3.1.91 \(\int \frac {(A+B \log (\frac {e (a+b x)}{c+d x}))^2}{(a g+b g x)^4 (c i+d i x)} \, dx\) [91]

Optimal. Leaf size=507 \[ -\frac {6 b B^2 d^2 (c+d x)}{(b c-a d)^4 g^4 i (a+b x)}+\frac {3 b^2 B^2 d (c+d x)^2}{4 (b c-a d)^4 g^4 i (a+b x)^2}-\frac {2 b^3 B^2 (c+d x)^3}{27 (b c-a d)^4 g^4 i (a+b x)^3}-\frac {6 b B d^2 (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^4 g^4 i (a+b x)}+\frac {3 b^2 B d (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 (b c-a d)^4 g^4 i (a+b x)^2}-\frac {2 b^3 B (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{9 (b c-a d)^4 g^4 i (a+b x)^3}-\frac {3 b d^2 (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(b c-a d)^4 g^4 i (a+b x)}+\frac {3 b^2 d (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2 (b c-a d)^4 g^4 i (a+b x)^2}-\frac {b^3 (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{3 (b c-a d)^4 g^4 i (a+b x)^3}-\frac {d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^3}{3 B (b c-a d)^4 g^4 i} \]

[Out]

-6*b*B^2*d^2*(d*x+c)/(-a*d+b*c)^4/g^4/i/(b*x+a)+3/4*b^2*B^2*d*(d*x+c)^2/(-a*d+b*c)^4/g^4/i/(b*x+a)^2-2/27*b^3*
B^2*(d*x+c)^3/(-a*d+b*c)^4/g^4/i/(b*x+a)^3-6*b*B*d^2*(d*x+c)*(A+B*ln(e*(b*x+a)/(d*x+c)))/(-a*d+b*c)^4/g^4/i/(b
*x+a)+3/2*b^2*B*d*(d*x+c)^2*(A+B*ln(e*(b*x+a)/(d*x+c)))/(-a*d+b*c)^4/g^4/i/(b*x+a)^2-2/9*b^3*B*(d*x+c)^3*(A+B*
ln(e*(b*x+a)/(d*x+c)))/(-a*d+b*c)^4/g^4/i/(b*x+a)^3-3*b*d^2*(d*x+c)*(A+B*ln(e*(b*x+a)/(d*x+c)))^2/(-a*d+b*c)^4
/g^4/i/(b*x+a)+3/2*b^2*d*(d*x+c)^2*(A+B*ln(e*(b*x+a)/(d*x+c)))^2/(-a*d+b*c)^4/g^4/i/(b*x+a)^2-1/3*b^3*(d*x+c)^
3*(A+B*ln(e*(b*x+a)/(d*x+c)))^2/(-a*d+b*c)^4/g^4/i/(b*x+a)^3-1/3*d^3*(A+B*ln(e*(b*x+a)/(d*x+c)))^3/B/(-a*d+b*c
)^4/g^4/i

________________________________________________________________________________________

Rubi [A]
time = 0.32, antiderivative size = 507, normalized size of antiderivative = 1.00, number of steps used = 11, 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 {b^3 (c+d x)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{3 g^4 i (a+b x)^3 (b c-a d)^4}-\frac {2 b^3 B (c+d x)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{9 g^4 i (a+b x)^3 (b c-a d)^4}+\frac {3 b^2 d (c+d x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{2 g^4 i (a+b x)^2 (b c-a d)^4}+\frac {3 b^2 B d (c+d x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 g^4 i (a+b x)^2 (b c-a d)^4}-\frac {d^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^3}{3 B g^4 i (b c-a d)^4}-\frac {3 b d^2 (c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{g^4 i (a+b x) (b c-a d)^4}-\frac {6 b B d^2 (c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{g^4 i (a+b x) (b c-a d)^4}-\frac {2 b^3 B^2 (c+d x)^3}{27 g^4 i (a+b x)^3 (b c-a d)^4}+\frac {3 b^2 B^2 d (c+d x)^2}{4 g^4 i (a+b x)^2 (b c-a d)^4}-\frac {6 b B^2 d^2 (c+d x)}{g^4 i (a+b x) (b c-a d)^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(A + B*Log[(e*(a + b*x))/(c + d*x)])^2/((a*g + b*g*x)^4*(c*i + d*i*x)),x]

[Out]

(-6*b*B^2*d^2*(c + d*x))/((b*c - a*d)^4*g^4*i*(a + b*x)) + (3*b^2*B^2*d*(c + d*x)^2)/(4*(b*c - a*d)^4*g^4*i*(a
+ b*x)^2) - (2*b^3*B^2*(c + d*x)^3)/(27*(b*c - a*d)^4*g^4*i*(a + b*x)^3) - (6*b*B*d^2*(c + d*x)*(A + B*Log[(e
*(a + b*x))/(c + d*x)]))/((b*c - a*d)^4*g^4*i*(a + b*x)) + (3*b^2*B*d*(c + d*x)^2*(A + B*Log[(e*(a + b*x))/(c
+ d*x)]))/(2*(b*c - a*d)^4*g^4*i*(a + b*x)^2) - (2*b^3*B*(c + d*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(9*
(b*c - a*d)^4*g^4*i*(a + b*x)^3) - (3*b*d^2*(c + d*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/((b*c - a*d)^4*g
^4*i*(a + b*x)) + (3*b^2*d*(c + d*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(2*(b*c - a*d)^4*g^4*i*(a + b*x
)^2) - (b^3*(c + d*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(3*(b*c - a*d)^4*g^4*i*(a + b*x)^3) - (d^3*(A
+ B*Log[(e*(a + b*x))/(c + d*x)])^3)/(3*B*(b*c - a*d)^4*g^4*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}{(91 c+91 d x) (a g+b g x)^4} \, dx &=\int \left (\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{91 (b c-a d) g^4 (a+b x)^4}-\frac {b d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{91 (b c-a d)^2 g^4 (a+b x)^3}+\frac {b d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{91 (b c-a d)^3 g^4 (a+b x)^2}-\frac {b d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{91 (b c-a d)^4 g^4 (a+b x)}+\frac {d^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{91 (b c-a d)^4 g^4 (c+d x)}\right ) \, dx\\ &=-\frac {\left (b d^3\right ) \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{a+b x} \, dx}{91 (b c-a d)^4 g^4}+\frac {d^4 \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{c+d x} \, dx}{91 (b c-a d)^4 g^4}+\frac {\left (b d^2\right ) \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a+b x)^2} \, dx}{91 (b c-a d)^3 g^4}-\frac {(b d) \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a+b x)^3} \, dx}{91 (b c-a d)^2 g^4}+\frac {b \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a+b x)^4} \, dx}{91 (b c-a d) g^4}\\ &=-\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{273 (b c-a d) g^4 (a+b x)^3}+\frac {d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{182 (b c-a d)^2 g^4 (a+b x)^2}-\frac {d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{91 (b c-a d)^3 g^4 (a+b x)}-\frac {d^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{91 (b c-a d)^4 g^4}+\frac {d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{91 (b c-a d)^4 g^4}+\frac {\left (2 B d^3\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) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{e (a+b x)} \, dx}{91 (b c-a d)^4 g^4}-\frac {\left (2 B d^3\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 (c+d x)}{e (a+b x)} \, dx}{91 (b c-a d)^4 g^4}+\frac {\left (2 B d^2\right ) \int \frac {(b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a+b x)^2 (c+d x)} \, dx}{91 (b c-a d)^3 g^4}-\frac {(B d) \int \frac {(b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a+b x)^3 (c+d x)} \, dx}{91 (b c-a d)^2 g^4}+\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)^4 (c+d x)} \, dx}{273 (b c-a d) g^4}\\ &=-\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{273 (b c-a d) g^4 (a+b x)^3}+\frac {d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{182 (b c-a d)^2 g^4 (a+b x)^2}-\frac {d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{91 (b c-a d)^3 g^4 (a+b x)}-\frac {d^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{91 (b c-a d)^4 g^4}+\frac {d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{91 (b c-a d)^4 g^4}+\frac {(2 B) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^4 (c+d x)} \, dx}{273 g^4}+\frac {\left (2 B d^2\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^2 (c+d x)} \, dx}{91 (b c-a d)^2 g^4}-\frac {(B d) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^3 (c+d x)} \, dx}{91 (b c-a d) g^4}+\frac {\left (2 B d^3\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) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{a+b x} \, dx}{91 (b c-a d)^4 e g^4}-\frac {\left (2 B d^3\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 (c+d x)}{a+b x} \, dx}{91 (b c-a d)^4 e g^4}\\ &=-\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{273 (b c-a d) g^4 (a+b x)^3}+\frac {d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{182 (b c-a d)^2 g^4 (a+b x)^2}-\frac {d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{91 (b c-a d)^3 g^4 (a+b x)}-\frac {d^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{91 (b c-a d)^4 g^4}+\frac {d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{91 (b c-a d)^4 g^4}+\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)^4}-\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)^3}+\frac {b d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^3 (a+b x)^2}-\frac {b d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^4 (a+b x)}+\frac {d^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^4 (c+d x)}\right ) \, dx}{273 g^4}+\frac {\left (2 B d^2\right ) \int \left (\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d) (a+b x)^2}-\frac {b d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^2 (a+b x)}+\frac {d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^2 (c+d x)}\right ) \, dx}{91 (b c-a d)^2 g^4}-\frac {(B d) \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)^3}-\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)^2}+\frac {b d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^3 (a+b x)}-\frac {d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^3 (c+d x)}\right ) \, dx}{91 (b c-a d) g^4}+\frac {\left (2 B d^3\right ) \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}{91 (b c-a d)^4 e g^4}-\frac {\left (2 B d^3\right ) \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}{91 (b c-a d)^4 e g^4}\\ &=-\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{273 (b c-a d) g^4 (a+b x)^3}+\frac {d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{182 (b c-a d)^2 g^4 (a+b x)^2}-\frac {d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{91 (b c-a d)^3 g^4 (a+b x)}-\frac {d^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{91 (b c-a d)^4 g^4}+\frac {d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{91 (b c-a d)^4 g^4}-\frac {\left (2 b B d^3\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{273 (b c-a d)^4 g^4}-\frac {\left (b B d^3\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{91 (b c-a d)^4 g^4}-\frac {\left (2 b B d^3\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{91 (b c-a d)^4 g^4}+\frac {\left (2 B d^4\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{c+d x} \, dx}{273 (b c-a d)^4 g^4}+\frac {\left (B d^4\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{c+d x} \, dx}{91 (b c-a d)^4 g^4}+\frac {\left (2 B d^4\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{c+d x} \, dx}{91 (b c-a d)^4 g^4}+\frac {\left (2 b B d^2\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^2} \, dx}{273 (b c-a d)^3 g^4}+\frac {\left (b B d^2\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^2} \, dx}{91 (b c-a d)^3 g^4}+\frac {\left (2 b B d^2\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^2} \, dx}{91 (b c-a d)^3 g^4}+\frac {\left (2 B d^3\right ) \int \frac {\log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a+b x) (c+d x)} \, dx}{91 (b c-a d)^3 g^4}-\frac {\left (2 B d^3\right ) \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}{91 (b c-a d)^3 g^4}-\frac {(2 b B d) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^3} \, dx}{273 (b c-a d)^2 g^4}-\frac {(b B d) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^3} \, dx}{91 (b c-a d)^2 g^4}+\frac {(2 b B) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^4} \, dx}{273 (b c-a d) g^4}\\ &=-\frac {2 B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{819 (b c-a d) g^4 (a+b x)^3}+\frac {5 B d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{546 (b c-a d)^2 g^4 (a+b x)^2}-\frac {11 B d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{273 (b c-a d)^3 g^4 (a+b x)}-\frac {11 B d^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{273 (b c-a d)^4 g^4}-\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{273 (b c-a d) g^4 (a+b x)^3}+\frac {d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{182 (b c-a d)^2 g^4 (a+b x)^2}-\frac {d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{91 (b c-a d)^3 g^4 (a+b x)}-\frac {d^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{91 (b c-a d)^4 g^4}+\frac {11 B d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{273 (b c-a d)^4 g^4}+\frac {d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{91 (b c-a d)^4 g^4}+\frac {\left (2 B^2 d^3\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}{273 (b c-a d)^4 g^4}-\frac {\left (2 B^2 d^3\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}{273 (b c-a d)^4 g^4}+\frac {\left (B^2 d^3\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}{91 (b c-a d)^4 g^4}-\frac {\left (B^2 d^3\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}{91 (b c-a d)^4 g^4}+\frac {\left (2 B^2 d^3\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}{91 (b c-a d)^4 g^4}-\frac {\left (2 B^2 d^3\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}{91 (b c-a d)^4 g^4}+\frac {\left (2 B^2 d^2\right ) \int \frac {b c-a d}{(a+b x)^2 (c+d x)} \, dx}{273 (b c-a d)^3 g^4}+\frac {\left (B^2 d^2\right ) \int \frac {b c-a d}{(a+b x)^2 (c+d x)} \, dx}{91 (b c-a d)^3 g^4}+\frac {\left (2 B^2 d^2\right ) \int \frac {b c-a d}{(a+b x)^2 (c+d x)} \, dx}{91 (b c-a d)^3 g^4}+\frac {\left (2 B d^3\right ) \int \left (\frac {A \log (a+b x)}{(a+b x) (c+d x)}+\frac {B \log (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x) (c+d x)}\right ) \, dx}{91 (b c-a d)^3 g^4}-\frac {\left (2 B d^3\right ) \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}{91 (b c-a d)^3 g^4}-\frac {\left (B^2 d\right ) \int \frac {b c-a d}{(a+b x)^3 (c+d x)} \, dx}{273 (b c-a d)^2 g^4}-\frac {\left (B^2 d\right ) \int \frac {b c-a d}{(a+b x)^3 (c+d x)} \, dx}{182 (b c-a d)^2 g^4}+\frac {\left (2 B^2\right ) \int \frac {b c-a d}{(a+b x)^4 (c+d x)} \, dx}{819 (b c-a d) g^4}\\ &=-\frac {2 B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{819 (b c-a d) g^4 (a+b x)^3}+\frac {5 B d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{546 (b c-a d)^2 g^4 (a+b x)^2}-\frac {11 B d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{273 (b c-a d)^3 g^4 (a+b x)}-\frac {11 B d^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{273 (b c-a d)^4 g^4}-\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{273 (b c-a d) g^4 (a+b x)^3}+\frac {d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{182 (b c-a d)^2 g^4 (a+b x)^2}-\frac {d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{91 (b c-a d)^3 g^4 (a+b x)}-\frac {d^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{91 (b c-a d)^4 g^4}+\frac {11 B d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{273 (b c-a d)^4 g^4}+\frac {d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{91 (b c-a d)^4 g^4}+\frac {\left (2 B^2\right ) \int \frac {1}{(a+b x)^4 (c+d x)} \, dx}{819 g^4}-\frac {\left (2 b B d^3\right ) \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}{91 (b c-a d)^4 g^4}+\frac {\left (2 B d^4\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}{91 (b c-a d)^4 g^4}+\frac {\left (2 A B d^3\right ) \int \frac {\log (a+b x)}{(a+b x) (c+d x)} \, dx}{91 (b c-a d)^3 g^4}+\frac {\left (2 B^2 d^3\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}{91 (b c-a d)^3 g^4}+\frac {\left (2 B^2 d^2\right ) \int \frac {1}{(a+b x)^2 (c+d x)} \, dx}{273 (b c-a d)^2 g^4}+\frac {\left (B^2 d^2\right ) \int \frac {1}{(a+b x)^2 (c+d x)} \, dx}{91 (b c-a d)^2 g^4}+\frac {\left (2 B^2 d^2\right ) \int \frac {1}{(a+b x)^2 (c+d x)} \, dx}{91 (b c-a d)^2 g^4}-\frac {\left (B^2 d\right ) \int \frac {1}{(a+b x)^3 (c+d x)} \, dx}{273 (b c-a d) g^4}-\frac {\left (B^2 d\right ) \int \frac {1}{(a+b x)^3 (c+d x)} \, dx}{182 (b c-a d) g^4}+\frac {\left (2 B^2 d^3\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}{273 (b c-a d)^4 e g^4}-\frac {\left (2 B^2 d^3\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}{273 (b c-a d)^4 e g^4}+\frac {\left (B^2 d^3\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}{91 (b c-a d)^4 e g^4}-\frac {\left (B^2 d^3\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}{91 (b c-a d)^4 e g^4}+\frac {\left (2 B^2 d^3\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}{91 (b c-a d)^4 e g^4}-\frac {\left (2 B^2 d^3\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}{91 (b c-a d)^4 e g^4}\\ &=\frac {B^2 d^3 \log (a+b x) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{91 (b c-a d)^4 g^4}-\frac {2 B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{819 (b c-a d) g^4 (a+b x)^3}+\frac {5 B d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{546 (b c-a d)^2 g^4 (a+b x)^2}-\frac {11 B d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{273 (b c-a d)^3 g^4 (a+b x)}-\frac {11 B d^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{273 (b c-a d)^4 g^4}-\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{273 (b c-a d) g^4 (a+b x)^3}+\frac {d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{182 (b c-a d)^2 g^4 (a+b x)^2}-\frac {d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{91 (b c-a d)^3 g^4 (a+b x)}-\frac {d^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{91 (b c-a d)^4 g^4}+\frac {11 B d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{273 (b c-a d)^4 g^4}+\frac {d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{91 (b c-a d)^4 g^4}+\frac {\left (2 B^2\right ) \int \left (\frac {b}{(b c-a d) (a+b x)^4}-\frac {b d}{(b c-a d)^2 (a+b x)^3}+\frac {b d^2}{(b c-a d)^3 (a+b x)^2}-\frac {b d^3}{(b c-a d)^4 (a+b x)}+\frac {d^4}{(b c-a d)^4 (c+d x)}\right ) \, dx}{819 g^4}-\frac {\left (2 b B d^3\right ) \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}{91 (b c-a d)^4 g^4}-\frac {\left (b B^2 d^3\right ) \int \frac {\log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{91 (b c-a d)^4 g^4}+\frac {\left (2 B d^4\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}{91 (b c-a d)^4 g^4}+\frac {\left (2 A B d^3\right ) \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 )}{91 b (b c-a d)^3 g^4}+\frac {\left (2 B^2 d^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}{273 (b c-a d)^2 g^4}+\frac {\left (B^2 d^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}{91 (b c-a d)^2 g^4}+\frac {\left (2 B^2 d^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}{91 (b c-a d)^2 g^4}-\frac {\left (B^2 d\right ) \int \left (\frac {b}{(b c-a d) (a+b x)^3}-\frac {b d}{(b c-a d)^2 (a+b x)^2}+\frac {b d^2}{(b c-a d)^3 (a+b x)}-\frac {d^3}{(b c-a d)^3 (c+d x)}\right ) \, dx}{273 (b c-a d) g^4}-\frac {\left (B^2 d\right ) \int \left (\frac {b}{(b c-a d) (a+b x)^3}-\frac {b d}{(b c-a d)^2 (a+b x)^2}+\frac {b d^2}{(b c-a d)^3 (a+b x)}-\frac {d^3}{(b c-a d)^3 (c+d x)}\right ) \, dx}{182 (b c-a d) g^4}+\frac {\left (2 B^2 d^3\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}{273 (b c-a d)^4 e g^4}-\frac {\left (2 B^2 d^3\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}{273 (b c-a d)^4 e g^4}+\frac {\left (B^2 d^3\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}{91 (b c-a d)^4 e g^4}-\frac {\left (B^2 d^3\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}{91 (b c-a d)^4 e g^4}+\frac {\left (2 B^2 d^3\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}{91 (b c-a d)^4 e g^4}-\frac {\left (2 B^2 d^3\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}{91 (b c-a d)^4 e g^4}\\ &=-\frac {2 B^2}{2457 (b c-a d) g^4 (a+b x)^3}+\frac {19 B^2 d}{3276 (b c-a d)^2 g^4 (a+b x)^2}-\frac {85 B^2 d^2}{1638 (b c-a d)^3 g^4 (a+b x)}-\frac {85 B^2 d^3 \log (a+b x)}{1638 (b c-a d)^4 g^4}+\frac {B^2 d^3 \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{91 (b c-a d)^4 g^4}+\frac {B^2 d^3 \log (a+b x) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{91 (b c-a d)^4 g^4}-\frac {2 B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{819 (b c-a d) g^4 (a+b x)^3}+\frac {5 B d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{546 (b c-a d)^2 g^4 (a+b x)^2}-\frac {11 B d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{273 (b c-a d)^3 g^4 (a+b x)}-\frac {11 B d^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{273 (b c-a d)^4 g^4}-\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{273 (b c-a d) g^4 (a+b x)^3}+\frac {d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{182 (b c-a d)^2 g^4 (a+b x)^2}-\frac {d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{91 (b c-a d)^3 g^4 (a+b x)}-\frac {d^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{91 (b c-a d)^4 g^4}+\frac {85 B^2 d^3 \log (c+d x)}{1638 (b c-a d)^4 g^4}+\frac {11 B d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{273 (b c-a d)^4 g^4}+\frac {d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{91 (b c-a d)^4 g^4}+\frac {\left (2 A B d^3\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{91 (b c-a d)^4 g^4}-\frac {\left (2 A b B d^3\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{91 (b c-a d)^4 g^4}+\frac {\left (2 b B^2 d^3\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{273 (b c-a d)^4 g^4}-\frac {\left (2 b B^2 d^3\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{273 (b c-a d)^4 g^4}+\frac {\left (b B^2 d^3\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{91 (b c-a d)^4 g^4}-\frac {\left (b B^2 d^3\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{91 (b c-a d)^4 g^4}+\frac {\left (2 b B^2 d^3\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{91 (b c-a d)^4 g^4}-\frac {\left (2 b B^2 d^3\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{91 (b c-a d)^4 g^4}-\frac {\left (2 b B^2 d^3\right ) \int \frac {\log \left (\frac {e (a+b x)}{c+d x}\right ) \log (c+d x)}{a+b x} \, dx}{91 (b c-a d)^4 g^4}+\frac {\left (2 A B d^4\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{91 (b c-a d)^4 g^4}-\frac {\left (2 A B d^4\right ) \text {Subst}\left (\int \frac {\log (x)}{\frac {b c-a d}{b}+\frac {d x}{b}} \, dx,x,a+b x\right )}{91 b (b c-a d)^4 g^4}-\frac {\left (2 B^2 d^4\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{273 (b c-a d)^4 g^4}+\frac {\left (2 B^2 d^4\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{273 (b c-a d)^4 g^4}-\frac {\left (B^2 d^4\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{91 (b c-a d)^4 g^4}+\frac {\left (B^2 d^4\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{91 (b c-a d)^4 g^4}-\frac {\left (2 B^2 d^4\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{91 (b c-a d)^4 g^4}+\frac {\left (2 B^2 d^4\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{91 (b c-a d)^4 g^4}+\frac {\left (2 B^2 d^4\right ) \int \frac {\log \left (\frac {e (a+b x)}{c+d x}\right ) \log (c+d x)}{c+d x} \, dx}{91 (b c-a d)^4 g^4}-\frac {\left (2 B^2 d^3\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}{91 (b c-a d)^3 g^4}\\ &=-\frac {2 B^2}{2457 (b c-a d) g^4 (a+b x)^3}+\frac {19 B^2 d}{3276 (b c-a d)^2 g^4 (a+b x)^2}-\frac {85 B^2 d^2}{1638 (b c-a d)^3 g^4 (a+b x)}-\frac {85 B^2 d^3 \log (a+b x)}{1638 (b c-a d)^4 g^4}+\frac {A B d^3 \log ^2(a+b x)}{91 (b c-a d)^4 g^4}+\frac {B^2 d^3 \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{91 (b c-a d)^4 g^4}+\frac {B^2 d^3 \log (a+b x) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{91 (b c-a d)^4 g^4}-\frac {2 B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{819 (b c-a d) g^4 (a+b x)^3}+\frac {5 B d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{546 (b c-a d)^2 g^4 (a+b x)^2}-\frac {11 B d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{273 (b c-a d)^3 g^4 (a+b x)}-\frac {11 B d^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{273 (b c-a d)^4 g^4}-\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{273 (b c-a d) g^4 (a+b x)^3}+\frac {d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{182 (b c-a d)^2 g^4 (a+b x)^2}-\frac {d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{91 (b c-a d)^3 g^4 (a+b x)}-\frac {d^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{91 (b c-a d)^4 g^4}+\frac {85 B^2 d^3 \log (c+d x)}{1638 (b c-a d)^4 g^4}-\frac {2 A B d^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{91 (b c-a d)^4 g^4}-\frac {11 B^2 d^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{273 (b c-a d)^4 g^4}+\frac {11 B d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{273 (b c-a d)^4 g^4}+\frac {d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{91 (b c-a d)^4 g^4}+\frac {B^2 d^3 \log \left (\frac {e (a+b x)}{c+d x}\right ) \log ^2(c+d x)}{91 (b c-a d)^4 g^4}-\frac {2 A B d^3 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{91 (b c-a d)^4 g^4}-\frac {11 B^2 d^3 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{273 (b c-a d)^4 g^4}-\frac {2 B^2 d^3 \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 )}{91 (b c-a d)^4 g^4}+\frac {\left (2 A B d^3\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{91 (b c-a d)^4 g^4}+\frac {\left (2 A B d^3\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{91 (b c-a d)^4 g^4}+\frac {\left (2 B^2 d^3\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{273 (b c-a d)^4 g^4}+\frac {\left (2 B^2 d^3\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{273 (b c-a d)^4 g^4}+\frac {\left (B^2 d^3\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{91 (b c-a d)^4 g^4}+\frac {\left (B^2 d^3\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{91 (b c-a d)^4 g^4}+\frac {\left (2 B^2 d^3\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{91 (b c-a d)^4 g^4}+\frac {\left (2 B^2 d^3\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{91 (b c-a d)^4 g^4}+\frac {\left (2 b B^2 d^3\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{273 (b c-a d)^4 g^4}-\frac {\left (b B^2 d^3\right ) \int \frac {\log ^2(c+d x)}{a+b x} \, dx}{91 (b c-a d)^4 g^4}+\frac {\left (b B^2 d^3\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{91 (b c-a d)^4 g^4}-\frac {\left (2 b B^2 d^3\right ) \int \frac {\log (a+b x) \log (c+d x)}{a+b x} \, dx}{91 (b c-a d)^4 g^4}-\frac {\left (2 b B^2 d^3\right ) \int \frac {\log \left (\frac {1}{c+d x}\right ) \log (c+d x)}{a+b x} \, dx}{91 (b c-a d)^4 g^4}+\frac {\left (2 b B^2 d^3\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{91 (b c-a d)^4 g^4}+\frac {\left (2 A B d^4\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{91 (b c-a d)^4 g^4}+\frac {\left (2 B^2 d^4\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{273 (b c-a d)^4 g^4}+\frac {\left (B^2 d^4\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{91 (b c-a d)^4 g^4}+\frac {\left (B^2 d^4\right ) \int \frac {\log ^2(c+d x)}{c+d x} \, dx}{91 (b c-a d)^4 g^4}+\frac {\left (2 B^2 d^4\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{91 (b c-a d)^4 g^4}+\frac {\left (2 B^2 d^3\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}{91 (b c-a d)^3 g^4}-\frac {\left (2 b B^2 d^3 \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}{91 (b c-a d)^4 g^4}\\ &=-\frac {2 B^2}{2457 (b c-a d) g^4 (a+b x)^3}+\frac {19 B^2 d}{3276 (b c-a d)^2 g^4 (a+b x)^2}-\frac {85 B^2 d^2}{1638 (b c-a d)^3 g^4 (a+b x)}-\frac {85 B^2 d^3 \log (a+b x)}{1638 (b c-a d)^4 g^4}+\frac {A B d^3 \log ^2(a+b x)}{91 (b c-a d)^4 g^4}+\frac {11 B^2 d^3 \log ^2(a+b x)}{546 (b c-a d)^4 g^4}+\frac {B^2 d^3 \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{91 (b c-a d)^4 g^4}+\frac {B^2 d^3 \log (a+b x) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{91 (b c-a d)^4 g^4}-\frac {2 B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{819 (b c-a d) g^4 (a+b x)^3}+\frac {5 B d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{546 (b c-a d)^2 g^4 (a+b x)^2}-\frac {11 B d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{273 (b c-a d)^3 g^4 (a+b x)}-\frac {11 B d^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{273 (b c-a d)^4 g^4}-\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{273 (b c-a d) g^4 (a+b x)^3}+\frac {d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{182 (b c-a d)^2 g^4 (a+b x)^2}-\frac {d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{91 (b c-a d)^3 g^4 (a+b x)}-\frac {d^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{91 (b c-a d)^4 g^4}+\frac {85 B^2 d^3 \log (c+d x)}{1638 (b c-a d)^4 g^4}-\frac {2 A B d^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{91 (b c-a d)^4 g^4}-\frac {11 B^2 d^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{273 (b c-a d)^4 g^4}+\frac {2 B^2 d^3 \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)}{91 (b c-a d)^4 g^4}+\frac {11 B d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{273 (b c-a d)^4 g^4}+\frac {d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{91 (b c-a d)^4 g^4}+\frac {A B d^3 \log ^2(c+d x)}{91 (b c-a d)^4 g^4}+\frac {11 B^2 d^3 \log ^2(c+d x)}{546 (b c-a d)^4 g^4}-\frac {B^2 d^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(c+d x)}{91 (b c-a d)^4 g^4}+\frac {B^2 d^3 \log \left (\frac {e (a+b x)}{c+d x}\right ) \log ^2(c+d x)}{91 (b c-a d)^4 g^4}-\frac {2 A B d^3 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{91 (b c-a d)^4 g^4}-\frac {11 B^2 d^3 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{273 (b c-a d)^4 g^4}-\frac {2 A B d^3 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{91 (b c-a d)^4 g^4}-\frac {2 B^2 d^3 \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 )}{91 (b c-a d)^4 g^4}-\frac {2 B^2 d^3 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{91 (b c-a d)^4 g^4}+\frac {\left (2 A B d^3\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{91 (b c-a d)^4 g^4}+\frac {\left (2 B^2 d^3\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{273 (b c-a d)^4 g^4}+\frac {\left (2 B^2 d^3\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{273 (b c-a d)^4 g^4}+\frac {\left (B^2 d^3\right ) \text {Subst}\left (\int \frac {\log ^2(x)}{x} \, dx,x,c+d x\right )}{91 (b c-a d)^4 g^4}+\frac {\left (B^2 d^3\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{91 (b c-a d)^4 g^4}+\frac {\left (B^2 d^3\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{91 (b c-a d)^4 g^4}-\frac {\left (2 B^2 d^3\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 )}{91 (b c-a d)^4 g^4}-\frac {\left (2 B^2 d^3\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 )}{91 (b c-a d)^4 g^4}+\frac {\left (2 B^2 d^3\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{91 (b c-a d)^4 g^4}+\frac {\left (2 B^2 d^3\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{91 (b c-a d)^4 g^4}+\frac {\left (2 B^2 d^4\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right ) \log (c+d x)}{c+d x} \, dx}{91 (b c-a d)^4 g^4}+\frac {\left (2 B^2 d^4 \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}{91 (b c-a d)^4 g^4}\\ &=-\frac {2 B^2}{2457 (b c-a d) g^4 (a+b x)^3}+\frac {19 B^2 d}{3276 (b c-a d)^2 g^4 (a+b x)^2}-\frac {85 B^2 d^2}{1638 (b c-a d)^3 g^4 (a+b x)}-\frac {85 B^2 d^3 \log (a+b x)}{1638 (b c-a d)^4 g^4}+\frac {A B d^3 \log ^2(a+b x)}{91 (b c-a d)^4 g^4}+\frac {11 B^2 d^3 \log ^2(a+b x)}{546 (b c-a d)^4 g^4}+\frac {B^2 d^3 \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{91 (b c-a d)^4 g^4}+\frac {B^2 d^3 \log (a+b x) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{91 (b c-a d)^4 g^4}-\frac {2 B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{819 (b c-a d) g^4 (a+b x)^3}+\frac {5 B d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{546 (b c-a d)^2 g^4 (a+b x)^2}-\frac {11 B d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{273 (b c-a d)^3 g^4 (a+b x)}-\frac {11 B d^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{273 (b c-a d)^4 g^4}-\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{273 (b c-a d) g^4 (a+b x)^3}+\frac {d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{182 (b c-a d)^2 g^4 (a+b x)^2}-\frac {d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{91 (b c-a d)^3 g^4 (a+b x)}-\frac {d^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{91 (b c-a d)^4 g^4}+\frac {85 B^2 d^3 \log (c+d x)}{1638 (b c-a d)^4 g^4}-\frac {B^2 d^3 \log ^2(a+b x) \log (c+d x)}{91 (b c-a d)^4 g^4}-\frac {2 A B d^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{91 (b c-a d)^4 g^4}-\frac {11 B^2 d^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{273 (b c-a d)^4 g^4}-\frac {2 B^2 d^3 \log (a+b x) \log \left (\frac {1}{c+d x}\right ) \log (c+d x)}{91 (b c-a d)^4 g^4}+\frac {2 B^2 d^3 \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)}{91 (b c-a d)^4 g^4}+\frac {11 B d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{273 (b c-a d)^4 g^4}+\frac {d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{91 (b c-a d)^4 g^4}+\frac {A B d^3 \log ^2(c+d x)}{91 (b c-a d)^4 g^4}+\frac {11 B^2 d^3 \log ^2(c+d x)}{546 (b c-a d)^4 g^4}-\frac {B^2 d^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(c+d x)}{91 (b c-a d)^4 g^4}+\frac {B^2 d^3 \log \left (\frac {e (a+b x)}{c+d x}\right ) \log ^2(c+d x)}{91 (b c-a d)^4 g^4}-\frac {2 A B d^3 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{91 (b c-a d)^4 g^4}-\frac {11 B^2 d^3 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{273 (b c-a d)^4 g^4}-\frac {2 A B d^3 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{91 (b c-a d)^4 g^4}-\frac {11 B^2 d^3 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{273 (b c-a d)^4 g^4}-\frac {2 A B d^3 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{91 (b c-a d)^4 g^4}-\frac {11 B^2 d^3 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{273 (b c-a d)^4 g^4}-\frac {2 B^2 d^3 \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 )}{91 (b c-a d)^4 g^4}-\frac {2 B^2 d^3 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{91 (b c-a d)^4 g^4}+\frac {\left (B^2 d^3\right ) \text {Subst}\left (\int x^2 \, dx,x,\log (c+d x)\right )}{91 (b c-a d)^4 g^4}+\frac {\left (2 B^2 d^3\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 )}{91 (b c-a d)^4 g^4}+\frac {\left (B^2 d^4\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 )}{91 b (b c-a d)^4 g^4}+\frac {\left (2 B^2 d^4\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 )}{91 b (b c-a d)^4 g^4}-\frac {\left (2 B^2 d^4\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 )}{91 b (b c-a d)^4 g^4}+\frac {\left (2 B^2 d^3 \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 )}{91 (b c-a d)^4 g^4}\\ &=-\frac {2 B^2}{2457 (b c-a d) g^4 (a+b x)^3}+\frac {19 B^2 d}{3276 (b c-a d)^2 g^4 (a+b x)^2}-\frac {85 B^2 d^2}{1638 (b c-a d)^3 g^4 (a+b x)}-\frac {85 B^2 d^3 \log (a+b x)}{1638 (b c-a d)^4 g^4}+\frac {A B d^3 \log ^2(a+b x)}{91 (b c-a d)^4 g^4}+\frac {11 B^2 d^3 \log ^2(a+b x)}{546 (b c-a d)^4 g^4}+\frac {B^2 d^3 \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{91 (b c-a d)^4 g^4}+\frac {B^2 d^3 \log (a+b x) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{91 (b c-a d)^4 g^4}-\frac {2 B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{819 (b c-a d) g^4 (a+b x)^3}+\frac {5 B d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{546 (b c-a d)^2 g^4 (a+b x)^2}-\frac {11 B d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{273 (b c-a d)^3 g^4 (a+b x)}-\frac {11 B d^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{273 (b c-a d)^4 g^4}-\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{273 (b c-a d) g^4 (a+b x)^3}+\frac {d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{182 (b c-a d)^2 g^4 (a+b x)^2}-\frac {d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{91 (b c-a d)^3 g^4 (a+b x)}-\frac {d^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{91 (b c-a d)^4 g^4}+\frac {85 B^2 d^3 \log (c+d x)}{1638 (b c-a d)^4 g^4}-\frac {B^2 d^3 \log ^2(a+b x) \log (c+d x)}{91 (b c-a d)^4 g^4}-\frac {2 A B d^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{91 (b c-a d)^4 g^4}-\frac {11 B^2 d^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{273 (b c-a d)^4 g^4}-\frac {2 B^2 d^3 \log (a+b x) \log \left (\frac {1}{c+d x}\right ) \log (c+d x)}{91 (b c-a d)^4 g^4}+\frac {2 B^2 d^3 \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)}{91 (b c-a d)^4 g^4}+\frac {11 B d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{273 (b c-a d)^4 g^4}+\frac {d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{91 (b c-a d)^4 g^4}+\frac {A B d^3 \log ^2(c+d x)}{91 (b c-a d)^4 g^4}+\frac {11 B^2 d^3 \log ^2(c+d x)}{546 (b c-a d)^4 g^4}-\frac {B^2 d^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(c+d x)}{91 (b c-a d)^4 g^4}+\frac {B^2 d^3 \log \left (\frac {e (a+b x)}{c+d x}\right ) \log ^2(c+d x)}{91 (b c-a d)^4 g^4}+\frac {B^2 d^3 \log ^3(c+d x)}{273 (b c-a d)^4 g^4}-\frac {2 A B d^3 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{91 (b c-a d)^4 g^4}-\frac {11 B^2 d^3 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{273 (b c-a d)^4 g^4}+\frac {B^2 d^3 \log ^2(a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{91 (b c-a d)^4 g^4}-\frac {2 A B d^3 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{91 (b c-a d)^4 g^4}-\frac {11 B^2 d^3 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{273 (b c-a d)^4 g^4}-\frac {2 A B d^3 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{91 (b c-a d)^4 g^4}-\frac {11 B^2 d^3 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{273 (b c-a d)^4 g^4}+\frac {2 B^2 d^3 \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 )}{91 (b c-a d)^4 g^4}-\frac {2 B^2 d^3 \log (c+d x) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{91 (b c-a d)^4 g^4}-\frac {2 B^2 d^3 \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 )}{91 (b c-a d)^4 g^4}-\frac {2 B^2 d^3 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{91 (b c-a d)^4 g^4}+\frac {\left (2 B^2 d^3\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 )}{91 (b c-a d)^4 g^4}-\frac {\left (2 B^2 d^3\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 )}{91 (b c-a d)^4 g^4}-\frac {\left (2 B^2 d^3\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 )}{91 (b c-a d)^4 g^4}+\frac {\left (2 B^2 d^3\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 )}{91 (b c-a d)^4 g^4}\\ &=-\frac {2 B^2}{2457 (b c-a d) g^4 (a+b x)^3}+\frac {19 B^2 d}{3276 (b c-a d)^2 g^4 (a+b x)^2}-\frac {85 B^2 d^2}{1638 (b c-a d)^3 g^4 (a+b x)}-\frac {85 B^2 d^3 \log (a+b x)}{1638 (b c-a d)^4 g^4}+\frac {A B d^3 \log ^2(a+b x)}{91 (b c-a d)^4 g^4}+\frac {11 B^2 d^3 \log ^2(a+b x)}{546 (b c-a d)^4 g^4}-\frac {B^2 d^3 \log (a+b x) \log ^2\left (\frac {1}{c+d x}\right )}{91 (b c-a d)^4 g^4}+\frac {B^2 d^3 \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{91 (b c-a d)^4 g^4}+\frac {B^2 d^3 \log (a+b x) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{91 (b c-a d)^4 g^4}-\frac {2 B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{819 (b c-a d) g^4 (a+b x)^3}+\frac {5 B d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{546 (b c-a d)^2 g^4 (a+b x)^2}-\frac {11 B d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{273 (b c-a d)^3 g^4 (a+b x)}-\frac {11 B d^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{273 (b c-a d)^4 g^4}-\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{273 (b c-a d) g^4 (a+b x)^3}+\frac {d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{182 (b c-a d)^2 g^4 (a+b x)^2}-\frac {d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{91 (b c-a d)^3 g^4 (a+b x)}-\frac {d^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{91 (b c-a d)^4 g^4}+\frac {85 B^2 d^3 \log (c+d x)}{1638 (b c-a d)^4 g^4}-\frac {B^2 d^3 \log ^2(a+b x) \log (c+d x)}{91 (b c-a d)^4 g^4}-\frac {2 A B d^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{91 (b c-a d)^4 g^4}-\frac {11 B^2 d^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{273 (b c-a d)^4 g^4}-\frac {2 B^2 d^3 \log (a+b x) \log \left (\frac {1}{c+d x}\right ) \log (c+d x)}{91 (b c-a d)^4 g^4}+\frac {2 B^2 d^3 \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)}{91 (b c-a d)^4 g^4}+\frac {11 B d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{273 (b c-a d)^4 g^4}+\frac {d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{91 (b c-a d)^4 g^4}+\frac {A B d^3 \log ^2(c+d x)}{91 (b c-a d)^4 g^4}+\frac {11 B^2 d^3 \log ^2(c+d x)}{546 (b c-a d)^4 g^4}-\frac {B^2 d^3 \log (a+b x) \log ^2(c+d x)}{91 (b c-a d)^4 g^4}-\frac {B^2 d^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(c+d x)}{91 (b c-a d)^4 g^4}+\frac {B^2 d^3 \log \left (\frac {e (a+b x)}{c+d x}\right ) \log ^2(c+d x)}{91 (b c-a d)^4 g^4}+\frac {B^2 d^3 \log ^3(c+d x)}{273 (b c-a d)^4 g^4}-\frac {2 A B d^3 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{91 (b c-a d)^4 g^4}-\frac {11 B^2 d^3 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{273 (b c-a d)^4 g^4}+\frac {B^2 d^3 \log ^2(a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{91 (b c-a d)^4 g^4}-\frac {2 A B d^3 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{91 (b c-a d)^4 g^4}-\frac {11 B^2 d^3 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{273 (b c-a d)^4 g^4}+\frac {2 B^2 d^3 \log (a+b x) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{91 (b c-a d)^4 g^4}-\frac {2 A B d^3 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{91 (b c-a d)^4 g^4}-\frac {11 B^2 d^3 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{273 (b c-a d)^4 g^4}+\frac {2 B^2 d^3 \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 )}{91 (b c-a d)^4 g^4}-\frac {2 B^2 d^3 \log (c+d x) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{91 (b c-a d)^4 g^4}-\frac {2 B^2 d^3 \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 )}{91 (b c-a d)^4 g^4}+\frac {2 B^2 d^3 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{91 (b c-a d)^4 g^4}-\frac {2 B^2 d^3 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{91 (b c-a d)^4 g^4}+\frac {\left (b B^2 d^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 )}{91 (b c-a d)^4 g^4}+\frac {\left (b B^2 d^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 )}{91 (b c-a d)^4 g^4}-\frac {\left (2 B^2 d^3\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 )}{91 (b c-a d)^4 g^4}\\ &=-\frac {2 B^2}{2457 (b c-a d) g^4 (a+b x)^3}+\frac {19 B^2 d}{3276 (b c-a d)^2 g^4 (a+b x)^2}-\frac {85 B^2 d^2}{1638 (b c-a d)^3 g^4 (a+b x)}-\frac {85 B^2 d^3 \log (a+b x)}{1638 (b c-a d)^4 g^4}+\frac {A B d^3 \log ^2(a+b x)}{91 (b c-a d)^4 g^4}+\frac {11 B^2 d^3 \log ^2(a+b x)}{546 (b c-a d)^4 g^4}-\frac {B^2 d^3 \log (a+b x) \log ^2\left (\frac {1}{c+d x}\right )}{91 (b c-a d)^4 g^4}+\frac {B^2 d^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2\left (\frac {1}{c+d x}\right )}{91 (b c-a d)^4 g^4}+\frac {B^2 d^3 \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{91 (b c-a d)^4 g^4}+\frac {B^2 d^3 \log (a+b x) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{91 (b c-a d)^4 g^4}-\frac {2 B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{819 (b c-a d) g^4 (a+b x)^3}+\frac {5 B d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{546 (b c-a d)^2 g^4 (a+b x)^2}-\frac {11 B d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{273 (b c-a d)^3 g^4 (a+b x)}-\frac {11 B d^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{273 (b c-a d)^4 g^4}-\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{273 (b c-a d) g^4 (a+b x)^3}+\frac {d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{182 (b c-a d)^2 g^4 (a+b x)^2}-\frac {d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{91 (b c-a d)^3 g^4 (a+b x)}-\frac {d^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{91 (b c-a d)^4 g^4}+\frac {85 B^2 d^3 \log (c+d x)}{1638 (b c-a d)^4 g^4}-\frac {B^2 d^3 \log ^2(a+b x) \log (c+d x)}{91 (b c-a d)^4 g^4}-\frac {2 A B d^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{91 (b c-a d)^4 g^4}-\frac {11 B^2 d^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{273 (b c-a d)^4 g^4}-\frac {2 B^2 d^3 \log (a+b x) \log \left (\frac {1}{c+d x}\right ) \log (c+d x)}{91 (b c-a d)^4 g^4}+\frac {2 B^2 d^3 \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)}{91 (b c-a d)^4 g^4}+\frac {11 B d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{273 (b c-a d)^4 g^4}+\frac {d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{91 (b c-a d)^4 g^4}+\frac {A B d^3 \log ^2(c+d x)}{91 (b c-a d)^4 g^4}+\frac {11 B^2 d^3 \log ^2(c+d x)}{546 (b c-a d)^4 g^4}-\frac {B^2 d^3 \log (a+b x) \log ^2(c+d x)}{91 (b c-a d)^4 g^4}+\frac {B^2 d^3 \log \left (\frac {e (a+b x)}{c+d x}\right ) \log ^2(c+d x)}{91 (b c-a d)^4 g^4}+\frac {B^2 d^3 \log ^3(c+d x)}{273 (b c-a d)^4 g^4}-\frac {2 A B d^3 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{91 (b c-a d)^4 g^4}-\frac {11 B^2 d^3 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{273 (b c-a d)^4 g^4}+\frac {B^2 d^3 \log ^2(a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{91 (b c-a d)^4 g^4}-\frac {2 A B d^3 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{91 (b c-a d)^4 g^4}-\frac {11 B^2 d^3 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{273 (b c-a d)^4 g^4}+\frac {2 B^2 d^3 \log (a+b x) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{91 (b c-a d)^4 g^4}-\frac {2 A B d^3 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{91 (b c-a d)^4 g^4}-\frac {11 B^2 d^3 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{273 (b c-a d)^4 g^4}+\frac {2 B^2 d^3 \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 )}{91 (b c-a d)^4 g^4}-\frac {2 B^2 d^3 \log (c+d x) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{91 (b c-a d)^4 g^4}-\frac {2 B^2 d^3 \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 )}{91 (b c-a d)^4 g^4}-\frac {2 B^2 d^3 \text {Li}_3\left (-\frac {d (a+b x)}{b c-a d}\right )}{91 (b c-a d)^4 g^4}+\frac {2 B^2 d^3 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{91 (b c-a d)^4 g^4}-\frac {2 B^2 d^3 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{91 (b c-a d)^4 g^4}+\frac {\left (2 B^2 d^3\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 )}{91 (b c-a d)^4 g^4}-\frac {\left (2 B^2 d^3\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 )}{91 (b c-a d)^4 g^4}\\ &=-\frac {2 B^2}{2457 (b c-a d) g^4 (a+b x)^3}+\frac {19 B^2 d}{3276 (b c-a d)^2 g^4 (a+b x)^2}-\frac {85 B^2 d^2}{1638 (b c-a d)^3 g^4 (a+b x)}-\frac {85 B^2 d^3 \log (a+b x)}{1638 (b c-a d)^4 g^4}+\frac {A B d^3 \log ^2(a+b x)}{91 (b c-a d)^4 g^4}+\frac {11 B^2 d^3 \log ^2(a+b x)}{546 (b c-a d)^4 g^4}-\frac {B^2 d^3 \log (a+b x) \log ^2\left (\frac {1}{c+d x}\right )}{91 (b c-a d)^4 g^4}+\frac {B^2 d^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2\left (\frac {1}{c+d x}\right )}{91 (b c-a d)^4 g^4}+\frac {B^2 d^3 \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{91 (b c-a d)^4 g^4}+\frac {B^2 d^3 \log (a+b x) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{91 (b c-a d)^4 g^4}-\frac {2 B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{819 (b c-a d) g^4 (a+b x)^3}+\frac {5 B d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{546 (b c-a d)^2 g^4 (a+b x)^2}-\frac {11 B d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{273 (b c-a d)^3 g^4 (a+b x)}-\frac {11 B d^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{273 (b c-a d)^4 g^4}-\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{273 (b c-a d) g^4 (a+b x)^3}+\frac {d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{182 (b c-a d)^2 g^4 (a+b x)^2}-\frac {d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{91 (b c-a d)^3 g^4 (a+b x)}-\frac {d^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{91 (b c-a d)^4 g^4}+\frac {85 B^2 d^3 \log (c+d x)}{1638 (b c-a d)^4 g^4}-\frac {B^2 d^3 \log ^2(a+b x) \log (c+d x)}{91 (b c-a d)^4 g^4}-\frac {2 A B d^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{91 (b c-a d)^4 g^4}-\frac {11 B^2 d^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{273 (b c-a d)^4 g^4}-\frac {2 B^2 d^3 \log (a+b x) \log \left (\frac {1}{c+d x}\right ) \log (c+d x)}{91 (b c-a d)^4 g^4}+\frac {2 B^2 d^3 \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)}{91 (b c-a d)^4 g^4}+\frac {11 B d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{273 (b c-a d)^4 g^4}+\frac {d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{91 (b c-a d)^4 g^4}+\frac {A B d^3 \log ^2(c+d x)}{91 (b c-a d)^4 g^4}+\frac {11 B^2 d^3 \log ^2(c+d x)}{546 (b c-a d)^4 g^4}-\frac {B^2 d^3 \log (a+b x) \log ^2(c+d x)}{91 (b c-a d)^4 g^4}+\frac {B^2 d^3 \log \left (\frac {e (a+b x)}{c+d x}\right ) \log ^2(c+d x)}{91 (b c-a d)^4 g^4}+\frac {B^2 d^3 \log ^3(c+d x)}{273 (b c-a d)^4 g^4}-\frac {2 A B d^3 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{91 (b c-a d)^4 g^4}-\frac {11 B^2 d^3 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{273 (b c-a d)^4 g^4}+\frac {B^2 d^3 \log ^2(a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{91 (b c-a d)^4 g^4}-\frac {2 A B d^3 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{91 (b c-a d)^4 g^4}-\frac {11 B^2 d^3 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{273 (b c-a d)^4 g^4}+\frac {2 B^2 d^3 \log (a+b x) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{91 (b c-a d)^4 g^4}-\frac {2 A B d^3 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{91 (b c-a d)^4 g^4}-\frac {11 B^2 d^3 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{273 (b c-a d)^4 g^4}-\frac {2 B^2 d^3 \log \left (\frac {1}{c+d x}\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{91 (b c-a d)^4 g^4}+\frac {2 B^2 d^3 \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 )}{91 (b c-a d)^4 g^4}-\frac {2 B^2 d^3 \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 )}{91 (b c-a d)^4 g^4}-\frac {2 B^2 d^3 \text {Li}_3\left (-\frac {d (a+b x)}{b c-a d}\right )}{91 (b c-a d)^4 g^4}+\frac {2 B^2 d^3 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{91 (b c-a d)^4 g^4}-\frac {2 B^2 d^3 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{91 (b c-a d)^4 g^4}-2 \frac {\left (2 B^2 d^3\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 )}{91 (b c-a d)^4 g^4}\\ &=-\frac {2 B^2}{2457 (b c-a d) g^4 (a+b x)^3}+\frac {19 B^2 d}{3276 (b c-a d)^2 g^4 (a+b x)^2}-\frac {85 B^2 d^2}{1638 (b c-a d)^3 g^4 (a+b x)}-\frac {85 B^2 d^3 \log (a+b x)}{1638 (b c-a d)^4 g^4}+\frac {A B d^3 \log ^2(a+b x)}{91 (b c-a d)^4 g^4}+\frac {11 B^2 d^3 \log ^2(a+b x)}{546 (b c-a d)^4 g^4}-\frac {B^2 d^3 \log (a+b x) \log ^2\left (\frac {1}{c+d x}\right )}{91 (b c-a d)^4 g^4}+\frac {B^2 d^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2\left (\frac {1}{c+d x}\right )}{91 (b c-a d)^4 g^4}+\frac {B^2 d^3 \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{91 (b c-a d)^4 g^4}+\frac {B^2 d^3 \log (a+b x) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{91 (b c-a d)^4 g^4}-\frac {2 B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{819 (b c-a d) g^4 (a+b x)^3}+\frac {5 B d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{546 (b c-a d)^2 g^4 (a+b x)^2}-\frac {11 B d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{273 (b c-a d)^3 g^4 (a+b x)}-\frac {11 B d^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{273 (b c-a d)^4 g^4}-\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{273 (b c-a d) g^4 (a+b x)^3}+\frac {d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{182 (b c-a d)^2 g^4 (a+b x)^2}-\frac {d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{91 (b c-a d)^3 g^4 (a+b x)}-\frac {d^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{91 (b c-a d)^4 g^4}+\frac {85 B^2 d^3 \log (c+d x)}{1638 (b c-a d)^4 g^4}-\frac {B^2 d^3 \log ^2(a+b x) \log (c+d x)}{91 (b c-a d)^4 g^4}-\frac {2 A B d^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{91 (b c-a d)^4 g^4}-\frac {11 B^2 d^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{273 (b c-a d)^4 g^4}-\frac {2 B^2 d^3 \log (a+b x) \log \left (\frac {1}{c+d x}\right ) \log (c+d x)}{91 (b c-a d)^4 g^4}+\frac {2 B^2 d^3 \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)}{91 (b c-a d)^4 g^4}+\frac {11 B d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{273 (b c-a d)^4 g^4}+\frac {d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{91 (b c-a d)^4 g^4}+\frac {A B d^3 \log ^2(c+d x)}{91 (b c-a d)^4 g^4}+\frac {11 B^2 d^3 \log ^2(c+d x)}{546 (b c-a d)^4 g^4}-\frac {B^2 d^3 \log (a+b x) \log ^2(c+d x)}{91 (b c-a d)^4 g^4}+\frac {B^2 d^3 \log \left (\frac {e (a+b x)}{c+d x}\right ) \log ^2(c+d x)}{91 (b c-a d)^4 g^4}+\frac {B^2 d^3 \log ^3(c+d x)}{273 (b c-a d)^4 g^4}-\frac {2 A B d^3 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{91 (b c-a d)^4 g^4}-\frac {11 B^2 d^3 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{273 (b c-a d)^4 g^4}+\frac {B^2 d^3 \log ^2(a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{91 (b c-a d)^4 g^4}-\frac {2 A B d^3 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{91 (b c-a d)^4 g^4}-\frac {11 B^2 d^3 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{273 (b c-a d)^4 g^4}+\frac {2 B^2 d^3 \log (a+b x) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{91 (b c-a d)^4 g^4}-\frac {2 A B d^3 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{91 (b c-a d)^4 g^4}-\frac {11 B^2 d^3 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{273 (b c-a d)^4 g^4}-\frac {2 B^2 d^3 \log \left (\frac {1}{c+d x}\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{91 (b c-a d)^4 g^4}+\frac {2 B^2 d^3 \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 )}{91 (b c-a d)^4 g^4}-\frac {2 B^2 d^3 \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 )}{91 (b c-a d)^4 g^4}-\frac {2 B^2 d^3 \text {Li}_3\left (-\frac {d (a+b x)}{b c-a d}\right )}{91 (b c-a d)^4 g^4}-\frac {2 B^2 d^3 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{91 (b c-a d)^4 g^4}-\frac {2 B^2 d^3 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{91 (b c-a d)^4 g^4}\\ \end {align*}

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Mathematica [A]
time = 0.91, size = 442, normalized size = 0.87 \begin {gather*} -\frac {4 \left (9 A^2+6 A B+2 B^2\right ) (b c-a d)^3-3 \left (18 A^2+30 A B+19 B^2\right ) d (b c-a d)^2 (a+b x)-6 \left (18 A^2+66 A B+85 B^2\right ) d^2 (-b c+a d) (a+b x)^2+6 \left (18 A^2+66 A B+85 B^2\right ) d^3 (a+b x)^3 \log (a+b x)+6 B (b c-a d) \left (4 (3 A+B) (b c-a d)^2+3 (6 A+5 B) d (-b c+a d) (a+b x)+6 (6 A+11 B) d^2 (a+b x)^2\right ) \log \left (\frac {e (a+b x)}{c+d x}\right )+18 B \left (6 a^3 A d^3+18 a^2 b d^2 (A d x+B (c+d x))+9 a b^2 d \left (2 A d^2 x^2+B \left (-c^2+2 c d x+3 d^2 x^2\right )\right )+b^3 \left (6 A d^3 x^3+B \left (2 c^3-3 c^2 d x+6 c d^2 x^2+11 d^3 x^3\right )\right )\right ) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )+36 B^2 d^3 (a+b x)^3 \log ^3\left (\frac {e (a+b x)}{c+d x}\right )-6 \left (18 A^2+66 A B+85 B^2\right ) d^3 (a+b x)^3 \log (c+d x)}{108 (b c-a d)^4 g^4 i (a+b x)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(A + B*Log[(e*(a + b*x))/(c + d*x)])^2/((a*g + b*g*x)^4*(c*i + d*i*x)),x]

[Out]

-1/108*(4*(9*A^2 + 6*A*B + 2*B^2)*(b*c - a*d)^3 - 3*(18*A^2 + 30*A*B + 19*B^2)*d*(b*c - a*d)^2*(a + b*x) - 6*(
18*A^2 + 66*A*B + 85*B^2)*d^2*(-(b*c) + a*d)*(a + b*x)^2 + 6*(18*A^2 + 66*A*B + 85*B^2)*d^3*(a + b*x)^3*Log[a
+ b*x] + 6*B*(b*c - a*d)*(4*(3*A + B)*(b*c - a*d)^2 + 3*(6*A + 5*B)*d*(-(b*c) + a*d)*(a + b*x) + 6*(6*A + 11*B
)*d^2*(a + b*x)^2)*Log[(e*(a + b*x))/(c + d*x)] + 18*B*(6*a^3*A*d^3 + 18*a^2*b*d^2*(A*d*x + B*(c + d*x)) + 9*a
*b^2*d*(2*A*d^2*x^2 + B*(-c^2 + 2*c*d*x + 3*d^2*x^2)) + b^3*(6*A*d^3*x^3 + B*(2*c^3 - 3*c^2*d*x + 6*c*d^2*x^2
+ 11*d^3*x^3)))*Log[(e*(a + b*x))/(c + d*x)]^2 + 36*B^2*d^3*(a + b*x)^3*Log[(e*(a + b*x))/(c + d*x)]^3 - 6*(18
*A^2 + 66*A*B + 85*B^2)*d^3*(a + b*x)^3*Log[c + d*x])/((b*c - a*d)^4*g^4*i*(a + b*x)^3)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1243\) vs. \(2(493)=986\).
time = 0.80, size = 1244, normalized size = 2.45

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(1244\)
default	\(\text {Expression too large to display}\)	\(1244\)
risch	\(\text {Expression too large to display}\)	\(1422\)
norman	\(\text {Expression too large to display}\)	\(1693\)

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)^4/(d*i*x+c*i),x,method=_RETURNVERBOSE)

[Out]

-1/d^2*e*(a*d-b*c)*(1/3*d^2*e^2/i/(a*d-b*c)^5/g^4*A^2*b^3/(b*e/d+(a*d-b*c)*e/d/(d*x+c))^3-3/2*d^3*e/i/(a*d-b*c
)^5/g^4*A^2*b^2/(b*e/d+(a*d-b*c)*e/d/(d*x+c))^2+3*d^4/i/(a*d-b*c)^5/g^4*A^2*b/(b*e/d+(a*d-b*c)*e/d/(d*x+c))+d^
5/e/i/(a*d-b*c)^5/g^4*A^2*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))-2*d^2*e^2/i/(a*d-b*c)^5/g^4*A*B*b^3*(-1/3/(b*e/d+(a*
d-b*c)*e/d/(d*x+c))^3*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))-1/9/(b*e/d+(a*d-b*c)*e/d/(d*x+c))^3)+6*d^3*e/i/(a*d-b*c)
^5/g^4*A*B*b^2*(-1/2/(b*e/d+(a*d-b*c)*e/d/(d*x+c))^2*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))-1/4/(b*e/d+(a*d-b*c)*e/d/
(d*x+c))^2)-6*d^4/i/(a*d-b*c)^5/g^4*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^5/e/i/(a*d-b*c)^5/g^4*A*B*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))^2-d^2*e^2/i/(a*d-b*
c)^5/g^4*B^2*b^3*(-1/3/(b*e/d+(a*d-b*c)*e/d/(d*x+c))^3*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))^2-2/9/(b*e/d+(a*d-b*c)*
e/d/(d*x+c))^3*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))-2/27/(b*e/d+(a*d-b*c)*e/d/(d*x+c))^3)+3*d^3*e/i/(a*d-b*c)^5/g^4
*B^2*b^2*(-1/2/(b*e/d+(a*d-b*c)*e/d/(d*x+c))^2*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))^2-1/2/(b*e/d+(a*d-b*c)*e/d/(d*x
+c))^2*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))-1/4/(b*e/d+(a*d-b*c)*e/d/(d*x+c))^2)-3*d^4/i/(a*d-b*c)^5/g^4*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^5/e/i/(a*d-b*c)^5/g^4*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. 3407 vs. \(2 (470) = 940\).
time = 0.96, size = 3407, normalized size = 6.72 \begin {gather*} \text {Too large to display} \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)^4/(d*i*x+c*i),x, algorithm="maxima")

[Out]

-1/6*B^2*((6*b^2*d^2*x^2 + 2*b^2*c^2 - 7*a*b*c*d + 11*a^2*d^2 - 3*(b^2*c*d - 5*a*b*d^2)*x)/((I*b^6*c^3 - 3*I*a
*b^5*c^2*d + 3*I*a^2*b^4*c*d^2 - I*a^3*b^3*d^3)*g^4*x^3 + 3*(I*a*b^5*c^3 - 3*I*a^2*b^4*c^2*d + 3*I*a^3*b^3*c*d
^2 - I*a^4*b^2*d^3)*g^4*x^2 + 3*(I*a^2*b^4*c^3 - 3*I*a^3*b^3*c^2*d + 3*I*a^4*b^2*c*d^2 - I*a^5*b*d^3)*g^4*x +
(I*a^3*b^3*c^3 - 3*I*a^4*b^2*c^2*d + 3*I*a^5*b*c*d^2 - I*a^6*d^3)*g^4) + 6*d^3*log(b*x + a)/((I*b^4*c^4 - 4*I*
a*b^3*c^3*d + 6*I*a^2*b^2*c^2*d^2 - 4*I*a^3*b*c*d^3 + I*a^4*d^4)*g^4) - 6*d^3*log(d*x + c)/((I*b^4*c^4 - 4*I*a
*b^3*c^3*d + 6*I*a^2*b^2*c^2*d^2 - 4*I*a^3*b*c*d^3 + I*a^4*d^4)*g^4))*log(b*x*e/(d*x + c) + a*e/(d*x + c))^2 -
1/3*A*B*((6*b^2*d^2*x^2 + 2*b^2*c^2 - 7*a*b*c*d + 11*a^2*d^2 - 3*(b^2*c*d - 5*a*b*d^2)*x)/((I*b^6*c^3 - 3*I*a
*b^5*c^2*d + 3*I*a^2*b^4*c*d^2 - I*a^3*b^3*d^3)*g^4*x^3 + 3*(I*a*b^5*c^3 - 3*I*a^2*b^4*c^2*d + 3*I*a^3*b^3*c*d
^2 - I*a^4*b^2*d^3)*g^4*x^2 + 3*(I*a^2*b^4*c^3 - 3*I*a^3*b^3*c^2*d + 3*I*a^4*b^2*c*d^2 - I*a^5*b*d^3)*g^4*x +
(I*a^3*b^3*c^3 - 3*I*a^4*b^2*c^2*d + 3*I*a^5*b*c*d^2 - I*a^6*d^3)*g^4) + 6*d^3*log(b*x + a)/((I*b^4*c^4 - 4*I*
a*b^3*c^3*d + 6*I*a^2*b^2*c^2*d^2 - 4*I*a^3*b*c*d^3 + I*a^4*d^4)*g^4) - 6*d^3*log(d*x + c)/((I*b^4*c^4 - 4*I*a
*b^3*c^3*d + 6*I*a^2*b^2*c^2*d^2 - 4*I*a^3*b*c*d^3 + I*a^4*d^4)*g^4))*log(b*x*e/(d*x + c) + a*e/(d*x + c)) - 1
/108*B^2*(6*(-4*I*b^3*c^3 + 27*I*a*b^2*c^2*d - 108*I*a^2*b*c*d^2 + 85*I*a^3*d^3 - 66*(I*b^3*c*d^2 - I*a*b^2*d^
3)*x^2 - 18*(-I*b^3*d^3*x^3 - 3*I*a*b^2*d^3*x^2 - 3*I*a^2*b*d^3*x - I*a^3*d^3)*log(b*x + a)^2 - 18*(-I*b^3*d^3
*x^3 - 3*I*a*b^2*d^3*x^2 - 3*I*a^2*b*d^3*x - I*a^3*d^3)*log(d*x + c)^2 - 3*(-5*I*b^3*c^2*d + 54*I*a*b^2*c*d^2
- 49*I*a^2*b*d^3)*x - 66*(I*b^3*d^3*x^3 + 3*I*a*b^2*d^3*x^2 + 3*I*a^2*b*d^3*x + I*a^3*d^3)*log(b*x + a) - 6*(-
11*I*b^3*d^3*x^3 - 33*I*a*b^2*d^3*x^2 - 33*I*a^2*b*d^3*x - 11*I*a^3*d^3 + 6*(I*b^3*d^3*x^3 + 3*I*a*b^2*d^3*x^2
+ 3*I*a^2*b*d^3*x + I*a^3*d^3)*log(b*x + a))*log(d*x + c))*log(b*x*e/(d*x + c) + a*e/(d*x + c))/(a^3*b^4*c^4*
g^4 - 4*a^4*b^3*c^3*d*g^4 + 6*a^5*b^2*c^2*d^2*g^4 - 4*a^6*b*c*d^3*g^4 + a^7*d^4*g^4 + (b^7*c^4*g^4 - 4*a*b^6*c
^3*d*g^4 + 6*a^2*b^5*c^2*d^2*g^4 - 4*a^3*b^4*c*d^3*g^4 + a^4*b^3*d^4*g^4)*x^3 + 3*(a*b^6*c^4*g^4 - 4*a^2*b^5*c
^3*d*g^4 + 6*a^3*b^4*c^2*d^2*g^4 - 4*a^4*b^3*c*d^3*g^4 + a^5*b^2*d^4*g^4)*x^2 + 3*(a^2*b^5*c^4*g^4 - 4*a^3*b^4
*c^3*d*g^4 + 6*a^4*b^3*c^2*d^2*g^4 - 4*a^5*b^2*c*d^3*g^4 + a^6*b*d^4*g^4)*x) + (-8*I*b^3*c^3 + 81*I*a*b^2*c^2*
d - 648*I*a^2*b*c*d^2 + 575*I*a^3*d^3 - 36*(I*b^3*d^3*x^3 + 3*I*a*b^2*d^3*x^2 + 3*I*a^2*b*d^3*x + I*a^3*d^3)*l
og(b*x + a)^3 - 36*(-I*b^3*d^3*x^3 - 3*I*a*b^2*d^3*x^2 - 3*I*a^2*b*d^3*x - I*a^3*d^3)*log(d*x + c)^3 - 510*(I*
b^3*c*d^2 - I*a*b^2*d^3)*x^2 - 198*(-I*b^3*d^3*x^3 - 3*I*a*b^2*d^3*x^2 - 3*I*a^2*b*d^3*x - I*a^3*d^3)*log(b*x
+ a)^2 - 18*(-11*I*b^3*d^3*x^3 - 33*I*a*b^2*d^3*x^2 - 33*I*a^2*b*d^3*x - 11*I*a^3*d^3 + 6*(I*b^3*d^3*x^3 + 3*I
*a*b^2*d^3*x^2 + 3*I*a^2*b*d^3*x + I*a^3*d^3)*log(b*x + a))*log(d*x + c)^2 - 3*(-19*I*b^3*c^2*d + 378*I*a*b^2*
c*d^2 - 359*I*a^2*b*d^3)*x - 510*(I*b^3*d^3*x^3 + 3*I*a*b^2*d^3*x^2 + 3*I*a^2*b*d^3*x + I*a^3*d^3)*log(b*x + a
) - 6*(-85*I*b^3*d^3*x^3 - 255*I*a*b^2*d^3*x^2 - 255*I*a^2*b*d^3*x - 85*I*a^3*d^3 + 18*(-I*b^3*d^3*x^3 - 3*I*a
*b^2*d^3*x^2 - 3*I*a^2*b*d^3*x - I*a^3*d^3)*log(b*x + a)^2 + 66*(I*b^3*d^3*x^3 + 3*I*a*b^2*d^3*x^2 + 3*I*a^2*b
*d^3*x + I*a^3*d^3)*log(b*x + a))*log(d*x + c))/(a^3*b^4*c^4*g^4 - 4*a^4*b^3*c^3*d*g^4 + 6*a^5*b^2*c^2*d^2*g^4
- 4*a^6*b*c*d^3*g^4 + a^7*d^4*g^4 + (b^7*c^4*g^4 - 4*a*b^6*c^3*d*g^4 + 6*a^2*b^5*c^2*d^2*g^4 - 4*a^3*b^4*c*d^
3*g^4 + a^4*b^3*d^4*g^4)*x^3 + 3*(a*b^6*c^4*g^4 - 4*a^2*b^5*c^3*d*g^4 + 6*a^3*b^4*c^2*d^2*g^4 - 4*a^4*b^3*c*d^
3*g^4 + a^5*b^2*d^4*g^4)*x^2 + 3*(a^2*b^5*c^4*g^4 - 4*a^3*b^4*c^3*d*g^4 + 6*a^4*b^3*c^2*d^2*g^4 - 4*a^5*b^2*c*
d^3*g^4 + a^6*b*d^4*g^4)*x)) - 1/6*A^2*((6*b^2*d^2*x^2 + 2*b^2*c^2 - 7*a*b*c*d + 11*a^2*d^2 - 3*(b^2*c*d - 5*a
*b*d^2)*x)/((I*b^6*c^3 - 3*I*a*b^5*c^2*d + 3*I*a^2*b^4*c*d^2 - I*a^3*b^3*d^3)*g^4*x^3 + 3*(I*a*b^5*c^3 - 3*I*a
^2*b^4*c^2*d + 3*I*a^3*b^3*c*d^2 - I*a^4*b^2*d^3)*g^4*x^2 + 3*(I*a^2*b^4*c^3 - 3*I*a^3*b^3*c^2*d + 3*I*a^4*b^2
*c*d^2 - I*a^5*b*d^3)*g^4*x + (I*a^3*b^3*c^3 - 3*I*a^4*b^2*c^2*d + 3*I*a^5*b*c*d^2 - I*a^6*d^3)*g^4) + 6*d^3*l
og(b*x + a)/((I*b^4*c^4 - 4*I*a*b^3*c^3*d + 6*I*a^2*b^2*c^2*d^2 - 4*I*a^3*b*c*d^3 + I*a^4*d^4)*g^4) - 6*d^3*lo
g(d*x + c)/((I*b^4*c^4 - 4*I*a*b^3*c^3*d + 6*I*a^2*b^2*c^2*d^2 - 4*I*a^3*b*c*d^3 + I*a^4*d^4)*g^4)) - 1/18*(-4
*I*b^3*c^3 + 27*I*a*b^2*c^2*d - 108*I*a^2*b*c*d^2 + 85*I*a^3*d^3 - 66*(I*b^3*c*d^2 - I*a*b^2*d^3)*x^2 - 18*(-I
*b^3*d^3*x^3 - 3*I*a*b^2*d^3*x^2 - 3*I*a^2*b*d^3*x - I*a^3*d^3)*log(b*x + a)^2 - 18*(-I*b^3*d^3*x^3 - 3*I*a*b^
2*d^3*x^2 - 3*I*a^2*b*d^3*x - I*a^3*d^3)*log(d*x + c)^2 - 3*(-5*I*b^3*c^2*d + 54*I*a*b^2*c*d^2 - 49*I*a^2*b*d^
3)*x - 66*(I*b^3*d^3*x^3 + 3*I*a*b^2*d^3*x^2 + 3*I*a^2*b*d^3*x + I*a^3*d^3)*log(b*x + a) - 6*(-11*I*b^3*d^3*x^
3 - 33*I*a*b^2*d^3*x^2 - 33*I*a^2*b*d^3*x - 11*I*a^3*d^3 + 6*(I*b^3*d^3*x^3 + 3*I*a*b^2*d^3*x^2 + 3*I*a^2*b*d^
3*x + I*a^3*d^3)*log(b*x + a))*log(d*x + c))*A*...

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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 951 vs. \(2 (470) = 940\).
time = 0.42, size = 951, normalized size = 1.88 \begin {gather*} -\frac {4 \, {\left (-9 i \, A^{2} - 6 i \, A B - 2 i \, B^{2}\right )} b^{3} c^{3} + 81 \, {\left (2 i \, A^{2} + 2 i \, A B + i \, B^{2}\right )} a b^{2} c^{2} d + 324 \, {\left (-i \, A^{2} - 2 i \, A B - 2 i \, B^{2}\right )} a^{2} b c d^{2} - {\left (-198 i \, A^{2} - 510 i \, A B - 575 i \, B^{2}\right )} a^{3} d^{3} + 36 \, {\left (-i \, B^{2} b^{3} d^{3} x^{3} - 3 i \, B^{2} a b^{2} d^{3} x^{2} - 3 i \, B^{2} a^{2} b d^{3} x - i \, B^{2} a^{3} d^{3}\right )} \log \left (\frac {{\left (b x + a\right )} e}{d x + c}\right )^{3} + 6 \, {\left ({\left (-18 i \, A^{2} - 66 i \, A B - 85 i \, B^{2}\right )} b^{3} c d^{2} + {\left (18 i \, A^{2} + 66 i \, A B + 85 i \, B^{2}\right )} a b^{2} d^{3}\right )} x^{2} + 18 \, {\left ({\left (-6 i \, A B - 11 i \, B^{2}\right )} b^{3} d^{3} x^{3} - 2 i \, B^{2} b^{3} c^{3} + 9 i \, B^{2} a b^{2} c^{2} d - 18 i \, B^{2} a^{2} b c d^{2} - 6 i \, A B a^{3} d^{3} + 3 \, {\left (-2 i \, B^{2} b^{3} c d^{2} + 3 \, {\left (-2 i \, A B - 3 i \, B^{2}\right )} a b^{2} d^{3}\right )} x^{2} + 3 \, {\left (i \, B^{2} b^{3} c^{2} d - 6 i \, B^{2} a b^{2} c d^{2} + 6 \, {\left (-i \, A B - i \, B^{2}\right )} a^{2} b d^{3}\right )} x\right )} \log \left (\frac {{\left (b x + a\right )} e}{d x + c}\right )^{2} + 3 \, {\left ({\left (18 i \, A^{2} + 30 i \, A B + 19 i \, B^{2}\right )} b^{3} c^{2} d + 54 \, {\left (-2 i \, A^{2} - 6 i \, A B - 7 i \, B^{2}\right )} a b^{2} c d^{2} + {\left (90 i \, A^{2} + 294 i \, A B + 359 i \, B^{2}\right )} a^{2} b d^{3}\right )} x + 6 \, {\left ({\left (-18 i \, A^{2} - 66 i \, A B - 85 i \, B^{2}\right )} b^{3} d^{3} x^{3} - 18 i \, A^{2} a^{3} d^{3} + 4 \, {\left (-3 i \, A B - i \, B^{2}\right )} b^{3} c^{3} + 27 \, {\left (2 i \, A B + i \, B^{2}\right )} a b^{2} c^{2} d + 108 \, {\left (-i \, A B - i \, B^{2}\right )} a^{2} b c d^{2} + 3 \, {\left (2 \, {\left (-6 i \, A B - 11 i \, B^{2}\right )} b^{3} c d^{2} + 9 \, {\left (-2 i \, A^{2} - 6 i \, A B - 7 i \, B^{2}\right )} a b^{2} d^{3}\right )} x^{2} + 3 \, {\left ({\left (6 i \, A B + 5 i \, B^{2}\right )} b^{3} c^{2} d + 18 \, {\left (-2 i \, A B - 3 i \, B^{2}\right )} a b^{2} c d^{2} + 18 \, {\left (-i \, A^{2} - 2 i \, A B - 2 i \, B^{2}\right )} a^{2} b d^{3}\right )} x\right )} \log \left (\frac {{\left (b x + a\right )} e}{d x + c}\right )}{108 \, {\left ({\left (b^{7} c^{4} - 4 \, a b^{6} c^{3} d + 6 \, a^{2} b^{5} c^{2} d^{2} - 4 \, a^{3} b^{4} c d^{3} + a^{4} b^{3} d^{4}\right )} g^{4} x^{3} + 3 \, {\left (a b^{6} c^{4} - 4 \, a^{2} b^{5} c^{3} d + 6 \, a^{3} b^{4} c^{2} d^{2} - 4 \, a^{4} b^{3} c d^{3} + a^{5} b^{2} d^{4}\right )} g^{4} x^{2} + 3 \, {\left (a^{2} b^{5} c^{4} - 4 \, a^{3} b^{4} c^{3} d + 6 \, a^{4} b^{3} c^{2} d^{2} - 4 \, a^{5} b^{2} c d^{3} + a^{6} b d^{4}\right )} g^{4} x + {\left (a^{3} b^{4} c^{4} - 4 \, a^{4} b^{3} c^{3} d + 6 \, a^{5} b^{2} c^{2} d^{2} - 4 \, a^{6} b c d^{3} + a^{7} d^{4}\right )} g^{4}\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)^4/(d*i*x+c*i),x, algorithm="fricas")

[Out]

-1/108*(4*(-9*I*A^2 - 6*I*A*B - 2*I*B^2)*b^3*c^3 + 81*(2*I*A^2 + 2*I*A*B + I*B^2)*a*b^2*c^2*d + 324*(-I*A^2 -
2*I*A*B - 2*I*B^2)*a^2*b*c*d^2 - (-198*I*A^2 - 510*I*A*B - 575*I*B^2)*a^3*d^3 + 36*(-I*B^2*b^3*d^3*x^3 - 3*I*B
^2*a*b^2*d^3*x^2 - 3*I*B^2*a^2*b*d^3*x - I*B^2*a^3*d^3)*log((b*x + a)*e/(d*x + c))^3 + 6*((-18*I*A^2 - 66*I*A*
B - 85*I*B^2)*b^3*c*d^2 + (18*I*A^2 + 66*I*A*B + 85*I*B^2)*a*b^2*d^3)*x^2 + 18*((-6*I*A*B - 11*I*B^2)*b^3*d^3*
x^3 - 2*I*B^2*b^3*c^3 + 9*I*B^2*a*b^2*c^2*d - 18*I*B^2*a^2*b*c*d^2 - 6*I*A*B*a^3*d^3 + 3*(-2*I*B^2*b^3*c*d^2 +
3*(-2*I*A*B - 3*I*B^2)*a*b^2*d^3)*x^2 + 3*(I*B^2*b^3*c^2*d - 6*I*B^2*a*b^2*c*d^2 + 6*(-I*A*B - I*B^2)*a^2*b*d
^3)*x)*log((b*x + a)*e/(d*x + c))^2 + 3*((18*I*A^2 + 30*I*A*B + 19*I*B^2)*b^3*c^2*d + 54*(-2*I*A^2 - 6*I*A*B -
7*I*B^2)*a*b^2*c*d^2 + (90*I*A^2 + 294*I*A*B + 359*I*B^2)*a^2*b*d^3)*x + 6*((-18*I*A^2 - 66*I*A*B - 85*I*B^2)
*b^3*d^3*x^3 - 18*I*A^2*a^3*d^3 + 4*(-3*I*A*B - I*B^2)*b^3*c^3 + 27*(2*I*A*B + I*B^2)*a*b^2*c^2*d + 108*(-I*A*
B - I*B^2)*a^2*b*c*d^2 + 3*(2*(-6*I*A*B - 11*I*B^2)*b^3*c*d^2 + 9*(-2*I*A^2 - 6*I*A*B - 7*I*B^2)*a*b^2*d^3)*x^
2 + 3*((6*I*A*B + 5*I*B^2)*b^3*c^2*d + 18*(-2*I*A*B - 3*I*B^2)*a*b^2*c*d^2 + 18*(-I*A^2 - 2*I*A*B - 2*I*B^2)*a
^2*b*d^3)*x)*log((b*x + a)*e/(d*x + c)))/((b^7*c^4 - 4*a*b^6*c^3*d + 6*a^2*b^5*c^2*d^2 - 4*a^3*b^4*c*d^3 + a^4
*b^3*d^4)*g^4*x^3 + 3*(a*b^6*c^4 - 4*a^2*b^5*c^3*d + 6*a^3*b^4*c^2*d^2 - 4*a^4*b^3*c*d^3 + a^5*b^2*d^4)*g^4*x^
2 + 3*(a^2*b^5*c^4 - 4*a^3*b^4*c^3*d + 6*a^4*b^3*c^2*d^2 - 4*a^5*b^2*c*d^3 + a^6*b*d^4)*g^4*x + (a^3*b^4*c^4 -
4*a^4*b^3*c^3*d + 6*a^5*b^2*c^2*d^2 - 4*a^6*b*c*d^3 + a^7*d^4)*g^4)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 2388 vs. \(2 (459) = 918\).
time = 36.67, size = 2388, normalized size = 4.71 \begin {gather*} \text {Too large to display} \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)**4/(d*i*x+c*i),x)

[Out]

-B**2*d**3*log(e*(a + b*x)/(c + d*x))**3/(3*a**4*d**4*g**4*i - 12*a**3*b*c*d**3*g**4*i + 18*a**2*b**2*c**2*d**
2*g**4*i - 12*a*b**3*c**3*d*g**4*i + 3*b**4*c**4*g**4*i) + d**3*(18*A**2 + 66*A*B + 85*B**2)*log(x + (18*A**2*
a*d**4 + 18*A**2*b*c*d**3 + 66*A*B*a*d**4 + 66*A*B*b*c*d**3 + 85*B**2*a*d**4 + 85*B**2*b*c*d**3 - a**5*d**8*(1
8*A**2 + 66*A*B + 85*B**2)/(a*d - b*c)**4 + 5*a**4*b*c*d**7*(18*A**2 + 66*A*B + 85*B**2)/(a*d - b*c)**4 - 10*a
**3*b**2*c**2*d**6*(18*A**2 + 66*A*B + 85*B**2)/(a*d - b*c)**4 + 10*a**2*b**3*c**3*d**5*(18*A**2 + 66*A*B + 85
*B**2)/(a*d - b*c)**4 - 5*a*b**4*c**4*d**4*(18*A**2 + 66*A*B + 85*B**2)/(a*d - b*c)**4 + b**5*c**5*d**3*(18*A*
*2 + 66*A*B + 85*B**2)/(a*d - b*c)**4)/(36*A**2*b*d**4 + 132*A*B*b*d**4 + 170*B**2*b*d**4))/(18*g**4*i*(a*d -
b*c)**4) - d**3*(18*A**2 + 66*A*B + 85*B**2)*log(x + (18*A**2*a*d**4 + 18*A**2*b*c*d**3 + 66*A*B*a*d**4 + 66*A
*B*b*c*d**3 + 85*B**2*a*d**4 + 85*B**2*b*c*d**3 + a**5*d**8*(18*A**2 + 66*A*B + 85*B**2)/(a*d - b*c)**4 - 5*a*
*4*b*c*d**7*(18*A**2 + 66*A*B + 85*B**2)/(a*d - b*c)**4 + 10*a**3*b**2*c**2*d**6*(18*A**2 + 66*A*B + 85*B**2)/
(a*d - b*c)**4 - 10*a**2*b**3*c**3*d**5*(18*A**2 + 66*A*B + 85*B**2)/(a*d - b*c)**4 + 5*a*b**4*c**4*d**4*(18*A
**2 + 66*A*B + 85*B**2)/(a*d - b*c)**4 - b**5*c**5*d**3*(18*A**2 + 66*A*B + 85*B**2)/(a*d - b*c)**4)/(36*A**2*
b*d**4 + 132*A*B*b*d**4 + 170*B**2*b*d**4))/(18*g**4*i*(a*d - b*c)**4) + (66*A*B*a**2*d**2 - 42*A*B*a*b*c*d +
90*A*B*a*b*d**2*x + 12*A*B*b**2*c**2 - 18*A*B*b**2*c*d*x + 36*A*B*b**2*d**2*x**2 + 85*B**2*a**2*d**2 - 23*B**2
*a*b*c*d + 147*B**2*a*b*d**2*x + 4*B**2*b**2*c**2 - 15*B**2*b**2*c*d*x + 66*B**2*b**2*d**2*x**2)*log(e*(a + b*
x)/(c + d*x))/(18*a**6*d**3*g**4*i - 54*a**5*b*c*d**2*g**4*i + 54*a**5*b*d**3*g**4*i*x + 54*a**4*b**2*c**2*d*g
**4*i - 162*a**4*b**2*c*d**2*g**4*i*x + 54*a**4*b**2*d**3*g**4*i*x**2 - 18*a**3*b**3*c**3*g**4*i + 162*a**3*b*
*3*c**2*d*g**4*i*x - 162*a**3*b**3*c*d**2*g**4*i*x**2 + 18*a**3*b**3*d**3*g**4*i*x**3 - 54*a**2*b**4*c**3*g**4
*i*x + 162*a**2*b**4*c**2*d*g**4*i*x**2 - 54*a**2*b**4*c*d**2*g**4*i*x**3 - 54*a*b**5*c**3*g**4*i*x**2 + 54*a*
b**5*c**2*d*g**4*i*x**3 - 18*b**6*c**3*g**4*i*x**3) + (-6*A*B*a**3*d**3 - 18*A*B*a**2*b*d**3*x - 18*A*B*a*b**2
*d**3*x**2 - 6*A*B*b**3*d**3*x**3 - 18*B**2*a**2*b*c*d**2 - 18*B**2*a**2*b*d**3*x + 9*B**2*a*b**2*c**2*d - 18*
B**2*a*b**2*c*d**2*x - 27*B**2*a*b**2*d**3*x**2 - 2*B**2*b**3*c**3 + 3*B**2*b**3*c**2*d*x - 6*B**2*b**3*c*d**2
*x**2 - 11*B**2*b**3*d**3*x**3)*log(e*(a + b*x)/(c + d*x))**2/(6*a**7*d**4*g**4*i - 24*a**6*b*c*d**3*g**4*i +
18*a**6*b*d**4*g**4*i*x + 36*a**5*b**2*c**2*d**2*g**4*i - 72*a**5*b**2*c*d**3*g**4*i*x + 18*a**5*b**2*d**4*g**
4*i*x**2 - 24*a**4*b**3*c**3*d*g**4*i + 108*a**4*b**3*c**2*d**2*g**4*i*x - 72*a**4*b**3*c*d**3*g**4*i*x**2 + 6
*a**4*b**3*d**4*g**4*i*x**3 + 6*a**3*b**4*c**4*g**4*i - 72*a**3*b**4*c**3*d*g**4*i*x + 108*a**3*b**4*c**2*d**2
*g**4*i*x**2 - 24*a**3*b**4*c*d**3*g**4*i*x**3 + 18*a**2*b**5*c**4*g**4*i*x - 72*a**2*b**5*c**3*d*g**4*i*x**2
+ 36*a**2*b**5*c**2*d**2*g**4*i*x**3 + 18*a*b**6*c**4*g**4*i*x**2 - 24*a*b**6*c**3*d*g**4*i*x**3 + 6*b**7*c**4
*g**4*i*x**3) + (198*A**2*a**2*d**2 - 126*A**2*a*b*c*d + 36*A**2*b**2*c**2 + 510*A*B*a**2*d**2 - 138*A*B*a*b*c
*d + 24*A*B*b**2*c**2 + 575*B**2*a**2*d**2 - 73*B**2*a*b*c*d + 8*B**2*b**2*c**2 + x**2*(108*A**2*b**2*d**2 + 3
96*A*B*b**2*d**2 + 510*B**2*b**2*d**2) + x*(270*A**2*a*b*d**2 - 54*A**2*b**2*c*d + 882*A*B*a*b*d**2 - 90*A*B*b
**2*c*d + 1077*B**2*a*b*d**2 - 57*B**2*b**2*c*d))/(108*a**6*d**3*g**4*i - 324*a**5*b*c*d**2*g**4*i + 324*a**4*
b**2*c**2*d*g**4*i - 108*a**3*b**3*c**3*g**4*i + x**3*(108*a**3*b**3*d**3*g**4*i - 324*a**2*b**4*c*d**2*g**4*i
+ 324*a*b**5*c**2*d*g**4*i - 108*b**6*c**3*g**4*i) + x**2*(324*a**4*b**2*d**3*g**4*i - 972*a**3*b**3*c*d**2*g
**4*i + 972*a**2*b**4*c**2*d*g**4*i - 324*a*b**5*c**3*g**4*i) + x*(324*a**5*b*d**3*g**4*i - 972*a**4*b**2*c*d*
*2*g**4*i + 972*a**3*b**3*c**2*d*g**4*i - 324*a**2*b**4*c**3*g**4*i))

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Giac [A]
time = 77.22, size = 427, normalized size = 0.84 \begin {gather*} -\frac {{\left (-36 i \, B^{2} b e^{4} \log \left (\frac {b x e + a e}{d x + c}\right )^{2} + \frac {54 i \, {\left (b x e + a e\right )} B^{2} d e^{3} \log \left (\frac {b x e + a e}{d x + c}\right )^{2}}{d x + c} - 72 i \, A B b e^{4} \log \left (\frac {b x e + a e}{d x + c}\right ) - 24 i \, B^{2} b e^{4} \log \left (\frac {b x e + a e}{d x + c}\right ) + \frac {108 i \, {\left (b x e + a e\right )} A B d e^{3} \log \left (\frac {b x e + a e}{d x + c}\right )}{d x + c} + \frac {54 i \, {\left (b x e + a e\right )} B^{2} d e^{3} \log \left (\frac {b x e + a e}{d x + c}\right )}{d x + c} - 36 i \, A^{2} b e^{4} - 24 i \, A B b e^{4} - 8 i \, B^{2} b e^{4} + \frac {54 i \, {\left (b x e + a e\right )} A^{2} d e^{3}}{d x + c} + \frac {54 i \, {\left (b x e + a e\right )} A B d e^{3}}{d x + c} + \frac {27 i \, {\left (b x e + a e\right )} B^{2} d e^{3}}{d x + c}\right )} {\left (\frac {b c}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}} - \frac {a d}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}}\right )}^{2}}{108 \, {\left (\frac {{\left (b x e + a e\right )}^{3} b c g^{4}}{{\left (d x + c\right )}^{3}} - \frac {{\left (b x e + a e\right )}^{3} a d g^{4}}{{\left (d x + c\right )}^{3}}\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)^4/(d*i*x+c*i),x, algorithm="giac")

[Out]

-1/108*(-36*I*B^2*b*e^4*log((b*x*e + a*e)/(d*x + c))^2 + 54*I*(b*x*e + a*e)*B^2*d*e^3*log((b*x*e + a*e)/(d*x +
c))^2/(d*x + c) - 72*I*A*B*b*e^4*log((b*x*e + a*e)/(d*x + c)) - 24*I*B^2*b*e^4*log((b*x*e + a*e)/(d*x + c)) +
108*I*(b*x*e + a*e)*A*B*d*e^3*log((b*x*e + a*e)/(d*x + c))/(d*x + c) + 54*I*(b*x*e + a*e)*B^2*d*e^3*log((b*x*
e + a*e)/(d*x + c))/(d*x + c) - 36*I*A^2*b*e^4 - 24*I*A*B*b*e^4 - 8*I*B^2*b*e^4 + 54*I*(b*x*e + a*e)*A^2*d*e^3
/(d*x + c) + 54*I*(b*x*e + a*e)*A*B*d*e^3/(d*x + c) + 27*I*(b*x*e + a*e)*B^2*d*e^3/(d*x + c))*(b*c/((b*c*e - a
*d*e)*(b*c - a*d)) - a*d/((b*c*e - a*d*e)*(b*c - a*d)))^2/((b*x*e + a*e)^3*b*c*g^4/(d*x + c)^3 - (b*x*e + a*e)
^3*a*d*g^4/(d*x + c)^3)

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Mupad [B]
time = 11.45, size = 1882, normalized size = 3.71 \begin {gather*} {\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}^2\,\left (\frac {\frac {B^2\,d^3\,\left (a\,\left (\frac {3\,a^2\,d^2-4\,a\,b\,c\,d+b^2\,c^2}{6\,b\,d^3}+\frac {a\,\left (a\,d-b\,c\right )}{3\,b\,d^2}\right )+\frac {3\,a^3\,d^3-6\,a^2\,b\,c\,d^2+4\,a\,b^2\,c^2\,d-b^3\,c^3}{3\,b\,d^4}\right )}{g^4\,i\,\left (a^4\,d^4-4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d+b^4\,c^4\right )}-\frac {B^2\,d^3\,x^2\,\left (\frac {b^2\,c-a\,b\,d}{3\,d^2}-\frac {2\,b\,\left (a\,d-b\,c\right )}{3\,d^2}\right )}{g^4\,i\,\left (a^4\,d^4-4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d+b^4\,c^4\right )}+\frac {B^2\,d^3\,x\,\left (b\,\left (\frac {3\,a^2\,d^2-4\,a\,b\,c\,d+b^2\,c^2}{6\,b\,d^3}+\frac {a\,\left (a\,d-b\,c\right )}{3\,b\,d^2}\right )+\frac {3\,a^2\,d^2-4\,a\,b\,c\,d+b^2\,c^2}{3\,d^3}+\frac {2\,a\,\left (a\,d-b\,c\right )}{3\,d^2}\right )}{g^4\,i\,\left (a^4\,d^4-4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d+b^4\,c^4\right )}}{\frac {3\,a^2\,x}{d}+\frac {a^3}{b\,d}+\frac {b^2\,x^3}{d}+\frac {3\,a\,b\,x^2}{d}}-\frac {d^3\,\left (11\,B^2+6\,A\,B\right )}{6\,g^4\,i\,\left (a^4\,d^4-4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d+b^4\,c^4\right )}\right )+\frac {\frac {198\,A^2\,a^2\,d^2-126\,A^2\,a\,b\,c\,d+36\,A^2\,b^2\,c^2+510\,A\,B\,a^2\,d^2-138\,A\,B\,a\,b\,c\,d+24\,A\,B\,b^2\,c^2+575\,B^2\,a^2\,d^2-73\,B^2\,a\,b\,c\,d+8\,B^2\,b^2\,c^2}{6\,\left (a\,d-b\,c\right )}+\frac {x\,\left (-18\,c\,A^2\,b^2\,d+90\,a\,A^2\,b\,d^2-30\,c\,A\,B\,b^2\,d+294\,a\,A\,B\,b\,d^2-19\,c\,B^2\,b^2\,d+359\,a\,B^2\,b\,d^2\right )}{2\,\left (a\,d-b\,c\right )}+\frac {d\,x^2\,\left (18\,d\,A^2\,b^2+66\,d\,A\,B\,b^2+85\,d\,B^2\,b^2\right )}{a\,d-b\,c}}{x\,\left (54\,i\,a^4\,b\,d^2\,g^4-108\,i\,a^3\,b^2\,c\,d\,g^4+54\,i\,a^2\,b^3\,c^2\,g^4\right )+x^2\,\left (54\,i\,a^3\,b^2\,d^2\,g^4-108\,i\,a^2\,b^3\,c\,d\,g^4+54\,i\,a\,b^4\,c^2\,g^4\right )+x^3\,\left (18\,i\,a^2\,b^3\,d^2\,g^4-36\,i\,a\,b^4\,c\,d\,g^4+18\,i\,b^5\,c^2\,g^4\right )+18\,a^5\,d^2\,g^4\,i+18\,a^3\,b^2\,c^2\,g^4\,i-36\,a^4\,b\,c\,d\,g^4\,i}-\frac {\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )\,\left (x\,\left (\frac {B^2}{g^4\,i\,{\left (a\,d-b\,c\right )}^2}-\frac {d^3\,\left (11\,B^2+6\,A\,B\right )\,\left (b\,\left (\frac {3\,a^2\,d^2-4\,a\,b\,c\,d+b^2\,c^2}{6\,b\,d^3}+\frac {a\,\left (a\,d-b\,c\right )}{3\,b\,d^2}\right )+\frac {3\,a^2\,d^2-4\,a\,b\,c\,d+b^2\,c^2}{3\,d^3}+\frac {2\,a\,\left (a\,d-b\,c\right )}{3\,d^2}\right )}{3\,g^4\,i\,\left (a^4\,d^4-4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d+b^4\,c^4\right )}\right )+\frac {5\,B^2\,a\,d-3\,B^2\,b\,c}{3\,b\,d\,g^4\,i\,{\left (a\,d-b\,c\right )}^2}+\frac {B^2\,a}{3\,b\,g^4\,i\,{\left (a\,d-b\,c\right )}^2}-\frac {d^3\,\left (11\,B^2+6\,A\,B\right )\,\left (a\,\left (\frac {3\,a^2\,d^2-4\,a\,b\,c\,d+b^2\,c^2}{6\,b\,d^3}+\frac {a\,\left (a\,d-b\,c\right )}{3\,b\,d^2}\right )+\frac {3\,a^3\,d^3-6\,a^2\,b\,c\,d^2+4\,a\,b^2\,c^2\,d-b^3\,c^3}{3\,b\,d^4}\right )}{3\,g^4\,i\,\left (a^4\,d^4-4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d+b^4\,c^4\right )}+\frac {d^3\,x^2\,\left (\frac {b^2\,c-a\,b\,d}{3\,d^2}-\frac {2\,b\,\left (a\,d-b\,c\right )}{3\,d^2}\right )\,\left (11\,B^2+6\,A\,B\right )}{3\,g^4\,i\,\left (a^4\,d^4-4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d+b^4\,c^4\right )}\right )}{\frac {3\,a^2\,x}{d}+\frac {a^3}{b\,d}+\frac {b^2\,x^3}{d}+\frac {3\,a\,b\,x^2}{d}}-\frac {B^2\,d^3\,{\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}^3}{3\,g^4\,i\,\left (a^4\,d^4-4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d+b^4\,c^4\right )}+\frac {d^3\,\mathrm {atan}\left (\frac {d^3\,\left (A^2+\frac {11\,A\,B}{3}+\frac {85\,B^2}{18}\right )\,\left (18\,i\,a^4\,d^4\,g^4-36\,i\,a^3\,b\,c\,d^3\,g^4+36\,i\,a\,b^3\,c^3\,d\,g^4-18\,i\,b^4\,c^4\,g^4\right )\,1{}\mathrm {i}}{g^4\,i\,{\left (a\,d-b\,c\right )}^4\,\left (18\,A^2\,d^3+66\,A\,B\,d^3+85\,B^2\,d^3\right )}+\frac {b\,d^4\,x\,\left (A^2+\frac {11\,A\,B}{3}+\frac {85\,B^2}{18}\right )\,\left (i\,a^3\,d^3\,g^4-3\,i\,a^2\,b\,c\,d^2\,g^4+3\,i\,a\,b^2\,c^2\,d\,g^4-i\,b^3\,c^3\,g^4\right )\,36{}\mathrm {i}}{g^4\,i\,{\left (a\,d-b\,c\right )}^4\,\left (18\,A^2\,d^3+66\,A\,B\,d^3+85\,B^2\,d^3\right )}\right )\,\left (A^2+\frac {11\,A\,B}{3}+\frac {85\,B^2}{18}\right )\,2{}\mathrm {i}}{g^4\,i\,{\left (a\,d-b\,c\right )}^4} \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)^4*(c*i + d*i*x)),x)

[Out]

log((e*(a + b*x))/(c + d*x))^2*(((B^2*d^3*(a*((3*a^2*d^2 + b^2*c^2 - 4*a*b*c*d)/(6*b*d^3) + (a*(a*d - b*c))/(3
*b*d^2)) + (3*a^3*d^3 - b^3*c^3 + 4*a*b^2*c^2*d - 6*a^2*b*c*d^2)/(3*b*d^4)))/(g^4*i*(a^4*d^4 + b^4*c^4 + 6*a^2
*b^2*c^2*d^2 - 4*a*b^3*c^3*d - 4*a^3*b*c*d^3)) - (B^2*d^3*x^2*((b^2*c - a*b*d)/(3*d^2) - (2*b*(a*d - b*c))/(3*
d^2)))/(g^4*i*(a^4*d^4 + b^4*c^4 + 6*a^2*b^2*c^2*d^2 - 4*a*b^3*c^3*d - 4*a^3*b*c*d^3)) + (B^2*d^3*x*(b*((3*a^2
*d^2 + b^2*c^2 - 4*a*b*c*d)/(6*b*d^3) + (a*(a*d - b*c))/(3*b*d^2)) + (3*a^2*d^2 + b^2*c^2 - 4*a*b*c*d)/(3*d^3)
+ (2*a*(a*d - b*c))/(3*d^2)))/(g^4*i*(a^4*d^4 + b^4*c^4 + 6*a^2*b^2*c^2*d^2 - 4*a*b^3*c^3*d - 4*a^3*b*c*d^3))
)/((3*a^2*x)/d + a^3/(b*d) + (b^2*x^3)/d + (3*a*b*x^2)/d) - (d^3*(11*B^2 + 6*A*B))/(6*g^4*i*(a^4*d^4 + b^4*c^4
+ 6*a^2*b^2*c^2*d^2 - 4*a*b^3*c^3*d - 4*a^3*b*c*d^3))) + ((198*A^2*a^2*d^2 + 36*A^2*b^2*c^2 + 575*B^2*a^2*d^2
+ 8*B^2*b^2*c^2 + 510*A*B*a^2*d^2 + 24*A*B*b^2*c^2 - 126*A^2*a*b*c*d - 73*B^2*a*b*c*d - 138*A*B*a*b*c*d)/(6*(
a*d - b*c)) + (x*(90*A^2*a*b*d^2 + 359*B^2*a*b*d^2 - 18*A^2*b^2*c*d - 19*B^2*b^2*c*d + 294*A*B*a*b*d^2 - 30*A*
B*b^2*c*d))/(2*(a*d - b*c)) + (d*x^2*(18*A^2*b^2*d + 85*B^2*b^2*d + 66*A*B*b^2*d))/(a*d - b*c))/(x*(54*a^4*b*d
^2*g^4*i + 54*a^2*b^3*c^2*g^4*i - 108*a^3*b^2*c*d*g^4*i) + x^2*(54*a*b^4*c^2*g^4*i + 54*a^3*b^2*d^2*g^4*i - 10
8*a^2*b^3*c*d*g^4*i) + x^3*(18*b^5*c^2*g^4*i + 18*a^2*b^3*d^2*g^4*i - 36*a*b^4*c*d*g^4*i) + 18*a^5*d^2*g^4*i +
18*a^3*b^2*c^2*g^4*i - 36*a^4*b*c*d*g^4*i) - (log((e*(a + b*x))/(c + d*x))*(x*(B^2/(g^4*i*(a*d - b*c)^2) - (d
^3*(11*B^2 + 6*A*B)*(b*((3*a^2*d^2 + b^2*c^2 - 4*a*b*c*d)/(6*b*d^3) + (a*(a*d - b*c))/(3*b*d^2)) + (3*a^2*d^2
+ b^2*c^2 - 4*a*b*c*d)/(3*d^3) + (2*a*(a*d - b*c))/(3*d^2)))/(3*g^4*i*(a^4*d^4 + b^4*c^4 + 6*a^2*b^2*c^2*d^2 -
4*a*b^3*c^3*d - 4*a^3*b*c*d^3))) + (5*B^2*a*d - 3*B^2*b*c)/(3*b*d*g^4*i*(a*d - b*c)^2) + (B^2*a)/(3*b*g^4*i*(
a*d - b*c)^2) - (d^3*(11*B^2 + 6*A*B)*(a*((3*a^2*d^2 + b^2*c^2 - 4*a*b*c*d)/(6*b*d^3) + (a*(a*d - b*c))/(3*b*d
^2)) + (3*a^3*d^3 - b^3*c^3 + 4*a*b^2*c^2*d - 6*a^2*b*c*d^2)/(3*b*d^4)))/(3*g^4*i*(a^4*d^4 + b^4*c^4 + 6*a^2*b
^2*c^2*d^2 - 4*a*b^3*c^3*d - 4*a^3*b*c*d^3)) + (d^3*x^2*((b^2*c - a*b*d)/(3*d^2) - (2*b*(a*d - b*c))/(3*d^2))*
(11*B^2 + 6*A*B))/(3*g^4*i*(a^4*d^4 + b^4*c^4 + 6*a^2*b^2*c^2*d^2 - 4*a*b^3*c^3*d - 4*a^3*b*c*d^3))))/((3*a^2*
x)/d + a^3/(b*d) + (b^2*x^3)/d + (3*a*b*x^2)/d) - (B^2*d^3*log((e*(a + b*x))/(c + d*x))^3)/(3*g^4*i*(a^4*d^4 +
b^4*c^4 + 6*a^2*b^2*c^2*d^2 - 4*a*b^3*c^3*d - 4*a^3*b*c*d^3)) + (d^3*atan((d^3*(A^2 + (85*B^2)/18 + (11*A*B)/
3)*(18*a^4*d^4*g^4*i - 18*b^4*c^4*g^4*i + 36*a*b^3*c^3*d*g^4*i - 36*a^3*b*c*d^3*g^4*i)*1i)/(g^4*i*(a*d - b*c)^
4*(18*A^2*d^3 + 85*B^2*d^3 + 66*A*B*d^3)) + (b*d^4*x*(A^2 + (85*B^2)/18 + (11*A*B)/3)*(a^3*d^3*g^4*i - b^3*c^3
*g^4*i + 3*a*b^2*c^2*d*g^4*i - 3*a^2*b*c*d^2*g^4*i)*36i)/(g^4*i*(a*d - b*c)^4*(18*A^2*d^3 + 85*B^2*d^3 + 66*A*
B*d^3)))*(A^2 + (85*B^2)/18 + (11*A*B)/3)*2i)/(g^4*i*(a*d - b*c)^4)

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