b*c - a*d)^2*g^2
*Log[a + b*x]*Log[(c + d*x)^(-1)]^2)/(d^3*i) + (B^2*(b*c - a*d)^2*g^2*Log[-((d*(a + b*x))/(b*c - a*d))]*Log[(c
+ d*x)^(-1)]^2)/(d^3*i) - (B^2*(b*c - a*d)*g^2*(a + b*x)*Log[(e*(a + b*x))/(c + d*x)])/(d^2*i) - (2*a*B*(b*c
- a*d)*g^2*Log[a + b*x]*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(d^2*i) - (b*(b*c - a*d)*g^2*x*(A + B*Log[(e*(a
+ b*x))/(c + d*x)])^2)/(d^2*i) + (g^2*(a + b*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(2*d*i) + (B^2*(b*c
- a*d)^2*g^2*Log[c + d*x])/(d^3*i) - (2*b*B^2*c*(b*c - a*d)*g^2*Log[-((d*(a + b*x))/(b*c - a*d))]*Log[c + d*x]
)/(d^3*i) - (B^2*(b*c - a*d)^2*g^2*Log[-((d*(a + b*x))/(b*c - a*d))]*Log[c + d*x])/(d^3*i) + (2*b*B*c*(b*c - a
*d)*g^2*(A + B*Log[(e*(a + b*x))/(c + d*x)])*Log[c + d*x])/(d^3*i) + (B*(b*c - a*d)^2*g^2*(A + B*Log[(e*(a + b
*x))/(c + d*x)])*Log[c + d*x])/(d^3*i) + (b*B^2*c*(b*c - a*d)*g^2*Log[c + d*x]^2)/(d^3*i) + (B^2*(b*c - a*d)^2
*g^2*Log[c + d*x]^2)/(2*d^3*i) - (2*a*B^2*(b*c - a*d)*g^2*Log[a + b*x]*Log[(b*(c + d*x))/(b*c - a*d)])/(d^2*i)
+ (B^2*(b*c - a*d)^2*g^2*Log[a + b*x]^2*Log[(b*(c + d*x))/(b*c - a*d)])/(d^3*i) - (B^2*(b*c - a*d)^2*g^2*Log[
a + b*x]^2*Log[i*(c + d*x)])/(d^3*i) - (2*B^2*(b*c - a*d)^2*g^2*Log[a + b*x]*Log[(c + d*x)^(-1)]*Log[i*(c + d*
x)])/(d^3*i) + (A*B*(b*c - a*d)^2*g^2*Log[i*(c + d*x)]^2)/(d^3*i) - (B^2*(b*c - a*d)^2*g^2*Log[a + b*x]*Log[i*
(c + d*x)]^2)/(d^3*i) + (B^2*(b*c - a*d)^2*g^2*Log[-((d*(a + b*x))/(b*c - a*d))]*Log[i*(c + d*x)]^2)/(d^3*i) +
(B^2*(b*c - a*d)^2*g^2*Log[i*(c + d*x)]^3)/(3*d^3*i) - (2*A*B*(b*c - a*d)^2*g^2*Log[-((d*(a + b*x))/(b*c - a*
d))]*Log[c*i + d*i*x])/(d^3*i) + (2*B^2*(b*c - a*d)^2*g^2*Log[-((d*(a + b*x))/(b*c - a*d))]*(Log[a + b*x] + Lo
g[(c + d*x)^(-1)] - Log[(e*(a + b*x))/(c + d*x)])*Log[c*i + d*i*x])/(d^3*i) + ((b*c - a*d)^2*g^2*(A + B*Log[(e
*(a + b*x))/(c + d*x)])^2*Log[c*i + d*i*x])/(d^3*i) - (B^2*(b*c - a*d)^2*g^2*Log[-((d*(a + b*x))/(b*c - a*d))]
*Log[c*i + d*i*x]^2)/(d^3*i) + (B^2*(b*c - a*d)^2*g^2*Log[(e*(a + b*x))/(c + d*x)]*Log[c*i + d*i*x]^2)/(d^3*i)
- (2*a*B^2*(b*c - a*d)*g^2*PolyLog[2, -((d*(a + b*x))/(b*c - a*d))])/(d^2*i) + (2*B^2*(b*c - a*d)^2*g^2*Log[a
+ b*x]*PolyLog[2, -((d*(a + b*x))/(b*c - a*d))])/(d^3*i) - (2*b*B^2*c*(b*c - a*d)*g^2*PolyLog[2, (b*(c + d*x)
)/(b*c - a*d)])/(d^3*i) - (2*A*B*(b*c - a*d)^2*g^2*PolyLog[2, (b*(c + d*x))/(b*c - a*d)])/(d^3*i) - (B^2*(b*c
- a*d)^2*g^2*PolyLog[2, (b*(c + d*x))/(b*c - a*d)])/(d^3*i) - (2*B^2*(b*c - a*d)^2*g^2*Log[(c + d*x)^(-1)]*Pol
yLog[2, (b*(c + d*x))/(b*c - a*d)])/(d^3*i) + (2*B^2*(b*c - a*d)^2*g^2*(Log[a + b*x] + Log[(c + d*x)^(-1)] - L
og[(e*(a + b*x))/(c + d*x)])*PolyLog[2, (b*(c + d*x))/(b*c - a*d)])/(d^3*i) - (2*B^2*(b*c - a*d)^2*g^2*PolyLog
[3, -((d*(a + b*x))/(b*c - a*d))])/(d^3*i) - (2*B^2*(b*c - a*d)^2*g^2*PolyLog[3, (b*(c + d*x))/(b*c - a*d)])/(
d^3*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 31

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

Rule 2301

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]

Rule 2302

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

Rule 2317

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

Rule 2374

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

Rule 2375

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

Rule 2390

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

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 2393

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

Rule 2394

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

Rule 2396

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

Rule 2418

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

Rule 2433

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

Rule 2434

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

Rule 2440

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

Rule 2486

Int[Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.))^(r_.)]^(s_.), x_Symbol] :> Simp[((
a + b*x)*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^s)/b, x] + Dist[(q*r*s*(b*c - a*d))/b, Int[Log[e*(f*(a + b*x)^p*
(c + d*x)^q)^r]^(s - 1)/(c + d*x), x], x] /; FreeQ[{a, b, c, d, e, f, p, q, r, s}, x] && NeQ[b*c - a*d, 0] &&
EqQ[p + q, 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 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 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 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 2528

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)*(RGx_), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*Log[c*
RFx^p])^n, RGx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, p}, x] && RationalFunctionQ[RFx, x] && RationalF
unctionQ[RGx, x] && IGtQ[n, 0]

Rule 6589

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

Rule 6688

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

Rule 6742

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

Rubi steps

\begin {align*} \int \frac {(a g+b g x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{85 c+85 d x} \, dx &=\int \left (-\frac {b (b c-a d) g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{85 d^2}+\frac {(b c-a d)^2 g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{d^2 (85 c+85 d x)}+\frac {b g (a g+b g x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{85 d}\right ) \, dx\\ &=\frac {(b g) \int (a g+b g x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \, dx}{85 d}-\frac {\left (b (b c-a d) g^2\right ) \int \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \, dx}{85 d^2}+\frac {\left ((b c-a d)^2 g^2\right ) \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{85 c+85 d x} \, dx}{d^2}\\ &=-\frac {b (b c-a d) g^2 x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{85 d^2}+\frac {g^2 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{170 d}+\frac {(b c-a d)^2 g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (85 c+85 d x)}{85 d^3}-\frac {B \int \frac {(b c-a d) g^2 (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{c+d x} \, dx}{85 d}+\frac {\left (2 b B (b c-a d) g^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}{85 d^2}-\frac {\left (2 B (b c-a d)^2 g^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 ) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (85 c+85 d x)}{e (a+b x)} \, dx}{85 d^3}\\ &=-\frac {b (b c-a d) g^2 x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{85 d^2}+\frac {g^2 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{170 d}+\frac {(b c-a d)^2 g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (85 c+85 d x)}{85 d^3}-\frac {\left (B (b c-a d) g^2\right ) \int \frac {(a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{c+d x} \, dx}{85 d}+\frac {\left (2 b B (b c-a d)^2 g^2\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}{85 d^2}-\frac {\left (2 B (b c-a d)^2 g^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 ) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (85 c+85 d x)}{a+b x} \, dx}{85 d^3 e}\\ &=-\frac {b (b c-a d) g^2 x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{85 d^2}+\frac {g^2 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{170 d}+\frac {(b c-a d)^2 g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (85 c+85 d x)}{85 d^3}-\frac {\left (B (b c-a d) g^2\right ) \int \left (\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{d}+\frac {(-b c+a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{d (c+d x)}\right ) \, dx}{85 d}+\frac {\left (2 b B (b c-a d)^2 g^2\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}{85 d^2}-\frac {\left (2 B (b c-a d)^2 g^2\right ) \int \frac {(b c-a d) e \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (85 c+85 d x)}{(a+b x) (c+d x)} \, dx}{85 d^3 e}\\ &=-\frac {b (b c-a d) g^2 x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{85 d^2}+\frac {g^2 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{170 d}+\frac {(b c-a d)^2 g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (85 c+85 d x)}{85 d^3}-\frac {\left (b B (b c-a d) g^2\right ) \int \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx}{85 d^2}-\frac {\left (2 a b B (b c-a d) g^2\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{85 d^2}+\frac {\left (2 b B c (b c-a d) g^2\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{c+d x} \, dx}{85 d^2}+\frac {\left (B (b c-a d)^2 g^2\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{c+d x} \, dx}{85 d^2}-\frac {\left (2 B (b c-a d)^3 g^2\right ) \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (85 c+85 d x)}{(a+b x) (c+d x)} \, dx}{85 d^3}\\ &=-\frac {A b B (b c-a d) g^2 x}{85 d^2}-\frac {2 a B (b c-a d) g^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{85 d^2}-\frac {b (b c-a d) g^2 x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{85 d^2}+\frac {g^2 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{170 d}+\frac {2 b B c (b c-a d) g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{85 d^3}+\frac {B (b c-a d)^2 g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{85 d^3}+\frac {(b c-a d)^2 g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (85 c+85 d x)}{85 d^3}-\frac {\left (2 b B^2 c (b c-a d) g^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 (c+d x)}{e (a+b x)} \, dx}{85 d^3}+\frac {\left (2 a B^2 (b c-a d) g^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}{85 d^2}-\frac {\left (b B^2 (b c-a d) g^2\right ) \int \log \left (\frac {e (a+b x)}{c+d x}\right ) \, dx}{85 d^2}-\frac {\left (B^2 (b c-a d)^2 g^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 (c+d x)}{e (a+b x)} \, dx}{85 d^3}-\frac {\left (2 B (b c-a d)^3 g^2\right ) \int \left (\frac {d \left (-A-B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (85 c+85 d x)}{(b c-a d) (c+d x)}+\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (85 c+85 d x)}{(b c-a d) (a+b x)}\right ) \, dx}{85 d^3}\\ &=-\frac {A b B (b c-a d) g^2 x}{85 d^2}-\frac {B^2 (b c-a d) g^2 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{85 d^2}-\frac {2 a B (b c-a d) g^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{85 d^2}-\frac {b (b c-a d) g^2 x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{85 d^2}+\frac {g^2 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{170 d}+\frac {2 b B c (b c-a d) g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{85 d^3}+\frac {B (b c-a d)^2 g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{85 d^3}+\frac {(b c-a d)^2 g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (85 c+85 d x)}{85 d^3}-\frac {\left (2 b B (b c-a d)^2 g^2\right ) \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (85 c+85 d x)}{a+b x} \, dx}{85 d^3}-\frac {\left (2 B (b c-a d)^2 g^2\right ) \int \frac {\left (-A-B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (85 c+85 d x)}{c+d x} \, dx}{85 d^2}+\frac {\left (B^2 (b c-a d)^2 g^2\right ) \int \frac {1}{c+d x} \, dx}{85 d^2}-\frac {\left (2 b B^2 c (b c-a d) g^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 (c+d x)}{a+b x} \, dx}{85 d^3 e}+\frac {\left (2 a B^2 (b c-a d) g^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}{85 d^2 e}-\frac {\left (B^2 (b c-a d)^2 g^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 (c+d x)}{a+b x} \, dx}{85 d^3 e}\\ &=-\frac {A b B (b c-a d) g^2 x}{85 d^2}-\frac {B^2 (b c-a d) g^2 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{85 d^2}-\frac {2 a B (b c-a d) g^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{85 d^2}-\frac {b (b c-a d) g^2 x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{85 d^2}+\frac {g^2 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{170 d}+\frac {B^2 (b c-a d)^2 g^2 \log (c+d x)}{85 d^3}+\frac {2 b B c (b c-a d) g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{85 d^3}+\frac {B (b c-a d)^2 g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{85 d^3}+\frac {(b c-a d)^2 g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (85 c+85 d x)}{85 d^3}-\frac {\left (2 b B (b c-a d)^2 g^2\right ) \int \left (\frac {A \log (85 c+85 d x)}{a+b x}+\frac {B \log \left (\frac {e (a+b x)}{c+d x}\right ) \log (85 c+85 d x)}{a+b x}\right ) \, dx}{85 d^3}-\frac {\left (2 B (b c-a d)^2 g^2\right ) \int \left (\frac {A \log (85 c+85 d x)}{-c-d x}+\frac {B \log \left (\frac {e (a+b x)}{c+d x}\right ) \log (85 c+85 d x)}{-c-d x}\right ) \, dx}{85 d^2}-\frac {\left (2 b B^2 c (b c-a d) g^2\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}{85 d^3 e}+\frac {\left (2 a B^2 (b c-a d) g^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}{85 d^2 e}-\frac {\left (B^2 (b c-a d)^2 g^2\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}{85 d^3 e}\\ &=-\frac {A b B (b c-a d) g^2 x}{85 d^2}-\frac {B^2 (b c-a d) g^2 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{85 d^2}-\frac {2 a B (b c-a d) g^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{85 d^2}-\frac {b (b c-a d) g^2 x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{85 d^2}+\frac {g^2 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{170 d}+\frac {B^2 (b c-a d)^2 g^2 \log (c+d x)}{85 d^3}+\frac {2 b B c (b c-a d) g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{85 d^3}+\frac {B (b c-a d)^2 g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{85 d^3}+\frac {(b c-a d)^2 g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (85 c+85 d x)}{85 d^3}-\frac {\left (2 b^2 B^2 c (b c-a d) g^2\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{85 d^3}+\frac {\left (2 a b B^2 (b c-a d) g^2\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{85 d^2}+\frac {\left (2 b B^2 c (b c-a d) g^2\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{85 d^2}-\frac {\left (2 a B^2 (b c-a d) g^2\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{85 d}-\frac {\left (2 A b B (b c-a d)^2 g^2\right ) \int \frac {\log (85 c+85 d x)}{a+b x} \, dx}{85 d^3}-\frac {\left (b B^2 (b c-a d)^2 g^2\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{85 d^3}-\frac {\left (2 b B^2 (b c-a d)^2 g^2\right ) \int \frac {\log \left (\frac {e (a+b x)}{c+d x}\right ) \log (85 c+85 d x)}{a+b x} \, dx}{85 d^3}-\frac {\left (2 A B (b c-a d)^2 g^2\right ) \int \frac {\log (85 c+85 d x)}{-c-d x} \, dx}{85 d^2}+\frac {\left (B^2 (b c-a d)^2 g^2\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{85 d^2}-\frac {\left (2 B^2 (b c-a d)^2 g^2\right ) \int \frac {\log \left (\frac {e (a+b x)}{c+d x}\right ) \log (85 c+85 d x)}{-c-d x} \, dx}{85 d^2}\\ &=-\frac {A b B (b c-a d) g^2 x}{85 d^2}-\frac {B^2 (b c-a d) g^2 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{85 d^2}-\frac {2 a B (b c-a d) g^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{85 d^2}-\frac {b (b c-a d) g^2 x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{85 d^2}+\frac {g^2 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{170 d}+\frac {B^2 (b c-a d)^2 g^2 \log (c+d x)}{85 d^3}-\frac {2 b B^2 c (b c-a d) g^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{85 d^3}-\frac {B^2 (b c-a d)^2 g^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{85 d^3}+\frac {2 b B c (b c-a d) g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{85 d^3}+\frac {B (b c-a d)^2 g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{85 d^3}-\frac {2 a B^2 (b c-a d) g^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{85 d^2}-\frac {2 A B (b c-a d)^2 g^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (85 c+85 d x)}{85 d^3}+\frac {(b c-a d)^2 g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (85 c+85 d x)}{85 d^3}+\frac {B^2 (b c-a d)^2 g^2 \log \left (\frac {e (a+b x)}{c+d x}\right ) \log ^2(85 c+85 d x)}{85 d^3}+\frac {\left (2 b B^2 c (b c-a d) g^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{85 d^3}+\frac {\left (2 a B^2 (b c-a d) g^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{85 d^2}+\frac {\left (2 a b B^2 (b c-a d) g^2\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{85 d^2}+\frac {\left (2 b B^2 c (b c-a d) g^2\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{85 d^2}-\frac {\left (2 A B (b c-a d)^2 g^2\right ) \operatorname {Subst}\left (\int -\frac {85 \log (x)}{x} \, dx,x,85 c+85 d x\right )}{7225 d^3}+\frac {\left (B^2 (b c-a d)^2 g^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{85 d^3}-\frac {\left (b B^2 (b c-a d)^2 g^2\right ) \int \frac {\log ^2(85 c+85 d x)}{a+b x} \, dx}{85 d^3}-\frac {\left (2 b B^2 (b c-a d)^2 g^2\right ) \int \frac {\log (a+b x) \log (85 c+85 d x)}{a+b x} \, dx}{85 d^3}-\frac {\left (2 b B^2 (b c-a d)^2 g^2\right ) \int \frac {\log \left (\frac {1}{c+d x}\right ) \log (85 c+85 d x)}{a+b x} \, dx}{85 d^3}+\frac {\left (2 A B (b c-a d)^2 g^2\right ) \int \frac {\log \left (\frac {85 d (a+b x)}{-85 b c+85 a d}\right )}{85 c+85 d x} \, dx}{d^2}+\frac {\left (B^2 (b c-a d)^2 g^2\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{85 d^2}+\frac {\left (B^2 (b c-a d)^2 g^2\right ) \int \frac {\log ^2(85 c+85 d x)}{c+d x} \, dx}{85 d^2}-\frac {\left (2 b B^2 (b c-a d)^2 g^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 (85 c+85 d x)}{a+b x} \, dx}{85 d^3}\\ &=-\frac {A b B (b c-a d) g^2 x}{85 d^2}+\frac {a B^2 (b c-a d) g^2 \log ^2(a+b x)}{85 d^2}-\frac {B^2 (b c-a d) g^2 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{85 d^2}-\frac {2 a B (b c-a d) g^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{85 d^2}-\frac {b (b c-a d) g^2 x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{85 d^2}+\frac {g^2 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{170 d}+\frac {B^2 (b c-a d)^2 g^2 \log (c+d x)}{85 d^3}-\frac {2 b B^2 c (b c-a d) g^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{85 d^3}-\frac {B^2 (b c-a d)^2 g^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{85 d^3}+\frac {2 b B c (b c-a d) g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{85 d^3}+\frac {B (b c-a d)^2 g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{85 d^3}+\frac {b B^2 c (b c-a d) g^2 \log ^2(c+d x)}{85 d^3}+\frac {B^2 (b c-a d)^2 g^2 \log ^2(c+d x)}{170 d^3}-\frac {2 a B^2 (b c-a d) g^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{85 d^2}-\frac {2 A B (b c-a d)^2 g^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (85 c+85 d x)}{85 d^3}+\frac {2 B^2 (b c-a d)^2 g^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 (85 c+85 d x)}{85 d^3}+\frac {(b c-a d)^2 g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (85 c+85 d x)}{85 d^3}-\frac {B^2 (b c-a d)^2 g^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(85 c+85 d x)}{85 d^3}+\frac {B^2 (b c-a d)^2 g^2 \log \left (\frac {e (a+b x)}{c+d x}\right ) \log ^2(85 c+85 d x)}{85 d^3}+\frac {\left (2 b B^2 c (b c-a d) g^2\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{85 d^3}+\frac {\left (2 a B^2 (b c-a d) g^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 )}{85 d^2}+\frac {\left (2 A B (b c-a d)^2 g^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,85 c+85 d x\right )}{85 d^3}+\frac {\left (2 A B (b c-a d)^2 g^2\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-85 b c+85 a d}\right )}{x} \, dx,x,85 c+85 d x\right )}{85 d^3}+\frac {\left (B^2 (b c-a d)^2 g^2\right ) \operatorname {Subst}\left (\int \frac {85 \log ^2(x)}{x} \, dx,x,85 c+85 d x\right )}{7225 d^3}+\frac {\left (B^2 (b c-a d)^2 g^2\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{85 d^3}-\frac {\left (2 B^2 (b c-a d)^2 g^2\right ) \operatorname {Subst}\left (\int \frac {\log (x) \log \left (\frac {85 b c-85 a d}{b}+\frac {85 d x}{b}\right )}{x} \, dx,x,a+b x\right )}{85 d^3}-\frac {\left (2 B^2 (b c-a d)^2 g^2\right ) \operatorname {Subst}\left (\int \frac {\log \left (\frac {1}{-\frac {-b c+a d}{b}+\frac {d x}{b}}\right ) \log \left (-\frac {-85 b c+85 a d}{b}+\frac {85 d x}{b}\right )}{x} \, dx,x,a+b x\right )}{85 d^3}+\frac {\left (2 B^2 (b c-a d)^2 g^2\right ) \int \frac {\log \left (\frac {85 d (a+b x)}{-85 b c+85 a d}\right ) \log (85 c+85 d x)}{85 c+85 d x} \, dx}{d^2}+\frac {\left (2 B^2 (b c-a d)^2 g^2 \left (-\log (a+b x)-\log \left (\frac {1}{c+d x}\right )+\log \left (\frac {e (a+b x)}{c+d x}\right )\right )\right ) \int \frac {\log \left (\frac {85 d (a+b x)}{-85 b c+85 a d}\right )}{85 c+85 d x} \, dx}{d^2}\\ &=-\frac {A b B (b c-a d) g^2 x}{85 d^2}+\frac {a B^2 (b c-a d) g^2 \log ^2(a+b x)}{85 d^2}-\frac {B^2 (b c-a d) g^2 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{85 d^2}-\frac {2 a B (b c-a d) g^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{85 d^2}-\frac {b (b c-a d) g^2 x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{85 d^2}+\frac {g^2 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{170 d}+\frac {B^2 (b c-a d)^2 g^2 \log (c+d x)}{85 d^3}-\frac {2 b B^2 c (b c-a d) g^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{85 d^3}-\frac {B^2 (b c-a d)^2 g^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{85 d^3}+\frac {2 b B c (b c-a d) g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{85 d^3}+\frac {B (b c-a d)^2 g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{85 d^3}+\frac {b B^2 c (b c-a d) g^2 \log ^2(c+d x)}{85 d^3}+\frac {B^2 (b c-a d)^2 g^2 \log ^2(c+d x)}{170 d^3}-\frac {B^2 (b c-a d)^2 g^2 \log ^2(a+b x) \log (85 (c+d x))}{85 d^3}-\frac {2 B^2 (b c-a d)^2 g^2 \log (a+b x) \log \left (\frac {1}{c+d x}\right ) \log (85 (c+d x))}{85 d^3}+\frac {A B (b c-a d)^2 g^2 \log ^2(85 (c+d x))}{85 d^3}-\frac {2 a B^2 (b c-a d) g^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{85 d^2}-\frac {2 A B (b c-a d)^2 g^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (85 c+85 d x)}{85 d^3}+\frac {2 B^2 (b c-a d)^2 g^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 (85 c+85 d x)}{85 d^3}+\frac {(b c-a d)^2 g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (85 c+85 d x)}{85 d^3}-\frac {B^2 (b c-a d)^2 g^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(85 c+85 d x)}{85 d^3}+\frac {B^2 (b c-a d)^2 g^2 \log \left (\frac {e (a+b x)}{c+d x}\right ) \log ^2(85 c+85 d x)}{85 d^3}-\frac {2 a B^2 (b c-a d) g^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{85 d^2}-\frac {2 b B^2 c (b c-a d) g^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{85 d^3}-\frac {2 A B (b c-a d)^2 g^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{85 d^3}-\frac {B^2 (b c-a d)^2 g^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{85 d^3}+\frac {\left (B^2 (b c-a d)^2 g^2\right ) \operatorname {Subst}\left (\int \frac {\log ^2(x)}{x} \, dx,x,85 c+85 d x\right )}{85 d^3}+\frac {\left (2 B^2 (b c-a d)^2 g^2\right ) \operatorname {Subst}\left (\int \frac {\log (x) \log \left (\frac {85 d \left (\frac {-85 b c+85 a d}{85 d}+\frac {b x}{85 d}\right )}{-85 b c+85 a d}\right )}{x} \, dx,x,85 c+85 d x\right )}{85 d^3}+\frac {\left (2 B^2 (b c-a d)^2 g^2\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 )}{85 b d^2}+\frac {\left (B^2 (b c-a d)^2 g^2\right ) \operatorname {Subst}\left (\int \frac {\log ^2(x)}{\frac {85 b c-85 a d}{b}+\frac {85 d x}{b}} \, dx,x,a+b x\right )}{b d^2}-\frac {\left (2 B^2 (b c-a d)^2 g^2\right ) \operatorname {Subst}\left (\int \frac {\log (x) \log \left (-\frac {-85 b c+85 a d}{b}+\frac {85 d x}{b}\right )}{-\frac {-85 b c+85 a d}{b}+\frac {85 d x}{b}} \, dx,x,a+b x\right )}{b d^2}+\frac {\left (2 B^2 (b c-a d)^2 g^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}{-85 b c+85 a d}\right )}{x} \, dx,x,85 c+85 d x\right )}{85 d^3}\\ &=-\frac {A b B (b c-a d) g^2 x}{85 d^2}+\frac {a B^2 (b c-a d) g^2 \log ^2(a+b x)}{85 d^2}-\frac {B^2 (b c-a d) g^2 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{85 d^2}-\frac {2 a B (b c-a d) g^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{85 d^2}-\frac {b (b c-a d) g^2 x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{85 d^2}+\frac {g^2 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{170 d}+\frac {B^2 (b c-a d)^2 g^2 \log (c+d x)}{85 d^3}-\frac {2 b B^2 c (b c-a d) g^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{85 d^3}-\frac {B^2 (b c-a d)^2 g^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{85 d^3}+\frac {2 b B c (b c-a d) g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{85 d^3}+\frac {B (b c-a d)^2 g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{85 d^3}+\frac {b B^2 c (b c-a d) g^2 \log ^2(c+d x)}{85 d^3}+\frac {B^2 (b c-a d)^2 g^2 \log ^2(c+d x)}{170 d^3}-\frac {B^2 (b c-a d)^2 g^2 \log ^2(a+b x) \log (85 (c+d x))}{85 d^3}-\frac {2 B^2 (b c-a d)^2 g^2 \log (a+b x) \log \left (\frac {1}{c+d x}\right ) \log (85 (c+d x))}{85 d^3}+\frac {A B (b c-a d)^2 g^2 \log ^2(85 (c+d x))}{85 d^3}-\frac {2 a B^2 (b c-a d) g^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{85 d^2}+\frac {B^2 (b c-a d)^2 g^2 \log ^2(a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{85 d^3}-\frac {2 A B (b c-a d)^2 g^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (85 c+85 d x)}{85 d^3}+\frac {2 B^2 (b c-a d)^2 g^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 (85 c+85 d x)}{85 d^3}+\frac {(b c-a d)^2 g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (85 c+85 d x)}{85 d^3}-\frac {B^2 (b c-a d)^2 g^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(85 c+85 d x)}{85 d^3}+\frac {B^2 (b c-a d)^2 g^2 \log \left (\frac {e (a+b x)}{c+d x}\right ) \log ^2(85 c+85 d x)}{85 d^3}-\frac {2 a B^2 (b c-a d) g^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{85 d^2}-\frac {2 b B^2 c (b c-a d) g^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{85 d^3}-\frac {2 A B (b c-a d)^2 g^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{85 d^3}-\frac {B^2 (b c-a d)^2 g^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{85 d^3}+\frac {2 B^2 (b c-a d)^2 g^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 )}{85 d^3}-\frac {2 B^2 (b c-a d)^2 g^2 \log (85 (c+d x)) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{85 d^3}+\frac {\left (B^2 (b c-a d)^2 g^2\right ) \operatorname {Subst}\left (\int x^2 \, dx,x,\log (85 (c+d x))\right )}{85 d^3}-\frac {\left (2 B^2 (b c-a d)^2 g^2\right ) \operatorname {Subst}\left (\int \frac {\log (x) \log \left (\frac {-85 b c+85 a d}{85 d}+\frac {b x}{85 d}\right )}{x} \, dx,x,85 (c+d x)\right )}{85 d^3}+\frac {\left (2 B^2 (b c-a d)^2 g^2\right ) \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 )}{85 d^3}-\frac {\left (2 B^2 (b c-a d)^2 g^2\right ) \operatorname {Subst}\left (\int \frac {\log (x) \log \left (1+\frac {85 d x}{85 b c-85 a d}\right )}{x} \, dx,x,a+b x\right )}{85 d^3}+\frac {\left (2 B^2 (b c-a d)^2 g^2\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {b x}{-85 b c+85 a d}\right )}{x} \, dx,x,85 c+85 d x\right )}{85 d^3}\\ &=-\frac {A b B (b c-a d) g^2 x}{85 d^2}+\frac {a B^2 (b c-a d) g^2 \log ^2(a+b x)}{85 d^2}-\frac {B^2 (b c-a d)^2 g^2 \log (a+b x) \log ^2\left (\frac {1}{c+d x}\right )}{85 d^3}-\frac {B^2 (b c-a d) g^2 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{85 d^2}-\frac {2 a B (b c-a d) g^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{85 d^2}-\frac {b (b c-a d) g^2 x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{85 d^2}+\frac {g^2 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{170 d}+\frac {B^2 (b c-a d)^2 g^2 \log (c+d x)}{85 d^3}-\frac {2 b B^2 c (b c-a d) g^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{85 d^3}-\frac {B^2 (b c-a d)^2 g^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{85 d^3}+\frac {2 b B c (b c-a d) g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{85 d^3}+\frac {B (b c-a d)^2 g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{85 d^3}+\frac {b B^2 c (b c-a d) g^2 \log ^2(c+d x)}{85 d^3}+\frac {B^2 (b c-a d)^2 g^2 \log ^2(c+d x)}{170 d^3}-\frac {B^2 (b c-a d)^2 g^2 \log ^2(a+b x) \log (85 (c+d x))}{85 d^3}-\frac {2 B^2 (b c-a d)^2 g^2 \log (a+b x) \log \left (\frac {1}{c+d x}\right ) \log (85 (c+d x))}{85 d^3}+\frac {A B (b c-a d)^2 g^2 \log ^2(85 (c+d x))}{85 d^3}-\frac {B^2 (b c-a d)^2 g^2 \log (a+b x) \log ^2(85 (c+d x))}{85 d^3}+\frac {B^2 (b c-a d)^2 g^2 \log ^3(85 (c+d x))}{255 d^3}-\frac {2 a B^2 (b c-a d) g^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{85 d^2}+\frac {B^2 (b c-a d)^2 g^2 \log ^2(a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{85 d^3}-\frac {2 A B (b c-a d)^2 g^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (85 c+85 d x)}{85 d^3}+\frac {2 B^2 (b c-a d)^2 g^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 (85 c+85 d x)}{85 d^3}+\frac {(b c-a d)^2 g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (85 c+85 d x)}{85 d^3}-\frac {B^2 (b c-a d)^2 g^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(85 c+85 d x)}{85 d^3}+\frac {B^2 (b c-a d)^2 g^2 \log \left (\frac {e (a+b x)}{c+d x}\right ) \log ^2(85 c+85 d x)}{85 d^3}-\frac {2 a B^2 (b c-a d) g^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{85 d^2}+\frac {2 B^2 (b c-a d)^2 g^2 \log (a+b x) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{85 d^3}-\frac {2 b B^2 c (b c-a d) g^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{85 d^3}-\frac {2 A B (b c-a d)^2 g^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{85 d^3}-\frac {B^2 (b c-a d)^2 g^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{85 d^3}+\frac {2 B^2 (b c-a d)^2 g^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 )}{85 d^3}-\frac {2 B^2 (b c-a d)^2 g^2 \log (85 (c+d x)) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{85 d^3}+\frac {2 B^2 (b c-a d)^2 g^2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{85 d^3}+\frac {\left (b B^2 (b c-a d)^2 g^2\right ) \operatorname {Subst}\left (\int \frac {\log ^2(x)}{\frac {-85 b c+85 a d}{85 d}+\frac {b x}{85 d}} \, dx,x,85 (c+d x)\right )}{7225 d^4}+\frac {\left (b B^2 (b c-a d)^2 g^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 )}{85 d^4}-\frac {\left (2 B^2 (b c-a d)^2 g^2\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {85 d x}{85 b c-85 a d}\right )}{x} \, dx,x,a+b x\right )}{85 d^3}\\ &=-\frac {A b B (b c-a d) g^2 x}{85 d^2}+\frac {a B^2 (b c-a d) g^2 \log ^2(a+b x)}{85 d^2}-\frac {B^2 (b c-a d)^2 g^2 \log (a+b x) \log ^2\left (\frac {1}{c+d x}\right )}{85 d^3}+\frac {B^2 (b c-a d)^2 g^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2\left (\frac {1}{c+d x}\right )}{85 d^3}-\frac {B^2 (b c-a d) g^2 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{85 d^2}-\frac {2 a B (b c-a d) g^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{85 d^2}-\frac {b (b c-a d) g^2 x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{85 d^2}+\frac {g^2 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{170 d}+\frac {B^2 (b c-a d)^2 g^2 \log (c+d x)}{85 d^3}-\frac {2 b B^2 c (b c-a d) g^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{85 d^3}-\frac {B^2 (b c-a d)^2 g^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{85 d^3}+\frac {2 b B c (b c-a d) g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{85 d^3}+\frac {B (b c-a d)^2 g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{85 d^3}+\frac {b B^2 c (b c-a d) g^2 \log ^2(c+d x)}{85 d^3}+\frac {B^2 (b c-a d)^2 g^2 \log ^2(c+d x)}{170 d^3}-\frac {B^2 (b c-a d)^2 g^2 \log ^2(a+b x) \log (85 (c+d x))}{85 d^3}-\frac {2 B^2 (b c-a d)^2 g^2 \log (a+b x) \log \left (\frac {1}{c+d x}\right ) \log (85 (c+d x))}{85 d^3}+\frac {A B (b c-a d)^2 g^2 \log ^2(85 (c+d x))}{85 d^3}-\frac {B^2 (b c-a d)^2 g^2 \log (a+b x) \log ^2(85 (c+d x))}{85 d^3}+\frac {B^2 (b c-a d)^2 g^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(85 (c+d x))}{85 d^3}+\frac {B^2 (b c-a d)^2 g^2 \log ^3(85 (c+d x))}{255 d^3}-\frac {2 a B^2 (b c-a d) g^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{85 d^2}+\frac {B^2 (b c-a d)^2 g^2 \log ^2(a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{85 d^3}-\frac {2 A B (b c-a d)^2 g^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (85 c+85 d x)}{85 d^3}+\frac {2 B^2 (b c-a d)^2 g^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 (85 c+85 d x)}{85 d^3}+\frac {(b c-a d)^2 g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (85 c+85 d x)}{85 d^3}-\frac {B^2 (b c-a d)^2 g^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(85 c+85 d x)}{85 d^3}+\frac {B^2 (b c-a d)^2 g^2 \log \left (\frac {e (a+b x)}{c+d x}\right ) \log ^2(85 c+85 d x)}{85 d^3}-\frac {2 a B^2 (b c-a d) g^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{85 d^2}+\frac {2 B^2 (b c-a d)^2 g^2 \log (a+b x) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{85 d^3}-\frac {2 b B^2 c (b c-a d) g^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{85 d^3}-\frac {2 A B (b c-a d)^2 g^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{85 d^3}-\frac {B^2 (b c-a d)^2 g^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{85 d^3}+\frac {2 B^2 (b c-a d)^2 g^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 )}{85 d^3}-\frac {2 B^2 (b c-a d)^2 g^2 \log (85 (c+d x)) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{85 d^3}-\frac {2 B^2 (b c-a d)^2 g^2 \text {Li}_3\left (-\frac {d (a+b x)}{b c-a d}\right )}{85 d^3}+\frac {2 B^2 (b c-a d)^2 g^2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{85 d^3}+\frac {\left (2 B^2 (b c-a d)^2 g^2\right ) \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 )}{85 d^3}-\frac {\left (2 B^2 (b c-a d)^2 g^2\right ) \operatorname {Subst}\left (\int \frac {\log (x) \log \left (1+\frac {b x}{-85 b c+85 a d}\right )}{x} \, dx,x,85 (c+d x)\right )}{85 d^3}\\ &=-\frac {A b B (b c-a d) g^2 x}{85 d^2}+\frac {a B^2 (b c-a d) g^2 \log ^2(a+b x)}{85 d^2}-\frac {B^2 (b c-a d)^2 g^2 \log (a+b x) \log ^2\left (\frac {1}{c+d x}\right )}{85 d^3}+\frac {B^2 (b c-a d)^2 g^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2\left (\frac {1}{c+d x}\right )}{85 d^3}-\frac {B^2 (b c-a d) g^2 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{85 d^2}-\frac {2 a B (b c-a d) g^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{85 d^2}-\frac {b (b c-a d) g^2 x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{85 d^2}+\frac {g^2 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{170 d}+\frac {B^2 (b c-a d)^2 g^2 \log (c+d x)}{85 d^3}-\frac {2 b B^2 c (b c-a d) g^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{85 d^3}-\frac {B^2 (b c-a d)^2 g^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{85 d^3}+\frac {2 b B c (b c-a d) g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{85 d^3}+\frac {B (b c-a d)^2 g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{85 d^3}+\frac {b B^2 c (b c-a d) g^2 \log ^2(c+d x)}{85 d^3}+\frac {B^2 (b c-a d)^2 g^2 \log ^2(c+d x)}{170 d^3}-\frac {B^2 (b c-a d)^2 g^2 \log ^2(a+b x) \log (85 (c+d x))}{85 d^3}-\frac {2 B^2 (b c-a d)^2 g^2 \log (a+b x) \log \left (\frac {1}{c+d x}\right ) \log (85 (c+d x))}{85 d^3}+\frac {A B (b c-a d)^2 g^2 \log ^2(85 (c+d x))}{85 d^3}-\frac {B^2 (b c-a d)^2 g^2 \log (a+b x) \log ^2(85 (c+d x))}{85 d^3}+\frac {B^2 (b c-a d)^2 g^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(85 (c+d x))}{85 d^3}+\frac {B^2 (b c-a d)^2 g^2 \log ^3(85 (c+d x))}{255 d^3}-\frac {2 a B^2 (b c-a d) g^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{85 d^2}+\frac {B^2 (b c-a d)^2 g^2 \log ^2(a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{85 d^3}-\frac {2 A B (b c-a d)^2 g^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (85 c+85 d x)}{85 d^3}+\frac {2 B^2 (b c-a d)^2 g^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 (85 c+85 d x)}{85 d^3}+\frac {(b c-a d)^2 g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (85 c+85 d x)}{85 d^3}-\frac {B^2 (b c-a d)^2 g^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(85 c+85 d x)}{85 d^3}+\frac {B^2 (b c-a d)^2 g^2 \log \left (\frac {e (a+b x)}{c+d x}\right ) \log ^2(85 c+85 d x)}{85 d^3}-\frac {2 a B^2 (b c-a d) g^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{85 d^2}+\frac {2 B^2 (b c-a d)^2 g^2 \log (a+b x) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{85 d^3}-\frac {2 b B^2 c (b c-a d) g^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{85 d^3}-\frac {2 A B (b c-a d)^2 g^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{85 d^3}-\frac {B^2 (b c-a d)^2 g^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{85 d^3}-\frac {2 B^2 (b c-a d)^2 g^2 \log \left (\frac {1}{c+d x}\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{85 d^3}+\frac {2 B^2 (b c-a d)^2 g^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 )}{85 d^3}-\frac {2 B^2 (b c-a d)^2 g^2 \text {Li}_3\left (-\frac {d (a+b x)}{b c-a d}\right )}{85 d^3}+\frac {2 B^2 (b c-a d)^2 g^2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{85 d^3}-\frac {\left (2 B^2 (b c-a d)^2 g^2\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{85 d^3}-\frac {\left (2 B^2 (b c-a d)^2 g^2\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {b x}{-85 b c+85 a d}\right )}{x} \, dx,x,85 (c+d x)\right )}{85 d^3}\\ &=-\frac {A b B (b c-a d) g^2 x}{85 d^2}+\frac {a B^2 (b c-a d) g^2 \log ^2(a+b x)}{85 d^2}-\frac {B^2 (b c-a d)^2 g^2 \log (a+b x) \log ^2\left (\frac {1}{c+d x}\right )}{85 d^3}+\frac {B^2 (b c-a d)^2 g^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2\left (\frac {1}{c+d x}\right )}{85 d^3}-\frac {B^2 (b c-a d) g^2 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{85 d^2}-\frac {2 a B (b c-a d) g^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{85 d^2}-\frac {b (b c-a d) g^2 x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{85 d^2}+\frac {g^2 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{170 d}+\frac {B^2 (b c-a d)^2 g^2 \log (c+d x)}{85 d^3}-\frac {2 b B^2 c (b c-a d) g^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{85 d^3}-\frac {B^2 (b c-a d)^2 g^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{85 d^3}+\frac {2 b B c (b c-a d) g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{85 d^3}+\frac {B (b c-a d)^2 g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{85 d^3}+\frac {b B^2 c (b c-a d) g^2 \log ^2(c+d x)}{85 d^3}+\frac {B^2 (b c-a d)^2 g^2 \log ^2(c+d x)}{170 d^3}-\frac {B^2 (b c-a d)^2 g^2 \log ^2(a+b x) \log (85 (c+d x))}{85 d^3}-\frac {2 B^2 (b c-a d)^2 g^2 \log (a+b x) \log \left (\frac {1}{c+d x}\right ) \log (85 (c+d x))}{85 d^3}+\frac {A B (b c-a d)^2 g^2 \log ^2(85 (c+d x))}{85 d^3}-\frac {B^2 (b c-a d)^2 g^2 \log (a+b x) \log ^2(85 (c+d x))}{85 d^3}+\frac {B^2 (b c-a d)^2 g^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(85 (c+d x))}{85 d^3}+\frac {B^2 (b c-a d)^2 g^2 \log ^3(85 (c+d x))}{255 d^3}-\frac {2 a B^2 (b c-a d) g^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{85 d^2}+\frac {B^2 (b c-a d)^2 g^2 \log ^2(a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{85 d^3}-\frac {2 A B (b c-a d)^2 g^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (85 c+85 d x)}{85 d^3}+\frac {2 B^2 (b c-a d)^2 g^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 (85 c+85 d x)}{85 d^3}+\frac {(b c-a d)^2 g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (85 c+85 d x)}{85 d^3}-\frac {B^2 (b c-a d)^2 g^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(85 c+85 d x)}{85 d^3}+\frac {B^2 (b c-a d)^2 g^2 \log \left (\frac {e (a+b x)}{c+d x}\right ) \log ^2(85 c+85 d x)}{85 d^3}-\frac {2 a B^2 (b c-a d) g^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{85 d^2}+\frac {2 B^2 (b c-a d)^2 g^2 \log (a+b x) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{85 d^3}-\frac {2 b B^2 c (b c-a d) g^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{85 d^3}-\frac {2 A B (b c-a d)^2 g^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{85 d^3}-\frac {B^2 (b c-a d)^2 g^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{85 d^3}-\frac {2 B^2 (b c-a d)^2 g^2 \log \left (\frac {1}{c+d x}\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{85 d^3}+\frac {2 B^2 (b c-a d)^2 g^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 )}{85 d^3}-\frac {2 B^2 (b c-a d)^2 g^2 \text {Li}_3\left (-\frac {d (a+b x)}{b c-a d}\right )}{85 d^3}-\frac {2 B^2 (b c-a d)^2 g^2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{85 d^3}\\ \end {align*}

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Mathematica [B]  time = 1.73, size = 1514, normalized size = 2.82 \[ \text {result too large to display} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [F]  time = 0.70, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {A^{2} b^{2} g^{2} x^{2} + 2 \, A^{2} a b g^{2} x + A^{2} a^{2} g^{2} + {\left (B^{2} b^{2} g^{2} x^{2} + 2 \, B^{2} a b g^{2} x + B^{2} a^{2} g^{2}\right )} \log \left (\frac {b e x + a e}{d x + c}\right )^{2} + 2 \, {\left (A B b^{2} g^{2} x^{2} + 2 \, A B a b g^{2} x + A B a^{2} g^{2}\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{d i x + c i}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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maple [F]  time = 1.96, size = 0, normalized size = 0.00 \[ \int \frac {\left (b g x +a g \right )^{2} \left (B \ln \left (\frac {\left (b x +a \right ) e}{d x +c}\right )+A \right )^{2}}{d i x +c i}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ 2 \, A^{2} a b g^{2} {\left (\frac {x}{d i} - \frac {c \log \left (d x + c\right )}{d^{2} i}\right )} + \frac {1}{2} \, A^{2} b^{2} g^{2} {\left (\frac {2 \, c^{2} \log \left (d x + c\right )}{d^{3} i} + \frac {d x^{2} - 2 \, c x}{d^{2} i}\right )} + \frac {A^{2} a^{2} g^{2} \log \left (d i x + c i\right )}{d i} + \frac {2 \, {\left (b^{2} c^{2} g^{2} - 2 \, a b c d g^{2} + a^{2} d^{2} g^{2}\right )} B^{2} \log \left (d x + c\right )^{3} + 3 \, {\left (B^{2} b^{2} d^{2} g^{2} x^{2} - 2 \, {\left (b^{2} c d g^{2} - 2 \, a b d^{2} g^{2}\right )} B^{2} x\right )} \log \left (d x + c\right )^{2}}{6 \, d^{3} i} - \int -\frac {B^{2} a^{2} d g^{2} \log \relax (e)^{2} + 2 \, A B a^{2} d g^{2} \log \relax (e) + {\left (B^{2} b^{2} d g^{2} \log \relax (e)^{2} + 2 \, A B b^{2} d g^{2} \log \relax (e)\right )} x^{2} + {\left (B^{2} b^{2} d g^{2} x^{2} + 2 \, B^{2} a b d g^{2} x + B^{2} a^{2} d g^{2}\right )} \log \left (b x + a\right )^{2} + 2 \, {\left (B^{2} a b d g^{2} \log \relax (e)^{2} + 2 \, A B a b d g^{2} \log \relax (e)\right )} x + 2 \, {\left (B^{2} a^{2} d g^{2} \log \relax (e) + A B a^{2} d g^{2} + {\left (B^{2} b^{2} d g^{2} \log \relax (e) + A B b^{2} d g^{2}\right )} x^{2} + 2 \, {\left (B^{2} a b d g^{2} \log \relax (e) + A B a b d g^{2}\right )} x\right )} \log \left (b x + a\right ) - {\left (2 \, B^{2} a^{2} d g^{2} \log \relax (e) + 2 \, A B a^{2} d g^{2} + {\left (2 \, A B b^{2} d g^{2} + {\left (2 \, g^{2} \log \relax (e) + g^{2}\right )} B^{2} b^{2} d\right )} x^{2} + 2 \, {\left (2 \, A B a b d g^{2} - {\left (b^{2} c g^{2} - 2 \, {\left (g^{2} \log \relax (e) + g^{2}\right )} a b d\right )} B^{2}\right )} x + 2 \, {\left (B^{2} b^{2} d g^{2} x^{2} + 2 \, B^{2} a b d g^{2} x + B^{2} a^{2} d g^{2}\right )} \log \left (b x + a\right )\right )} \log \left (d x + c\right )}{d^{2} i x + c d i}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {g^{2} \left (\int \frac {A^{2} a^{2}}{c + d x}\, dx + \int \frac {A^{2} b^{2} x^{2}}{c + d x}\, dx + \int \frac {B^{2} a^{2} \log {\left (\frac {a e}{c + d x} + \frac {b e x}{c + d x} \right )}^{2}}{c + d x}\, dx + \int \frac {2 A B a^{2} \log {\left (\frac {a e}{c + d x} + \frac {b e x}{c + d x} \right )}}{c + d x}\, dx + \int \frac {2 A^{2} a b x}{c + d x}\, dx + \int \frac {B^{2} b^{2} x^{2} \log {\left (\frac {a e}{c + d x} + \frac {b e x}{c + d x} \right )}^{2}}{c + d x}\, dx + \int \frac {2 A B b^{2} x^{2} \log {\left (\frac {a e}{c + d x} + \frac {b e x}{c + d x} \right )}}{c + d x}\, dx + \int \frac {2 B^{2} a b x \log {\left (\frac {a e}{c + d x} + \frac {b e x}{c + d x} \right )}^{2}}{c + d x}\, dx + \int \frac {4 A B a b x \log {\left (\frac {a e}{c + d x} + \frac {b e x}{c + d x} \right )}}{c + d x}\, dx\right )}{i} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

g**2*(Integral(A**2*a**2/(c + d*x), x) + Integral(A**2*b**2*x**2/(c + d*x), x) + Integral(B**2*a**2*log(a*e/(c
+ d*x) + b*e*x/(c + d*x))**2/(c + d*x), x) + Integral(2*A*B*a**2*log(a*e/(c + d*x) + b*e*x/(c + d*x))/(c + d*
x), x) + Integral(2*A**2*a*b*x/(c + d*x), x) + Integral(B**2*b**2*x**2*log(a*e/(c + d*x) + b*e*x/(c + d*x))**2
/(c + d*x), x) + Integral(2*A*B*b**2*x**2*log(a*e/(c + d*x) + b*e*x/(c + d*x))/(c + d*x), x) + Integral(2*B**2
*a*b*x*log(a*e/(c + d*x) + b*e*x/(c + d*x))**2/(c + d*x), x) + Integral(4*A*B*a*b*x*log(a*e/(c + d*x) + b*e*x/
(c + d*x))/(c + d*x), x))/i

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