3.1.89 \(\int \frac {(A+B \log (\frac {e (a+b x)}{c+d x}))^2}{(a g+b g x)^2 (c i+d i x)} \, dx\) [89]

Optimal. Leaf size=183 \[ -\frac {2 b B^2 (c+d x)}{(b c-a d)^2 g^2 i (a+b x)}-\frac {2 b B (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^2 g^2 i (a+b x)}-\frac {b (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(b c-a d)^2 g^2 i (a+b x)}-\frac {d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^3}{3 B (b c-a d)^2 g^2 i} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.19, antiderivative size = 183, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2562, 2395, 2342, 2341, 2339, 30} \begin {gather*} -\frac {d \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^3}{3 B g^2 i (b c-a d)^2}-\frac {b (c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{g^2 i (a+b x) (b c-a d)^2}-\frac {2 b B (c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{g^2 i (a+b x) (b c-a d)^2}-\frac {2 b B^2 (c+d x)}{g^2 i (a+b x) (b c-a d)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 30

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

Rule 2339

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 2341

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*Log[c*x^
n])/(d*(m + 1))), x] - Simp[b*n*((d*x)^(m + 1)/(d*(m + 1)^2)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1
]

Rule 2342

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

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 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]

Rubi steps

\begin {align*} \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(89 c+89 d x) (a g+b g x)^2} \, dx &=\int \left (\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{89 (b c-a d) g^2 (a+b x)^2}-\frac {b d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{89 (b c-a d)^2 g^2 (a+b x)}+\frac {d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{89 (b c-a d)^2 g^2 (c+d x)}\right ) \, dx\\ &=-\frac {(b d) \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{a+b x} \, dx}{89 (b c-a d)^2 g^2}+\frac {d^2 \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{c+d x} \, dx}{89 (b c-a d)^2 g^2}+\frac {b \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a+b x)^2} \, dx}{89 (b c-a d) g^2}\\ &=-\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{89 (b c-a d) g^2 (a+b x)}-\frac {d \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{89 (b c-a d)^2 g^2}+\frac {d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{89 (b c-a d)^2 g^2}+\frac {(2 B d) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{e (a+b x)} \, dx}{89 (b c-a d)^2 g^2}-\frac {(2 B d) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{e (a+b x)} \, dx}{89 (b c-a d)^2 g^2}+\frac {(2 B) \int \frac {(b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a+b x)^2 (c+d x)} \, dx}{89 (b c-a d) g^2}\\ &=-\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{89 (b c-a d) g^2 (a+b x)}-\frac {d \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{89 (b c-a d)^2 g^2}+\frac {d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{89 (b c-a d)^2 g^2}+\frac {(2 B) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^2 (c+d x)} \, dx}{89 g^2}+\frac {(2 B d) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{a+b x} \, dx}{89 (b c-a d)^2 e g^2}-\frac {(2 B d) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{a+b x} \, dx}{89 (b c-a d)^2 e g^2}\\ &=-\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{89 (b c-a d) g^2 (a+b x)}-\frac {d \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{89 (b c-a d)^2 g^2}+\frac {d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{89 (b c-a d)^2 g^2}+\frac {(2 B) \int \left (\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d) (a+b x)^2}-\frac {b d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^2 (a+b x)}+\frac {d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^2 (c+d x)}\right ) \, dx}{89 g^2}+\frac {(2 B d) \int \frac {(b c-a d) e \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a+b x) (c+d x)} \, dx}{89 (b c-a d)^2 e g^2}-\frac {(2 B d) \int \frac {(b c-a d) e \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{(a+b x) (c+d x)} \, dx}{89 (b c-a d)^2 e g^2}\\ &=-\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{89 (b c-a d) g^2 (a+b x)}-\frac {d \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{89 (b c-a d)^2 g^2}+\frac {d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{89 (b c-a d)^2 g^2}-\frac {(2 b B d) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{89 (b c-a d)^2 g^2}+\frac {\left (2 B d^2\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{c+d x} \, dx}{89 (b c-a d)^2 g^2}+\frac {(2 b B) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^2} \, dx}{89 (b c-a d) g^2}+\frac {(2 B d) \int \frac {\log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a+b x) (c+d x)} \, dx}{89 (b c-a d) g^2}-\frac {(2 B d) \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{(a+b x) (c+d x)} \, dx}{89 (b c-a d) g^2}\\ &=-\frac {2 B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{89 (b c-a d) g^2 (a+b x)}-\frac {2 B d \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{89 (b c-a d)^2 g^2}-\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{89 (b c-a d) g^2 (a+b x)}-\frac {d \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{89 (b c-a d)^2 g^2}+\frac {2 B d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{89 (b c-a d)^2 g^2}+\frac {d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{89 (b c-a d)^2 g^2}+\frac {\left (2 B^2 d\right ) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (a+b x)}{e (a+b x)} \, dx}{89 (b c-a d)^2 g^2}-\frac {\left (2 B^2 d\right ) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (c+d x)}{e (a+b x)} \, dx}{89 (b c-a d)^2 g^2}+\frac {\left (2 B^2\right ) \int \frac {b c-a d}{(a+b x)^2 (c+d x)} \, dx}{89 (b c-a d) g^2}+\frac {(2 B d) \int \left (\frac {A \log (a+b x)}{(a+b x) (c+d x)}+\frac {B \log (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x) (c+d x)}\right ) \, dx}{89 (b c-a d) g^2}-\frac {(2 B d) \int \left (\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{(b c-a d) (a+b x)}-\frac {d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{(b c-a d) (c+d x)}\right ) \, dx}{89 (b c-a d) g^2}\\ &=-\frac {2 B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{89 (b c-a d) g^2 (a+b x)}-\frac {2 B d \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{89 (b c-a d)^2 g^2}-\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{89 (b c-a d) g^2 (a+b x)}-\frac {d \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{89 (b c-a d)^2 g^2}+\frac {2 B d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{89 (b c-a d)^2 g^2}+\frac {d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{89 (b c-a d)^2 g^2}+\frac {\left (2 B^2\right ) \int \frac {1}{(a+b x)^2 (c+d x)} \, dx}{89 g^2}-\frac {(2 b B d) \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{a+b x} \, dx}{89 (b c-a d)^2 g^2}+\frac {\left (2 B d^2\right ) \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{c+d x} \, dx}{89 (b c-a d)^2 g^2}+\frac {(2 A B d) \int \frac {\log (a+b x)}{(a+b x) (c+d x)} \, dx}{89 (b c-a d) g^2}+\frac {\left (2 B^2 d\right ) \int \frac {\log (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x) (c+d x)} \, dx}{89 (b c-a d) g^2}+\frac {\left (2 B^2 d\right ) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (a+b x)}{a+b x} \, dx}{89 (b c-a d)^2 e g^2}-\frac {\left (2 B^2 d\right ) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (c+d x)}{a+b x} \, dx}{89 (b c-a d)^2 e g^2}\\ &=\frac {B^2 d \log (a+b x) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{89 (b c-a d)^2 g^2}-\frac {2 B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{89 (b c-a d) g^2 (a+b x)}-\frac {2 B d \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{89 (b c-a d)^2 g^2}-\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{89 (b c-a d) g^2 (a+b x)}-\frac {d \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{89 (b c-a d)^2 g^2}+\frac {2 B d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{89 (b c-a d)^2 g^2}+\frac {d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{89 (b c-a d)^2 g^2}+\frac {\left (2 B^2\right ) \int \left (\frac {b}{(b c-a d) (a+b x)^2}-\frac {b d}{(b c-a d)^2 (a+b x)}+\frac {d^2}{(b c-a d)^2 (c+d x)}\right ) \, dx}{89 g^2}-\frac {(2 b B d) \int \left (\frac {A \log (c+d x)}{a+b x}+\frac {B \log \left (\frac {e (a+b x)}{c+d x}\right ) \log (c+d x)}{a+b x}\right ) \, dx}{89 (b c-a d)^2 g^2}-\frac {\left (b B^2 d\right ) \int \frac {\log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{89 (b c-a d)^2 g^2}+\frac {\left (2 B d^2\right ) \int \left (\frac {A \log (c+d x)}{c+d x}+\frac {B \log \left (\frac {e (a+b x)}{c+d x}\right ) \log (c+d x)}{c+d x}\right ) \, dx}{89 (b c-a d)^2 g^2}+\frac {(2 A B d) \text {Subst}\left (\int \frac {\log (x)}{x \left (\frac {b c-a d}{b}+\frac {d x}{b}\right )} \, dx,x,a+b x\right )}{89 b (b c-a d) g^2}+\frac {\left (2 B^2 d\right ) \int \left (\frac {b e \log (a+b x)}{a+b x}-\frac {d e \log (a+b x)}{c+d x}\right ) \, dx}{89 (b c-a d)^2 e g^2}-\frac {\left (2 B^2 d\right ) \int \left (\frac {b e \log (c+d x)}{a+b x}-\frac {d e \log (c+d x)}{c+d x}\right ) \, dx}{89 (b c-a d)^2 e g^2}\\ &=-\frac {2 B^2}{89 (b c-a d) g^2 (a+b x)}-\frac {2 B^2 d \log (a+b x)}{89 (b c-a d)^2 g^2}+\frac {B^2 d \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{89 (b c-a d)^2 g^2}+\frac {B^2 d \log (a+b x) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{89 (b c-a d)^2 g^2}-\frac {2 B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{89 (b c-a d) g^2 (a+b x)}-\frac {2 B d \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{89 (b c-a d)^2 g^2}-\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{89 (b c-a d) g^2 (a+b x)}-\frac {d \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{89 (b c-a d)^2 g^2}+\frac {2 B^2 d \log (c+d x)}{89 (b c-a d)^2 g^2}+\frac {2 B d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{89 (b c-a d)^2 g^2}+\frac {d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{89 (b c-a d)^2 g^2}+\frac {(2 A B d) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{89 (b c-a d)^2 g^2}-\frac {(2 A b B d) \int \frac {\log (c+d x)}{a+b x} \, dx}{89 (b c-a d)^2 g^2}+\frac {\left (2 b B^2 d\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{89 (b c-a d)^2 g^2}-\frac {\left (2 b B^2 d\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{89 (b c-a d)^2 g^2}-\frac {\left (2 b B^2 d\right ) \int \frac {\log \left (\frac {e (a+b x)}{c+d x}\right ) \log (c+d x)}{a+b x} \, dx}{89 (b c-a d)^2 g^2}+\frac {\left (2 A B d^2\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{89 (b c-a d)^2 g^2}-\frac {\left (2 A B d^2\right ) \text {Subst}\left (\int \frac {\log (x)}{\frac {b c-a d}{b}+\frac {d x}{b}} \, dx,x,a+b x\right )}{89 b (b c-a d)^2 g^2}-\frac {\left (2 B^2 d^2\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{89 (b c-a d)^2 g^2}+\frac {\left (2 B^2 d^2\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{89 (b c-a d)^2 g^2}+\frac {\left (2 B^2 d^2\right ) \int \frac {\log \left (\frac {e (a+b x)}{c+d x}\right ) \log (c+d x)}{c+d x} \, dx}{89 (b c-a d)^2 g^2}-\frac {\left (2 B^2 d\right ) \int \frac {\log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x) (c+d x)} \, dx}{89 (b c-a d) g^2}\\ &=-\frac {2 B^2}{89 (b c-a d) g^2 (a+b x)}-\frac {2 B^2 d \log (a+b x)}{89 (b c-a d)^2 g^2}+\frac {A B d \log ^2(a+b x)}{89 (b c-a d)^2 g^2}+\frac {B^2 d \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{89 (b c-a d)^2 g^2}+\frac {B^2 d \log (a+b x) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{89 (b c-a d)^2 g^2}-\frac {2 B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{89 (b c-a d) g^2 (a+b x)}-\frac {2 B d \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{89 (b c-a d)^2 g^2}-\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{89 (b c-a d) g^2 (a+b x)}-\frac {d \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{89 (b c-a d)^2 g^2}+\frac {2 B^2 d \log (c+d x)}{89 (b c-a d)^2 g^2}-\frac {2 A B d \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{89 (b c-a d)^2 g^2}-\frac {2 B^2 d \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{89 (b c-a d)^2 g^2}+\frac {2 B d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{89 (b c-a d)^2 g^2}+\frac {d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{89 (b c-a d)^2 g^2}+\frac {B^2 d \log \left (\frac {e (a+b x)}{c+d x}\right ) \log ^2(c+d x)}{89 (b c-a d)^2 g^2}-\frac {2 A B d \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{89 (b c-a d)^2 g^2}-\frac {2 B^2 d \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{89 (b c-a d)^2 g^2}-\frac {2 B^2 d \log \left (\frac {e (a+b x)}{c+d x}\right ) \text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{89 (b c-a d)^2 g^2}+\frac {(2 A B d) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{89 (b c-a d)^2 g^2}+\frac {(2 A B d) \text {Subst}\left (\int \frac {\log \left (1+\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{89 (b c-a d)^2 g^2}+\frac {\left (2 B^2 d\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{89 (b c-a d)^2 g^2}+\frac {\left (2 B^2 d\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{89 (b c-a d)^2 g^2}-\frac {\left (b B^2 d\right ) \int \frac {\log ^2(c+d x)}{a+b x} \, dx}{89 (b c-a d)^2 g^2}-\frac {\left (2 b B^2 d\right ) \int \frac {\log (a+b x) \log (c+d x)}{a+b x} \, dx}{89 (b c-a d)^2 g^2}-\frac {\left (2 b B^2 d\right ) \int \frac {\log \left (\frac {1}{c+d x}\right ) \log (c+d x)}{a+b x} \, dx}{89 (b c-a d)^2 g^2}+\frac {\left (2 b B^2 d\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{89 (b c-a d)^2 g^2}+\frac {\left (2 A B d^2\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{89 (b c-a d)^2 g^2}+\frac {\left (B^2 d^2\right ) \int \frac {\log ^2(c+d x)}{c+d x} \, dx}{89 (b c-a d)^2 g^2}+\frac {\left (2 B^2 d^2\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{89 (b c-a d)^2 g^2}+\frac {\left (2 B^2 d\right ) \int \frac {\text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{(a+b x) (c+d x)} \, dx}{89 (b c-a d) g^2}-\frac {\left (2 b B^2 d \left (-\log (a+b x)-\log \left (\frac {1}{c+d x}\right )+\log \left (\frac {e (a+b x)}{c+d x}\right )\right )\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{89 (b c-a d)^2 g^2}\\ &=-\frac {2 B^2}{89 (b c-a d) g^2 (a+b x)}-\frac {2 B^2 d \log (a+b x)}{89 (b c-a d)^2 g^2}+\frac {A B d \log ^2(a+b x)}{89 (b c-a d)^2 g^2}+\frac {B^2 d \log ^2(a+b x)}{89 (b c-a d)^2 g^2}+\frac {B^2 d \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{89 (b c-a d)^2 g^2}+\frac {B^2 d \log (a+b x) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{89 (b c-a d)^2 g^2}-\frac {2 B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{89 (b c-a d) g^2 (a+b x)}-\frac {2 B d \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{89 (b c-a d)^2 g^2}-\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{89 (b c-a d) g^2 (a+b x)}-\frac {d \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{89 (b c-a d)^2 g^2}+\frac {2 B^2 d \log (c+d x)}{89 (b c-a d)^2 g^2}-\frac {2 A B d \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{89 (b c-a d)^2 g^2}-\frac {2 B^2 d \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{89 (b c-a d)^2 g^2}+\frac {2 B^2 d \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \left (\log (a+b x)+\log \left (\frac {1}{c+d x}\right )-\log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{89 (b c-a d)^2 g^2}+\frac {2 B d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{89 (b c-a d)^2 g^2}+\frac {d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{89 (b c-a d)^2 g^2}+\frac {A B d \log ^2(c+d x)}{89 (b c-a d)^2 g^2}+\frac {B^2 d \log ^2(c+d x)}{89 (b c-a d)^2 g^2}-\frac {B^2 d \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(c+d x)}{89 (b c-a d)^2 g^2}+\frac {B^2 d \log \left (\frac {e (a+b x)}{c+d x}\right ) \log ^2(c+d x)}{89 (b c-a d)^2 g^2}-\frac {2 A B d \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{89 (b c-a d)^2 g^2}-\frac {2 B^2 d \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{89 (b c-a d)^2 g^2}-\frac {2 A B d \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{89 (b c-a d)^2 g^2}-\frac {2 B^2 d \log \left (\frac {e (a+b x)}{c+d x}\right ) \text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{89 (b c-a d)^2 g^2}-\frac {2 B^2 d \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{89 (b c-a d)^2 g^2}+\frac {(2 A B d) \text {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{89 (b c-a d)^2 g^2}+\frac {\left (B^2 d\right ) \text {Subst}\left (\int \frac {\log ^2(x)}{x} \, dx,x,c+d x\right )}{89 (b c-a d)^2 g^2}-\frac {\left (2 B^2 d\right ) \text {Subst}\left (\int \frac {\log (x) \log \left (\frac {b c-a d}{b}+\frac {d x}{b}\right )}{x} \, dx,x,a+b x\right )}{89 (b c-a d)^2 g^2}-\frac {\left (2 B^2 d\right ) \text {Subst}\left (\int \frac {\log \left (\frac {1}{-\frac {-b c+a d}{b}+\frac {d x}{b}}\right ) \log \left (-\frac {-b c+a d}{b}+\frac {d x}{b}\right )}{x} \, dx,x,a+b x\right )}{89 (b c-a d)^2 g^2}+\frac {\left (2 B^2 d\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{89 (b c-a d)^2 g^2}+\frac {\left (2 B^2 d\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{89 (b c-a d)^2 g^2}+\frac {\left (2 B^2 d^2\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right ) \log (c+d x)}{c+d x} \, dx}{89 (b c-a d)^2 g^2}+\frac {\left (2 B^2 d^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 {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{89 (b c-a d)^2 g^2}\\ &=-\frac {2 B^2}{89 (b c-a d) g^2 (a+b x)}-\frac {2 B^2 d \log (a+b x)}{89 (b c-a d)^2 g^2}+\frac {A B d \log ^2(a+b x)}{89 (b c-a d)^2 g^2}+\frac {B^2 d \log ^2(a+b x)}{89 (b c-a d)^2 g^2}+\frac {B^2 d \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{89 (b c-a d)^2 g^2}+\frac {B^2 d \log (a+b x) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{89 (b c-a d)^2 g^2}-\frac {2 B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{89 (b c-a d) g^2 (a+b x)}-\frac {2 B d \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{89 (b c-a d)^2 g^2}-\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{89 (b c-a d) g^2 (a+b x)}-\frac {d \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{89 (b c-a d)^2 g^2}+\frac {2 B^2 d \log (c+d x)}{89 (b c-a d)^2 g^2}-\frac {B^2 d \log ^2(a+b x) \log (c+d x)}{89 (b c-a d)^2 g^2}-\frac {2 A B d \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{89 (b c-a d)^2 g^2}-\frac {2 B^2 d \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{89 (b c-a d)^2 g^2}-\frac {2 B^2 d \log (a+b x) \log \left (\frac {1}{c+d x}\right ) \log (c+d x)}{89 (b c-a d)^2 g^2}+\frac {2 B^2 d \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \left (\log (a+b x)+\log \left (\frac {1}{c+d x}\right )-\log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{89 (b c-a d)^2 g^2}+\frac {2 B d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{89 (b c-a d)^2 g^2}+\frac {d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{89 (b c-a d)^2 g^2}+\frac {A B d \log ^2(c+d x)}{89 (b c-a d)^2 g^2}+\frac {B^2 d \log ^2(c+d x)}{89 (b c-a d)^2 g^2}-\frac {B^2 d \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(c+d x)}{89 (b c-a d)^2 g^2}+\frac {B^2 d \log \left (\frac {e (a+b x)}{c+d x}\right ) \log ^2(c+d x)}{89 (b c-a d)^2 g^2}-\frac {2 A B d \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{89 (b c-a d)^2 g^2}-\frac {2 B^2 d \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{89 (b c-a d)^2 g^2}-\frac {2 A B d \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{89 (b c-a d)^2 g^2}-\frac {2 B^2 d \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{89 (b c-a d)^2 g^2}-\frac {2 A B d \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{89 (b c-a d)^2 g^2}-\frac {2 B^2 d \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{89 (b c-a d)^2 g^2}-\frac {2 B^2 d \log \left (\frac {e (a+b x)}{c+d x}\right ) \text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{89 (b c-a d)^2 g^2}-\frac {2 B^2 d \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{89 (b c-a d)^2 g^2}+\frac {\left (B^2 d\right ) \text {Subst}\left (\int x^2 \, dx,x,\log (c+d x)\right )}{89 (b c-a d)^2 g^2}+\frac {\left (2 B^2 d\right ) \text {Subst}\left (\int \frac {\log (x) \log \left (\frac {d \left (\frac {-b c+a d}{d}+\frac {b x}{d}\right )}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{89 (b c-a d)^2 g^2}+\frac {\left (B^2 d^2\right ) \text {Subst}\left (\int \frac {\log ^2(x)}{\frac {b c-a d}{b}+\frac {d x}{b}} \, dx,x,a+b x\right )}{89 b (b c-a d)^2 g^2}+\frac {\left (2 B^2 d^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 )}{89 b (b c-a d)^2 g^2}-\frac {\left (2 B^2 d^2\right ) \text {Subst}\left (\int \frac {\log (x) \log \left (-\frac {-b c+a d}{b}+\frac {d x}{b}\right )}{-\frac {-b c+a d}{b}+\frac {d x}{b}} \, dx,x,a+b x\right )}{89 b (b c-a d)^2 g^2}+\frac {\left (2 B^2 d \left (-\log (a+b x)-\log \left (\frac {1}{c+d x}\right )+\log \left (\frac {e (a+b x)}{c+d x}\right )\right )\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{89 (b c-a d)^2 g^2}\\ &=-\frac {2 B^2}{89 (b c-a d) g^2 (a+b x)}-\frac {2 B^2 d \log (a+b x)}{89 (b c-a d)^2 g^2}+\frac {A B d \log ^2(a+b x)}{89 (b c-a d)^2 g^2}+\frac {B^2 d \log ^2(a+b x)}{89 (b c-a d)^2 g^2}+\frac {B^2 d \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{89 (b c-a d)^2 g^2}+\frac {B^2 d \log (a+b x) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{89 (b c-a d)^2 g^2}-\frac {2 B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{89 (b c-a d) g^2 (a+b x)}-\frac {2 B d \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{89 (b c-a d)^2 g^2}-\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{89 (b c-a d) g^2 (a+b x)}-\frac {d \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{89 (b c-a d)^2 g^2}+\frac {2 B^2 d \log (c+d x)}{89 (b c-a d)^2 g^2}-\frac {B^2 d \log ^2(a+b x) \log (c+d x)}{89 (b c-a d)^2 g^2}-\frac {2 A B d \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{89 (b c-a d)^2 g^2}-\frac {2 B^2 d \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{89 (b c-a d)^2 g^2}-\frac {2 B^2 d \log (a+b x) \log \left (\frac {1}{c+d x}\right ) \log (c+d x)}{89 (b c-a d)^2 g^2}+\frac {2 B^2 d \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \left (\log (a+b x)+\log \left (\frac {1}{c+d x}\right )-\log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{89 (b c-a d)^2 g^2}+\frac {2 B d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{89 (b c-a d)^2 g^2}+\frac {d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{89 (b c-a d)^2 g^2}+\frac {A B d \log ^2(c+d x)}{89 (b c-a d)^2 g^2}+\frac {B^2 d \log ^2(c+d x)}{89 (b c-a d)^2 g^2}-\frac {B^2 d \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(c+d x)}{89 (b c-a d)^2 g^2}+\frac {B^2 d \log \left (\frac {e (a+b x)}{c+d x}\right ) \log ^2(c+d x)}{89 (b c-a d)^2 g^2}+\frac {B^2 d \log ^3(c+d x)}{267 (b c-a d)^2 g^2}-\frac {2 A B d \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{89 (b c-a d)^2 g^2}-\frac {2 B^2 d \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{89 (b c-a d)^2 g^2}+\frac {B^2 d \log ^2(a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{89 (b c-a d)^2 g^2}-\frac {2 A B d \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{89 (b c-a d)^2 g^2}-\frac {2 B^2 d \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{89 (b c-a d)^2 g^2}-\frac {2 A B d \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{89 (b c-a d)^2 g^2}-\frac {2 B^2 d \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{89 (b c-a d)^2 g^2}+\frac {2 B^2 d \left (\log (a+b x)+\log \left (\frac {1}{c+d x}\right )-\log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{89 (b c-a d)^2 g^2}-\frac {2 B^2 d \log (c+d x) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{89 (b c-a d)^2 g^2}-\frac {2 B^2 d \log \left (\frac {e (a+b x)}{c+d x}\right ) \text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{89 (b c-a d)^2 g^2}-\frac {2 B^2 d \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{89 (b c-a d)^2 g^2}+\frac {\left (2 B^2 d\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 )}{89 (b c-a d)^2 g^2}-\frac {\left (2 B^2 d\right ) \text {Subst}\left (\int \frac {\log (x) \log \left (\frac {-b c+a d}{d}+\frac {b x}{d}\right )}{x} \, dx,x,c+d x\right )}{89 (b c-a d)^2 g^2}-\frac {\left (2 B^2 d\right ) \text {Subst}\left (\int \frac {\log (x) \log \left (1+\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{89 (b c-a d)^2 g^2}+\frac {\left (2 B^2 d\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 )}{89 (b c-a d)^2 g^2}\\ &=-\frac {2 B^2}{89 (b c-a d) g^2 (a+b x)}-\frac {2 B^2 d \log (a+b x)}{89 (b c-a d)^2 g^2}+\frac {A B d \log ^2(a+b x)}{89 (b c-a d)^2 g^2}+\frac {B^2 d \log ^2(a+b x)}{89 (b c-a d)^2 g^2}-\frac {B^2 d \log (a+b x) \log ^2\left (\frac {1}{c+d x}\right )}{89 (b c-a d)^2 g^2}+\frac {B^2 d \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{89 (b c-a d)^2 g^2}+\frac {B^2 d \log (a+b x) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{89 (b c-a d)^2 g^2}-\frac {2 B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{89 (b c-a d) g^2 (a+b x)}-\frac {2 B d \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{89 (b c-a d)^2 g^2}-\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{89 (b c-a d) g^2 (a+b x)}-\frac {d \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{89 (b c-a d)^2 g^2}+\frac {2 B^2 d \log (c+d x)}{89 (b c-a d)^2 g^2}-\frac {B^2 d \log ^2(a+b x) \log (c+d x)}{89 (b c-a d)^2 g^2}-\frac {2 A B d \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{89 (b c-a d)^2 g^2}-\frac {2 B^2 d \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{89 (b c-a d)^2 g^2}-\frac {2 B^2 d \log (a+b x) \log \left (\frac {1}{c+d x}\right ) \log (c+d x)}{89 (b c-a d)^2 g^2}+\frac {2 B^2 d \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \left (\log (a+b x)+\log \left (\frac {1}{c+d x}\right )-\log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{89 (b c-a d)^2 g^2}+\frac {2 B d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{89 (b c-a d)^2 g^2}+\frac {d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{89 (b c-a d)^2 g^2}+\frac {A B d \log ^2(c+d x)}{89 (b c-a d)^2 g^2}+\frac {B^2 d \log ^2(c+d x)}{89 (b c-a d)^2 g^2}-\frac {B^2 d \log (a+b x) \log ^2(c+d x)}{89 (b c-a d)^2 g^2}-\frac {B^2 d \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(c+d x)}{89 (b c-a d)^2 g^2}+\frac {B^2 d \log \left (\frac {e (a+b x)}{c+d x}\right ) \log ^2(c+d x)}{89 (b c-a d)^2 g^2}+\frac {B^2 d \log ^3(c+d x)}{267 (b c-a d)^2 g^2}-\frac {2 A B d \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{89 (b c-a d)^2 g^2}-\frac {2 B^2 d \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{89 (b c-a d)^2 g^2}+\frac {B^2 d \log ^2(a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{89 (b c-a d)^2 g^2}-\frac {2 A B d \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{89 (b c-a d)^2 g^2}-\frac {2 B^2 d \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{89 (b c-a d)^2 g^2}+\frac {2 B^2 d \log (a+b x) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{89 (b c-a d)^2 g^2}-\frac {2 A B d \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{89 (b c-a d)^2 g^2}-\frac {2 B^2 d \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{89 (b c-a d)^2 g^2}+\frac {2 B^2 d \left (\log (a+b x)+\log \left (\frac {1}{c+d x}\right )-\log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{89 (b c-a d)^2 g^2}-\frac {2 B^2 d \log (c+d x) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{89 (b c-a d)^2 g^2}-\frac {2 B^2 d \log \left (\frac {e (a+b x)}{c+d x}\right ) \text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{89 (b c-a d)^2 g^2}+\frac {2 B^2 d \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{89 (b c-a d)^2 g^2}-\frac {2 B^2 d \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{89 (b c-a d)^2 g^2}+\frac {\left (b B^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 )}{89 (b c-a d)^2 g^2}+\frac {\left (b B^2\right ) \text {Subst}\left (\int \frac {\log ^2(x)}{\frac {-b c+a d}{d}+\frac {b x}{d}} \, dx,x,c+d x\right )}{89 (b c-a d)^2 g^2}-\frac {\left (2 B^2 d\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{89 (b c-a d)^2 g^2}\\ &=-\frac {2 B^2}{89 (b c-a d) g^2 (a+b x)}-\frac {2 B^2 d \log (a+b x)}{89 (b c-a d)^2 g^2}+\frac {A B d \log ^2(a+b x)}{89 (b c-a d)^2 g^2}+\frac {B^2 d \log ^2(a+b x)}{89 (b c-a d)^2 g^2}-\frac {B^2 d \log (a+b x) \log ^2\left (\frac {1}{c+d x}\right )}{89 (b c-a d)^2 g^2}+\frac {B^2 d \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2\left (\frac {1}{c+d x}\right )}{89 (b c-a d)^2 g^2}+\frac {B^2 d \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{89 (b c-a d)^2 g^2}+\frac {B^2 d \log (a+b x) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{89 (b c-a d)^2 g^2}-\frac {2 B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{89 (b c-a d) g^2 (a+b x)}-\frac {2 B d \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{89 (b c-a d)^2 g^2}-\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{89 (b c-a d) g^2 (a+b x)}-\frac {d \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{89 (b c-a d)^2 g^2}+\frac {2 B^2 d \log (c+d x)}{89 (b c-a d)^2 g^2}-\frac {B^2 d \log ^2(a+b x) \log (c+d x)}{89 (b c-a d)^2 g^2}-\frac {2 A B d \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{89 (b c-a d)^2 g^2}-\frac {2 B^2 d \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{89 (b c-a d)^2 g^2}-\frac {2 B^2 d \log (a+b x) \log \left (\frac {1}{c+d x}\right ) \log (c+d x)}{89 (b c-a d)^2 g^2}+\frac {2 B^2 d \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \left (\log (a+b x)+\log \left (\frac {1}{c+d x}\right )-\log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{89 (b c-a d)^2 g^2}+\frac {2 B d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{89 (b c-a d)^2 g^2}+\frac {d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{89 (b c-a d)^2 g^2}+\frac {A B d \log ^2(c+d x)}{89 (b c-a d)^2 g^2}+\frac {B^2 d \log ^2(c+d x)}{89 (b c-a d)^2 g^2}-\frac {B^2 d \log (a+b x) \log ^2(c+d x)}{89 (b c-a d)^2 g^2}+\frac {B^2 d \log \left (\frac {e (a+b x)}{c+d x}\right ) \log ^2(c+d x)}{89 (b c-a d)^2 g^2}+\frac {B^2 d \log ^3(c+d x)}{267 (b c-a d)^2 g^2}-\frac {2 A B d \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{89 (b c-a d)^2 g^2}-\frac {2 B^2 d \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{89 (b c-a d)^2 g^2}+\frac {B^2 d \log ^2(a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{89 (b c-a d)^2 g^2}-\frac {2 A B d \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{89 (b c-a d)^2 g^2}-\frac {2 B^2 d \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{89 (b c-a d)^2 g^2}+\frac {2 B^2 d \log (a+b x) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{89 (b c-a d)^2 g^2}-\frac {2 A B d \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{89 (b c-a d)^2 g^2}-\frac {2 B^2 d \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{89 (b c-a d)^2 g^2}+\frac {2 B^2 d \left (\log (a+b x)+\log \left (\frac {1}{c+d x}\right )-\log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{89 (b c-a d)^2 g^2}-\frac {2 B^2 d \log (c+d x) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{89 (b c-a d)^2 g^2}-\frac {2 B^2 d \log \left (\frac {e (a+b x)}{c+d x}\right ) \text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{89 (b c-a d)^2 g^2}-\frac {2 B^2 d \text {Li}_3\left (-\frac {d (a+b x)}{b c-a d}\right )}{89 (b c-a d)^2 g^2}+\frac {2 B^2 d \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{89 (b c-a d)^2 g^2}-\frac {2 B^2 d \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{89 (b c-a d)^2 g^2}+\frac {\left (2 B^2 d\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 )}{89 (b c-a d)^2 g^2}-\frac {\left (2 B^2 d\right ) \text {Subst}\left (\int \frac {\log (x) \log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{89 (b c-a d)^2 g^2}\\ &=-\frac {2 B^2}{89 (b c-a d) g^2 (a+b x)}-\frac {2 B^2 d \log (a+b x)}{89 (b c-a d)^2 g^2}+\frac {A B d \log ^2(a+b x)}{89 (b c-a d)^2 g^2}+\frac {B^2 d \log ^2(a+b x)}{89 (b c-a d)^2 g^2}-\frac {B^2 d \log (a+b x) \log ^2\left (\frac {1}{c+d x}\right )}{89 (b c-a d)^2 g^2}+\frac {B^2 d \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2\left (\frac {1}{c+d x}\right )}{89 (b c-a d)^2 g^2}+\frac {B^2 d \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{89 (b c-a d)^2 g^2}+\frac {B^2 d \log (a+b x) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{89 (b c-a d)^2 g^2}-\frac {2 B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{89 (b c-a d) g^2 (a+b x)}-\frac {2 B d \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{89 (b c-a d)^2 g^2}-\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{89 (b c-a d) g^2 (a+b x)}-\frac {d \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{89 (b c-a d)^2 g^2}+\frac {2 B^2 d \log (c+d x)}{89 (b c-a d)^2 g^2}-\frac {B^2 d \log ^2(a+b x) \log (c+d x)}{89 (b c-a d)^2 g^2}-\frac {2 A B d \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{89 (b c-a d)^2 g^2}-\frac {2 B^2 d \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{89 (b c-a d)^2 g^2}-\frac {2 B^2 d \log (a+b x) \log \left (\frac {1}{c+d x}\right ) \log (c+d x)}{89 (b c-a d)^2 g^2}+\frac {2 B^2 d \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \left (\log (a+b x)+\log \left (\frac {1}{c+d x}\right )-\log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{89 (b c-a d)^2 g^2}+\frac {2 B d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{89 (b c-a d)^2 g^2}+\frac {d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{89 (b c-a d)^2 g^2}+\frac {A B d \log ^2(c+d x)}{89 (b c-a d)^2 g^2}+\frac {B^2 d \log ^2(c+d x)}{89 (b c-a d)^2 g^2}-\frac {B^2 d \log (a+b x) \log ^2(c+d x)}{89 (b c-a d)^2 g^2}+\frac {B^2 d \log \left (\frac {e (a+b x)}{c+d x}\right ) \log ^2(c+d x)}{89 (b c-a d)^2 g^2}+\frac {B^2 d \log ^3(c+d x)}{267 (b c-a d)^2 g^2}-\frac {2 A B d \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{89 (b c-a d)^2 g^2}-\frac {2 B^2 d \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{89 (b c-a d)^2 g^2}+\frac {B^2 d \log ^2(a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{89 (b c-a d)^2 g^2}-\frac {2 A B d \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{89 (b c-a d)^2 g^2}-\frac {2 B^2 d \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{89 (b c-a d)^2 g^2}+\frac {2 B^2 d \log (a+b x) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{89 (b c-a d)^2 g^2}-\frac {2 A B d \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{89 (b c-a d)^2 g^2}-\frac {2 B^2 d \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{89 (b c-a d)^2 g^2}-\frac {2 B^2 d \log \left (\frac {1}{c+d x}\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{89 (b c-a d)^2 g^2}+\frac {2 B^2 d \left (\log (a+b x)+\log \left (\frac {1}{c+d x}\right )-\log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{89 (b c-a d)^2 g^2}-\frac {2 B^2 d \log \left (\frac {e (a+b x)}{c+d x}\right ) \text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{89 (b c-a d)^2 g^2}-\frac {2 B^2 d \text {Li}_3\left (-\frac {d (a+b x)}{b c-a d}\right )}{89 (b c-a d)^2 g^2}+\frac {2 B^2 d \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{89 (b c-a d)^2 g^2}-\frac {2 B^2 d \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{89 (b c-a d)^2 g^2}-2 \frac {\left (2 B^2 d\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 )}{89 (b c-a d)^2 g^2}\\ &=-\frac {2 B^2}{89 (b c-a d) g^2 (a+b x)}-\frac {2 B^2 d \log (a+b x)}{89 (b c-a d)^2 g^2}+\frac {A B d \log ^2(a+b x)}{89 (b c-a d)^2 g^2}+\frac {B^2 d \log ^2(a+b x)}{89 (b c-a d)^2 g^2}-\frac {B^2 d \log (a+b x) \log ^2\left (\frac {1}{c+d x}\right )}{89 (b c-a d)^2 g^2}+\frac {B^2 d \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2\left (\frac {1}{c+d x}\right )}{89 (b c-a d)^2 g^2}+\frac {B^2 d \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{89 (b c-a d)^2 g^2}+\frac {B^2 d \log (a+b x) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{89 (b c-a d)^2 g^2}-\frac {2 B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{89 (b c-a d) g^2 (a+b x)}-\frac {2 B d \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{89 (b c-a d)^2 g^2}-\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{89 (b c-a d) g^2 (a+b x)}-\frac {d \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{89 (b c-a d)^2 g^2}+\frac {2 B^2 d \log (c+d x)}{89 (b c-a d)^2 g^2}-\frac {B^2 d \log ^2(a+b x) \log (c+d x)}{89 (b c-a d)^2 g^2}-\frac {2 A B d \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{89 (b c-a d)^2 g^2}-\frac {2 B^2 d \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{89 (b c-a d)^2 g^2}-\frac {2 B^2 d \log (a+b x) \log \left (\frac {1}{c+d x}\right ) \log (c+d x)}{89 (b c-a d)^2 g^2}+\frac {2 B^2 d \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \left (\log (a+b x)+\log \left (\frac {1}{c+d x}\right )-\log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{89 (b c-a d)^2 g^2}+\frac {2 B d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{89 (b c-a d)^2 g^2}+\frac {d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{89 (b c-a d)^2 g^2}+\frac {A B d \log ^2(c+d x)}{89 (b c-a d)^2 g^2}+\frac {B^2 d \log ^2(c+d x)}{89 (b c-a d)^2 g^2}-\frac {B^2 d \log (a+b x) \log ^2(c+d x)}{89 (b c-a d)^2 g^2}+\frac {B^2 d \log \left (\frac {e (a+b x)}{c+d x}\right ) \log ^2(c+d x)}{89 (b c-a d)^2 g^2}+\frac {B^2 d \log ^3(c+d x)}{267 (b c-a d)^2 g^2}-\frac {2 A B d \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{89 (b c-a d)^2 g^2}-\frac {2 B^2 d \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{89 (b c-a d)^2 g^2}+\frac {B^2 d \log ^2(a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{89 (b c-a d)^2 g^2}-\frac {2 A B d \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{89 (b c-a d)^2 g^2}-\frac {2 B^2 d \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{89 (b c-a d)^2 g^2}+\frac {2 B^2 d \log (a+b x) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{89 (b c-a d)^2 g^2}-\frac {2 A B d \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{89 (b c-a d)^2 g^2}-\frac {2 B^2 d \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{89 (b c-a d)^2 g^2}-\frac {2 B^2 d \log \left (\frac {1}{c+d x}\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{89 (b c-a d)^2 g^2}+\frac {2 B^2 d \left (\log (a+b x)+\log \left (\frac {1}{c+d x}\right )-\log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{89 (b c-a d)^2 g^2}-\frac {2 B^2 d \log \left (\frac {e (a+b x)}{c+d x}\right ) \text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{89 (b c-a d)^2 g^2}-\frac {2 B^2 d \text {Li}_3\left (-\frac {d (a+b x)}{b c-a d}\right )}{89 (b c-a d)^2 g^2}-\frac {2 B^2 d \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{89 (b c-a d)^2 g^2}-\frac {2 B^2 d \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{89 (b c-a d)^2 g^2}\\ \end {align*}

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Mathematica [A]
time = 0.36, size = 186, normalized size = 1.02 \begin {gather*} -\frac {3 \left (A^2+2 A B+2 B^2\right ) d (a+b x) \log (a+b x)+6 B (A+B) (b c-a d) \log \left (\frac {e (a+b x)}{c+d x}\right )+3 B (a A d+A b d x+b B (c+d x)) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )+B^2 d (a+b x) \log ^3\left (\frac {e (a+b x)}{c+d x}\right )+3 \left (A^2+2 A B+2 B^2\right ) (b c-a d-d (a+b x) \log (c+d x))}{3 (b c-a d)^2 g^2 i (a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(526\) vs. \(2(181)=362\).
time = 0.58, size = 527, normalized size = 2.88 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1020 vs. \(2 (172) = 344\).
time = 0.43, size = 1020, normalized size = 5.57 \begin {gather*} B^{2} {\left (\frac {1}{{\left (-i \, b^{2} c + i \, a b d\right )} g^{2} x + {\left (-i \, a b c + i \, a^{2} d\right )} g^{2}} - \frac {d \log \left (b x + a\right )}{{\left (i \, b^{2} c^{2} - 2 i \, a b c d + i \, a^{2} d^{2}\right )} g^{2}} + \frac {d \log \left (d x + c\right )}{{\left (i \, b^{2} c^{2} - 2 i \, a b c d + i \, a^{2} d^{2}\right )} g^{2}}\right )} \log \left (\frac {b x e}{d x + c} + \frac {a e}{d x + c}\right )^{2} + 2 \, A B {\left (\frac {1}{{\left (-i \, b^{2} c + i \, a b d\right )} g^{2} x + {\left (-i \, a b c + i \, a^{2} d\right )} g^{2}} - \frac {d \log \left (b x + a\right )}{{\left (i \, b^{2} c^{2} - 2 i \, a b c d + i \, a^{2} d^{2}\right )} g^{2}} + \frac {d \log \left (d x + c\right )}{{\left (i \, b^{2} c^{2} - 2 i \, a b c d + i \, a^{2} d^{2}\right )} g^{2}}\right )} \log \left (\frac {b x e}{d x + c} + \frac {a e}{d x + c}\right ) - \frac {1}{3} \, B^{2} {\left (\frac {3 \, {\left ({\left (i \, b d x + i \, a d\right )} \log \left (b x + a\right )^{2} + {\left (i \, b d x + i \, a d\right )} \log \left (d x + c\right )^{2} - 2 i \, b c + 2 i \, a d - 2 \, {\left (i \, b d x + i \, a d\right )} \log \left (b x + a\right ) - 2 \, {\left (-i \, b d x - i \, a d + {\left (i \, b d x + i \, a d\right )} \log \left (b x + a\right )\right )} \log \left (d x + c\right )\right )} \log \left (\frac {b x e}{d x + c} + \frac {a e}{d x + c}\right )}{a b^{2} c^{2} g^{2} - 2 \, a^{2} b c d g^{2} + a^{3} d^{2} g^{2} + {\left (b^{3} c^{2} g^{2} - 2 \, a b^{2} c d g^{2} + a^{2} b d^{2} g^{2}\right )} x} + \frac {{\left (-i \, b d x - i \, a d\right )} \log \left (b x + a\right )^{3} + {\left (i \, b d x + i \, a d\right )} \log \left (d x + c\right )^{3} - 3 \, {\left (-i \, b d x - i \, a d\right )} \log \left (b x + a\right )^{2} - 3 \, {\left (-i \, b d x - i \, a d + {\left (i \, b d x + i \, a d\right )} \log \left (b x + a\right )\right )} \log \left (d x + c\right )^{2} - 6 i \, b c + 6 i \, a d - 6 \, {\left (i \, b d x + i \, a d\right )} \log \left (b x + a\right ) - 3 \, {\left (-2 i \, b d x + {\left (-i \, b d x - i \, a d\right )} \log \left (b x + a\right )^{2} - 2 i \, a d + 2 \, {\left (i \, b d x + i \, a d\right )} \log \left (b x + a\right )\right )} \log \left (d x + c\right )}{a b^{2} c^{2} g^{2} - 2 \, a^{2} b c d g^{2} + a^{3} d^{2} g^{2} + {\left (b^{3} c^{2} g^{2} - 2 \, a b^{2} c d g^{2} + a^{2} b d^{2} g^{2}\right )} x}\right )} + A^{2} {\left (\frac {1}{{\left (-i \, b^{2} c + i \, a b d\right )} g^{2} x + {\left (-i \, a b c + i \, a^{2} d\right )} g^{2}} - \frac {d \log \left (b x + a\right )}{{\left (i \, b^{2} c^{2} - 2 i \, a b c d + i \, a^{2} d^{2}\right )} g^{2}} + \frac {d \log \left (d x + c\right )}{{\left (i \, b^{2} c^{2} - 2 i \, a b c d + i \, a^{2} d^{2}\right )} g^{2}}\right )} - \frac {{\left ({\left (i \, b d x + i \, a d\right )} \log \left (b x + a\right )^{2} + {\left (i \, b d x + i \, a d\right )} \log \left (d x + c\right )^{2} - 2 i \, b c + 2 i \, a d - 2 \, {\left (i \, b d x + i \, a d\right )} \log \left (b x + a\right ) - 2 \, {\left (-i \, b d x - i \, a d + {\left (i \, b d x + i \, a d\right )} \log \left (b x + a\right )\right )} \log \left (d x + c\right )\right )} A B}{a b^{2} c^{2} g^{2} - 2 \, a^{2} b c d g^{2} + a^{3} d^{2} g^{2} + {\left (b^{3} c^{2} g^{2} - 2 \, a b^{2} c d g^{2} + a^{2} b d^{2} g^{2}\right )} x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.38, size = 243, normalized size = 1.33 \begin {gather*} \frac {{\left (i \, B^{2} b d x + i \, B^{2} a d\right )} \log \left (\frac {{\left (b x + a\right )} e}{d x + c}\right )^{3} - 3 \, {\left (-i \, A^{2} - 2 i \, A B - 2 i \, B^{2}\right )} b c - 3 \, {\left (i \, A^{2} + 2 i \, A B + 2 i \, B^{2}\right )} a d - 3 \, {\left (-i \, B^{2} b c - i \, A B a d + {\left (-i \, A B - i \, B^{2}\right )} b d x\right )} \log \left (\frac {{\left (b x + a\right )} e}{d x + c}\right )^{2} - 3 \, {\left (-i \, A^{2} a d + {\left (-i \, A^{2} - 2 i \, A B - 2 i \, B^{2}\right )} b d x + 2 \, {\left (-i \, A B - i \, B^{2}\right )} b c\right )} \log \left (\frac {{\left (b x + a\right )} e}{d x + c}\right )}{3 \, {\left ({\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} g^{2} x + {\left (a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2}\right )} g^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 541 vs. \(2 (158) = 316\).
time = 0.94, size = 541, normalized size = 2.96 \begin {gather*} - \frac {B^{2} d \log {\left (\frac {e \left (a + b x\right )}{c + d x} \right )}^{3}}{3 a^{2} d^{2} g^{2} i - 6 a b c d g^{2} i + 3 b^{2} c^{2} g^{2} i} + \frac {\left (2 A B + 2 B^{2}\right ) \log {\left (\frac {e \left (a + b x\right )}{c + d x} \right )}}{a^{2} d g^{2} i - a b c g^{2} i + a b d g^{2} i x - b^{2} c g^{2} i x} + \left (A^{2} + 2 A B + 2 B^{2}\right ) \left (\frac {d \log {\left (x + \frac {- \frac {a^{3} d^{4}}{\left (a d - b c\right )^{2}} + \frac {3 a^{2} b c d^{3}}{\left (a d - b c\right )^{2}} - \frac {3 a b^{2} c^{2} d^{2}}{\left (a d - b c\right )^{2}} + a d^{2} + \frac {b^{3} c^{3} d}{\left (a d - b c\right )^{2}} + b c d}{2 b d^{2}} \right )}}{g^{2} i \left (a d - b c\right )^{2}} - \frac {d \log {\left (x + \frac {\frac {a^{3} d^{4}}{\left (a d - b c\right )^{2}} - \frac {3 a^{2} b c d^{3}}{\left (a d - b c\right )^{2}} + \frac {3 a b^{2} c^{2} d^{2}}{\left (a d - b c\right )^{2}} + a d^{2} - \frac {b^{3} c^{3} d}{\left (a d - b c\right )^{2}} + b c d}{2 b d^{2}} \right )}}{g^{2} i \left (a d - b c\right )^{2}} + \frac {1}{a^{2} d g^{2} i - a b c g^{2} i + x \left (a b d g^{2} i - b^{2} c g^{2} i\right )}\right ) + \frac {\left (- A B a d - A B b d x - B^{2} b c - B^{2} b d x\right ) \log {\left (\frac {e \left (a + b x\right )}{c + d x} \right )}^{2}}{a^{3} d^{2} g^{2} i - 2 a^{2} b c d g^{2} i + a^{2} b d^{2} g^{2} i x + a b^{2} c^{2} g^{2} i - 2 a b^{2} c d g^{2} i x + b^{3} c^{2} g^{2} i x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-B**2*d*log(e*(a + b*x)/(c + d*x))**3/(3*a**2*d**2*g**2*i - 6*a*b*c*d*g**2*i + 3*b**2*c**2*g**2*i) + (2*A*B +
2*B**2)*log(e*(a + b*x)/(c + d*x))/(a**2*d*g**2*i - a*b*c*g**2*i + a*b*d*g**2*i*x - b**2*c*g**2*i*x) + (A**2 +
2*A*B + 2*B**2)*(d*log(x + (-a**3*d**4/(a*d - b*c)**2 + 3*a**2*b*c*d**3/(a*d - b*c)**2 - 3*a*b**2*c**2*d**2/(
a*d - b*c)**2 + a*d**2 + b**3*c**3*d/(a*d - b*c)**2 + b*c*d)/(2*b*d**2))/(g**2*i*(a*d - b*c)**2) - d*log(x + (
a**3*d**4/(a*d - b*c)**2 - 3*a**2*b*c*d**3/(a*d - b*c)**2 + 3*a*b**2*c**2*d**2/(a*d - b*c)**2 + a*d**2 - b**3*
c**3*d/(a*d - b*c)**2 + b*c*d)/(2*b*d**2))/(g**2*i*(a*d - b*c)**2) + 1/(a**2*d*g**2*i - a*b*c*g**2*i + x*(a*b*
d*g**2*i - b**2*c*g**2*i))) + (-A*B*a*d - A*B*b*d*x - B**2*b*c - B**2*b*d*x)*log(e*(a + b*x)/(c + d*x))**2/(a*
*3*d**2*g**2*i - 2*a**2*b*c*d*g**2*i + a**2*b*d**2*g**2*i*x + a*b**2*c**2*g**2*i - 2*a*b**2*c*d*g**2*i*x + b**
3*c**2*g**2*i*x)

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

[Out]

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

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Mupad [B]
time = 6.37, size = 419, normalized size = 2.29 \begin {gather*} \frac {A^2+2\,A\,B+2\,B^2}{\left (a\,d-b\,c\right )\,\left (a\,g^2\,i+b\,g^2\,i\,x\right )}-{\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}^2\,\left (\frac {B\,d\,\left (A+B\right )}{g^2\,i\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}-\frac {B^2\,\left (a\,d-b\,c\right )}{b\,d\,g^2\,i\,\left (\frac {x}{d}+\frac {a}{b\,d}\right )\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}\right )-\frac {B^2\,d\,{\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}^3}{3\,g^2\,i\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}+\frac {2\,B\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )\,\left (a\,d-b\,c\right )\,\left (A+B\right )}{b\,d\,g^2\,i\,\left (\frac {x}{d}+\frac {a}{b\,d}\right )\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}+\frac {d\,\mathrm {atan}\left (\frac {d\,\left (2\,b\,d\,x+\frac {a^2\,d^2\,g^2\,i-b^2\,c^2\,g^2\,i}{g^2\,i\,\left (a\,d-b\,c\right )}\right )\,\left (A^2+2\,A\,B+2\,B^2\right )\,1{}\mathrm {i}}{\left (a\,d-b\,c\right )\,\left (d\,A^2+2\,d\,A\,B+2\,d\,B^2\right )}\right )\,\left (A^2+2\,A\,B+2\,B^2\right )\,2{}\mathrm {i}}{g^2\,i\,{\left (a\,d-b\,c\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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