3.1.85 \(\int \frac {(a g+b g x)^2 (A+B \log (\frac {e (a+b x)}{c+d x}))^2}{c i+d i x} \, dx\) [85]

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

[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 [A]
time = 0.43, antiderivative size = 536, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 12, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2562, 2395, 2356, 2389, 2379, 2438, 2351, 31, 2355, 2354, 2421, 6724} \begin {gather*} -\frac {2 B g^2 (b c-a d)^2 \text {PolyLog}\left (2,\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 {4 B^2 g^2 (b c-a d)^2 \text {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{d^3 i}-\frac {B^2 g^2 (b c-a d)^2 \text {PolyLog}\left (2,\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 {PolyLog}\left (3,\frac {d (a+b x)}{b (c+d x)}\right )}{d^3 i}+\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 {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 {B^2 g^2 (b c-a d)^2 \log (c+d x)}{d^3 i} \end {gather*}

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]

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

Rule 31

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

Rule 2351

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

Rule 2354

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 2355

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

Rule 2356

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_.), x_Symbol] :> Simp[(d + e*x)^(q + 1)
*((a + b*Log[c*x^n])^p/(e*(q + 1))), x] - Dist[b*n*(p/(e*(q + 1))), Int[((d + e*x)^(q + 1)*(a + b*Log[c*x^n])^
(p - 1))/x, x], x] /; FreeQ[{a, b, c, d, e, n, p, q}, x] && GtQ[p, 0] && NeQ[q, -1] && (EqQ[p, 1] || (Integers
Q[2*p, 2*q] &&  !IGtQ[q, 0]) || (EqQ[p, 2] && NeQ[q, 1]))

Rule 2379

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

Rule 2389

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_))/(x_), x_Symbol] :> Dist[1/d, Int[(d
+ e*x)^(q + 1)*((a + b*Log[c*x^n])^p/x), x], x] - Dist[e/d, Int[(d + e*x)^q*(a + b*Log[c*x^n])^p, x], x] /; F
reeQ[{a, b, c, d, e, n}, x] && IGtQ[p, 0] && LtQ[q, -1] && IntegerQ[2*q]

Rule 2395

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol]
:> With[{u = ExpandIntegrand[(a + b*Log[c*x^n])^p, (f*x)^m*(d + e*x^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[
{a, b, c, d, e, f, m, n, p, q, r}, x] && IntegerQ[q] && (GtQ[q, 0] || (IGtQ[p, 0] && IntegerQ[m] && IntegerQ[r
]))

Rule 2421

Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.))/(x_), x_Symbol] :> Simp
[(-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*L
og[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 2438

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 2562

Int[((A_.) + Log[(e_.)*((a_.) + (b_.)*(x_))^(n_.)*((c_.) + (d_.)*(x_))^(mn_)]*(B_.))^(p_.)*((f_.) + (g_.)*(x_)
)^(m_.)*((h_.) + (i_.)*(x_))^(q_.), x_Symbol] :> Dist[(b*c - a*d)^(m + q + 1)*(g/b)^m*(i/d)^q, Subst[Int[x^m*(
(A + B*Log[e*x^n])^p/(b - d*x)^(m + q + 2)), x], x, (a + b*x)/(c + d*x)], x] /; FreeQ[{a, b, c, d, e, f, g, h,
i, A, B, n, p}, x] && EqQ[n + mn, 0] && IGtQ[n, 0] && NeQ[b*c - a*d, 0] && EqQ[b*f - a*g, 0] && EqQ[d*h - c*i
, 0] && IntegersQ[m, q]

Rule 6724

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]

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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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] Leaf count is larger than twice the leaf count of optimal. \(1514\) vs. \(2(536)=1072\).
time = 1.07, size = 1514, normalized size = 2.82 \begin {gather*} \frac {g^2 \left (-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 \left (-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 \left (\frac {c}{d}+x\right )-b^2 c^2 \log ^2\left (\frac {c}{d}+x\right )-a^2 d^2 \log (a+b x)-2 b^2 c d x \log \left (\frac {e (a+b x)}{c+d x}\right )+b^2 d^2 x^2 \log \left (\frac {e (a+b x)}{c+d x}\right )+b^2 c^2 \log (c+d x)+2 b^2 c^2 \log \left (\frac {c}{d}+x\right ) \log (c+d x)+2 b^2 c^2 \log \left (\frac {e (a+b x)}{c+d x}\right ) \log (c+d x)-2 b c \log \left (\frac {a}{b}+x\right ) \left (a d+b c \log (c+d x)-b c \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )+2 b^2 c^2 \text {Li}_2\left (\frac {d (a+b x)}{-b c+a d}\right )\right )-2 a^2 A B d^2 \left (\log ^2\left (\frac {c}{d}+x\right )+2 \left (\log \left (\frac {a}{b}+x\right )-\log \left (\frac {c}{d}+x\right )-\log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)-2 \left (\log \left (\frac {a}{b}+x\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )+\text {Li}_2\left (\frac {d (a+b x)}{-b c+a d}\right )\right )\right )-4 a A B d \left (-2 d (a+b x) \left (-1+\log \left (\frac {a}{b}+x\right )\right )+2 b (c+d x) \left (-1+\log \left (\frac {c}{d}+x\right )\right )-b c \log ^2\left (\frac {c}{d}+x\right )+2 b \left (\log \left (\frac {a}{b}+x\right )-\log \left (\frac {c}{d}+x\right )-\log \left (\frac {e (a+b x)}{c+d x}\right )\right ) (d x-c \log (c+d x))+2 b c \left (\log \left (\frac {a}{b}+x\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )+\text {Li}_2\left (\frac {d (a+b x)}{-b c+a d}\right )\right )\right )+4 a B^2 d \left (d (a+b x) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )+b c \log ^2\left (\frac {e (a+b x)}{c+d x}\right ) \log \left (\frac {b c-a d}{b c+b d x}\right )-(b c-a d) \left (\log \left (\frac {b c-a d}{b c+b d x}\right ) \left (2 \log \left (\frac {d (a+b x)}{-b c+a d}\right )-2 \log \left (\frac {e (a+b x)}{c+d x}\right )+\log \left (\frac {b c-a d}{b c+b d x}\right )\right )-2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )\right )+2 b c \left (\log \left (\frac {e (a+b x)}{c+d x}\right ) \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right )-\text {Li}_3\left (\frac {d (a+b x)}{b (c+d x)}\right )\right )\right )+B^2 \left (2 d (-b c+a d) (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )-2 a^2 d^2 \log (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )+b^2 d^2 x^2 \log ^2\left (\frac {e (a+b x)}{c+d x}\right )-2 b c d (a+b x) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )+2 (b c-a d)^2 \log (c+d x)-2 b^2 c^2 \log \left (\frac {e (a+b x)}{c+d x}\right ) \log \left (\frac {b c-a d}{b c+b d x}\right )-2 b^2 c^2 \log ^2\left (\frac {e (a+b x)}{c+d x}\right ) \log \left (\frac {b c-a d}{b c+b d x}\right )+a^2 d^2 \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 \text {Li}_2\left (\frac {d (a+b x)}{-b c+a d}\right )\right )+b^2 c^2 \left (\log \left (\frac {b c-a d}{b c+b d x}\right ) \left (2 \log \left (\frac {d (a+b x)}{-b c+a d}\right )+\log \left (\frac {b c-a d}{b c+b d x}\right )\right )-2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )\right )+2 b c (b c-a d) \left (\log \left (\frac {b c-a d}{b c+b d x}\right ) \left (2 \log \left (\frac {d (a+b x)}{-b c+a d}\right )-2 \log \left (\frac {e (a+b x)}{c+d x}\right )+\log \left (\frac {b c-a d}{b c+b d x}\right )\right )-2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )\right )-4 b^2 c^2 \left (\log \left (\frac {e (a+b x)}{c+d x}\right ) \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right )-\text {Li}_3\left (\frac {d (a+b x)}{b (c+d x)}\right )\right )\right )-2 a^2 B^2 d^2 \left (\log ^2\left (\frac {e (a+b x)}{c+d x}\right ) \log \left (\frac {b c-a d}{b c+b d x}\right )+2 \log \left (\frac {e (a+b x)}{c+d x}\right ) \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right )-2 \text {Li}_3\left (\frac {d (a+b x)}{b (c+d x)}\right )\right )\right )}{2 d^3 i} \end {gather*}

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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Maple [F]
time = 0.28, size = 0, normalized size = 0.00 \[\int \frac {\left (b g x +a g \right )^{2} \left (A +B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )\right )^{2}}{d i x +c i}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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*(-I*x/d + I*c*log(d*x + c)/d^2) - 1/2*A^2*b^2*g^2*(2*I*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(I*d*x + I*c)/d + 1/6*(2*(-I*b^2*c^2*g^2 + 2*I*a*b*c*d*g^2 - I*a^2*d^2*g^2)*B^2*log(d*
x + c)^3 + 3*(-I*B^2*b^2*d^2*g^2*x^2 + 2*(I*b^2*c*d*g^2 - 2*I*a*b*d^2*g^2)*B^2*x)*log(d*x + c)^2)/d^3 + integr
ate((-2*I*A*B*a^2*d*g^2 - I*B^2*a^2*d*g^2 + (-2*I*A*B*b^2*d*g^2 - I*B^2*b^2*d*g^2)*x^2 + (-I*B^2*b^2*d*g^2*x^2
- 2*I*B^2*a*b*d*g^2*x - I*B^2*a^2*d*g^2)*log(b*x + a)^2 - 2*(2*I*A*B*a*b*d*g^2 + I*B^2*a*b*d*g^2)*x - 2*(I*A*
B*a^2*d*g^2 + I*B^2*a^2*d*g^2 + (I*A*B*b^2*d*g^2 + I*B^2*b^2*d*g^2)*x^2 + 2*(I*A*B*a*b*d*g^2 + I*B^2*a*b*d*g^2
)*x)*log(b*x + a) + (2*I*A*B*a^2*d*g^2 + 2*I*B^2*a^2*d*g^2 + (2*I*A*B*b^2*d*g^2 + 3*I*B^2*b^2*d*g^2)*x^2 - 2*(
-2*I*A*B*a*b*d*g^2 + (I*b^2*c*g^2 - 4*I*a*b*d*g^2)*B^2)*x - 2*(-I*B^2*b^2*d*g^2*x^2 - 2*I*B^2*a*b*d*g^2*x - I*
B^2*a^2*d*g^2)*log(b*x + a))*log(d*x + c))/(d^2*x + c*d), x)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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]

Timed out

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \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 \end {gather*}

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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