3.1.84 \(\int \frac {(a g+b g x)^3 (A+B \log (\frac {e (a+b x)}{c+d x}))^2}{c i+d i x} \, dx\) [84]

Optimal. Leaf size=718 \[ \frac {b B^2 (b c-a d)^2 g^3 x}{3 d^3 i}+\frac {B^2 (b c-a d)^3 g^3 \log \left (\frac {a+b x}{c+d x}\right )}{3 d^4 i}+\frac {7 B (b c-a d)^2 g^3 (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 d^3 i}-\frac {b^2 B (b c-a d) g^3 (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 d^4 i}+\frac {6 B (b c-a d)^3 g^3 \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{d^4 i}+\frac {3 (b c-a d)^2 g^3 (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{d^3 i}-\frac {3 b^2 (b c-a d) g^3 (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2 d^4 i}+\frac {b^3 g^3 (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{3 d^4 i}+\frac {(b c-a d)^3 g^3 \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{d^4 i}-\frac {2 B^2 (b c-a d)^3 g^3 \log (c+d x)}{d^4 i}-\frac {7 B (b c-a d)^3 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log \left (1-\frac {b (c+d x)}{d (a+b x)}\right )}{3 d^4 i}+\frac {6 B^2 (b c-a d)^3 g^3 \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right )}{d^4 i}+\frac {2 B (b c-a d)^3 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right )}{d^4 i}+\frac {7 B^2 (b c-a d)^3 g^3 \text {Li}_2\left (\frac {b (c+d x)}{d (a+b x)}\right )}{3 d^4 i}-\frac {2 B^2 (b c-a d)^3 g^3 \text {Li}_3\left (\frac {d (a+b x)}{b (c+d x)}\right )}{d^4 i} \]

[Out]

1/3*b*B^2*(-a*d+b*c)^2*g^3*x/d^3/i+1/3*B^2*(-a*d+b*c)^3*g^3*ln((b*x+a)/(d*x+c))/d^4/i+7/3*B*(-a*d+b*c)^2*g^3*(
b*x+a)*(A+B*ln(e*(b*x+a)/(d*x+c)))/d^3/i-1/3*b^2*B*(-a*d+b*c)*g^3*(d*x+c)^2*(A+B*ln(e*(b*x+a)/(d*x+c)))/d^4/i+
6*B*(-a*d+b*c)^3*g^3*ln((-a*d+b*c)/b/(d*x+c))*(A+B*ln(e*(b*x+a)/(d*x+c)))/d^4/i+3*(-a*d+b*c)^2*g^3*(b*x+a)*(A+
B*ln(e*(b*x+a)/(d*x+c)))^2/d^3/i-3/2*b^2*(-a*d+b*c)*g^3*(d*x+c)^2*(A+B*ln(e*(b*x+a)/(d*x+c)))^2/d^4/i+1/3*b^3*
g^3*(d*x+c)^3*(A+B*ln(e*(b*x+a)/(d*x+c)))^2/d^4/i+(-a*d+b*c)^3*g^3*ln((-a*d+b*c)/b/(d*x+c))*(A+B*ln(e*(b*x+a)/
(d*x+c)))^2/d^4/i-2*B^2*(-a*d+b*c)^3*g^3*ln(d*x+c)/d^4/i-7/3*B*(-a*d+b*c)^3*g^3*(A+B*ln(e*(b*x+a)/(d*x+c)))*ln
(1-b*(d*x+c)/d/(b*x+a))/d^4/i+6*B^2*(-a*d+b*c)^3*g^3*polylog(2,d*(b*x+a)/b/(d*x+c))/d^4/i+2*B*(-a*d+b*c)^3*g^3
*(A+B*ln(e*(b*x+a)/(d*x+c)))*polylog(2,d*(b*x+a)/b/(d*x+c))/d^4/i+7/3*B^2*(-a*d+b*c)^3*g^3*polylog(2,b*(d*x+c)
/d/(b*x+a))/d^4/i-2*B^2*(-a*d+b*c)^3*g^3*polylog(3,d*(b*x+a)/b/(d*x+c))/d^4/i

________________________________________________________________________________________

Rubi [A]
time = 0.64, antiderivative size = 718, normalized size of antiderivative = 1.00, number of steps used = 25, number of rules used = 13, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.310, Rules used = {2562, 2395, 2356, 2389, 2379, 2438, 2351, 31, 46, 2355, 2354, 2421, 6724} \begin {gather*} \frac {2 B g^3 (b c-a d)^3 \text {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right ) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{d^4 i}+\frac {6 B^2 g^3 (b c-a d)^3 \text {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{d^4 i}+\frac {7 B^2 g^3 (b c-a d)^3 \text {PolyLog}\left (2,\frac {b (c+d x)}{d (a+b x)}\right )}{3 d^4 i}-\frac {2 B^2 g^3 (b c-a d)^3 \text {PolyLog}\left (3,\frac {d (a+b x)}{b (c+d x)}\right )}{d^4 i}+\frac {b^3 g^3 (c+d x)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{3 d^4 i}-\frac {3 b^2 g^3 (c+d x)^2 (b c-a d) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{2 d^4 i}-\frac {b^2 B g^3 (c+d x)^2 (b c-a d) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{3 d^4 i}+\frac {g^3 (b c-a d)^3 \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{d^4 i}+\frac {6 B g^3 (b c-a d)^3 \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{d^4 i}-\frac {7 B g^3 (b c-a d)^3 \log \left (1-\frac {b (c+d x)}{d (a+b x)}\right ) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{3 d^4 i}+\frac {3 g^3 (a+b x) (b c-a d)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{d^3 i}+\frac {7 B g^3 (a+b x) (b c-a d)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{3 d^3 i}+\frac {B^2 g^3 (b c-a d)^3 \log \left (\frac {a+b x}{c+d x}\right )}{3 d^4 i}-\frac {2 B^2 g^3 (b c-a d)^3 \log (c+d x)}{d^4 i}+\frac {b B^2 g^3 x (b c-a d)^2}{3 d^3 i} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(b*B^2*(b*c - a*d)^2*g^3*x)/(3*d^3*i) + (B^2*(b*c - a*d)^3*g^3*Log[(a + b*x)/(c + d*x)])/(3*d^4*i) + (7*B*(b*c
- a*d)^2*g^3*(a + b*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(3*d^3*i) - (b^2*B*(b*c - a*d)*g^3*(c + d*x)^2*(
A + B*Log[(e*(a + b*x))/(c + d*x)]))/(3*d^4*i) + (6*B*(b*c - a*d)^3*g^3*Log[(b*c - a*d)/(b*(c + d*x))]*(A + B*
Log[(e*(a + b*x))/(c + d*x)]))/(d^4*i) + (3*(b*c - a*d)^2*g^3*(a + b*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2
)/(d^3*i) - (3*b^2*(b*c - a*d)*g^3*(c + d*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(2*d^4*i) + (b^3*g^3*(c
+ d*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(3*d^4*i) + ((b*c - a*d)^3*g^3*Log[(b*c - a*d)/(b*(c + d*x))
]*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(d^4*i) - (2*B^2*(b*c - a*d)^3*g^3*Log[c + d*x])/(d^4*i) - (7*B*(b*c
- a*d)^3*g^3*(A + B*Log[(e*(a + b*x))/(c + d*x)])*Log[1 - (b*(c + d*x))/(d*(a + b*x))])/(3*d^4*i) + (6*B^2*(b
*c - a*d)^3*g^3*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(d^4*i) + (2*B*(b*c - a*d)^3*g^3*(A + B*Log[(e*(a + b
*x))/(c + d*x)])*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(d^4*i) + (7*B^2*(b*c - a*d)^3*g^3*PolyLog[2, (b*(c
+ d*x))/(d*(a + b*x))])/(3*d^4*i) - (2*B^2*(b*c - a*d)^3*g^3*PolyLog[3, (d*(a + b*x))/(b*(c + d*x))])/(d^4*i)

Rule 31

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

Rule 46

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

Rule 2351

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

Rule 2354

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

Rule 2355

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

Rule 2356

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

Rule 2379

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

Rule 2389

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

Rule 2395

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

Rule 2421

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

Rule 2438

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

Rule 2562

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

Rule 6724

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

Rubi steps

\begin {align*} \int \frac {(a g+b g x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{84 c+84 d x} \, dx &=\int \left (\frac {b (b c-a d)^2 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{84 d^3}+\frac {(-b c+a d)^3 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{d^3 (84 c+84 d x)}-\frac {b (b c-a d) g^2 (a g+b g x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{84 d^2}+\frac {b g (a g+b g x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{84 d}\right ) \, dx\\ &=\frac {(b g) \int (a g+b g x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \, dx}{84 d}-\frac {\left (b (b c-a d) g^2\right ) \int (a g+b g x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \, dx}{84 d^2}+\frac {\left (b (b c-a d)^2 g^3\right ) \int \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \, dx}{84 d^3}-\frac {\left ((b c-a d)^3 g^3\right ) \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{84 c+84 d x} \, dx}{d^3}\\ &=\frac {b (b c-a d)^2 g^3 x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{84 d^3}-\frac {(b c-a d) g^3 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{168 d^2}+\frac {g^3 (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{252 d}-\frac {(b c-a d)^3 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (84 c+84 d x)}{84 d^4}-\frac {B \int \frac {(b c-a d) g^3 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{c+d x} \, dx}{126 d}+\frac {(B (b c-a d) g) \int \frac {(b c-a d) g^2 (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{c+d x} \, dx}{84 d^2}-\frac {\left (b B (b c-a d)^2 g^3\right ) \int \frac {(b c-a d) x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a+b x) (c+d x)} \, dx}{42 d^3}+\frac {\left (B (b c-a d)^3 g^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 (84 c+84 d x)}{e (a+b x)} \, dx}{42 d^4}\\ &=\frac {b (b c-a d)^2 g^3 x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{84 d^3}-\frac {(b c-a d) g^3 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{168 d^2}+\frac {g^3 (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{252 d}-\frac {(b c-a d)^3 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (84 c+84 d x)}{84 d^4}-\frac {\left (B (b c-a d) g^3\right ) \int \frac {(a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{c+d x} \, dx}{126 d}+\frac {\left (B (b c-a d)^2 g^3\right ) \int \frac {(a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{c+d x} \, dx}{84 d^2}-\frac {\left (b B (b c-a d)^3 g^3\right ) \int \frac {x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a+b x) (c+d x)} \, dx}{42 d^3}+\frac {\left (B (b c-a d)^3 g^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 (84 c+84 d x)}{a+b x} \, dx}{42 d^4 e}\\ &=\frac {b (b c-a d)^2 g^3 x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{84 d^3}-\frac {(b c-a d) g^3 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{168 d^2}+\frac {g^3 (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{252 d}-\frac {(b c-a d)^3 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (84 c+84 d x)}{84 d^4}-\frac {\left (B (b c-a d) g^3\right ) \int \left (-\frac {b (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{d^2}+\frac {b (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{d}+\frac {(-b c+a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{d^2 (c+d x)}\right ) \, dx}{126 d}+\frac {\left (B (b c-a d)^2 g^3\right ) \int \left (\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{d}+\frac {(-b c+a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{d (c+d x)}\right ) \, dx}{84 d^2}-\frac {\left (b B (b c-a d)^3 g^3\right ) \int \left (-\frac {a \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d) (a+b x)}+\frac {c \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d) (c+d x)}\right ) \, dx}{42 d^3}+\frac {\left (B (b c-a d)^3 g^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 (84 c+84 d x)}{(a+b x) (c+d x)} \, dx}{42 d^4 e}\\ &=\frac {b (b c-a d)^2 g^3 x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{84 d^3}-\frac {(b c-a d) g^3 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{168 d^2}+\frac {g^3 (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{252 d}-\frac {(b c-a d)^3 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (84 c+84 d x)}{84 d^4}-\frac {\left (b B (b c-a d) g^3\right ) \int (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx}{126 d^2}+\frac {\left (b B (b c-a d)^2 g^3\right ) \int \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx}{126 d^3}+\frac {\left (b B (b c-a d)^2 g^3\right ) \int \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx}{84 d^3}+\frac {\left (a b B (b c-a d)^2 g^3\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{42 d^3}-\frac {\left (b B c (b c-a d)^2 g^3\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{c+d x} \, dx}{42 d^3}-\frac {\left (B (b c-a d)^3 g^3\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{c+d x} \, dx}{126 d^3}-\frac {\left (B (b c-a d)^3 g^3\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{c+d x} \, dx}{84 d^3}+\frac {\left (B (b c-a d)^4 g^3\right ) \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (84 c+84 d x)}{(a+b x) (c+d x)} \, dx}{42 d^4}\\ &=\frac {5 A b B (b c-a d)^2 g^3 x}{252 d^3}-\frac {B (b c-a d) g^3 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{252 d^2}+\frac {a B (b c-a d)^2 g^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{42 d^3}+\frac {b (b c-a d)^2 g^3 x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{84 d^3}-\frac {(b c-a d) g^3 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{168 d^2}+\frac {g^3 (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{252 d}-\frac {b B c (b c-a d)^2 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{42 d^4}-\frac {5 B (b c-a d)^3 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{252 d^4}-\frac {(b c-a d)^3 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (84 c+84 d x)}{84 d^4}+\frac {\left (B^2 (b c-a d) g^3\right ) \int \frac {(b c-a d) (a+b x)}{c+d x} \, dx}{252 d^2}+\frac {\left (b B^2 c (b c-a d)^2 g^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}{42 d^4}-\frac {\left (a B^2 (b c-a d)^2 g^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}{42 d^3}+\frac {\left (b B^2 (b c-a d)^2 g^3\right ) \int \log \left (\frac {e (a+b x)}{c+d x}\right ) \, dx}{126 d^3}+\frac {\left (b B^2 (b c-a d)^2 g^3\right ) \int \log \left (\frac {e (a+b x)}{c+d x}\right ) \, dx}{84 d^3}+\frac {\left (B^2 (b c-a d)^3 g^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}{126 d^4}+\frac {\left (B^2 (b c-a d)^3 g^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}{84 d^4}+\frac {\left (B (b c-a d)^4 g^3\right ) \int \left (\frac {d \left (-A-B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (84 c+84 d x)}{(b c-a d) (c+d x)}+\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (84 c+84 d x)}{(b c-a d) (a+b x)}\right ) \, dx}{42 d^4}\\ &=\frac {5 A b B (b c-a d)^2 g^3 x}{252 d^3}+\frac {5 B^2 (b c-a d)^2 g^3 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{252 d^3}-\frac {B (b c-a d) g^3 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{252 d^2}+\frac {a B (b c-a d)^2 g^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{42 d^3}+\frac {b (b c-a d)^2 g^3 x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{84 d^3}-\frac {(b c-a d) g^3 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{168 d^2}+\frac {g^3 (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{252 d}-\frac {b B c (b c-a d)^2 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{42 d^4}-\frac {5 B (b c-a d)^3 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{252 d^4}-\frac {(b c-a d)^3 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (84 c+84 d x)}{84 d^4}+\frac {\left (B^2 (b c-a d)^2 g^3\right ) \int \frac {a+b x}{c+d x} \, dx}{252 d^2}+\frac {\left (b B (b c-a d)^3 g^3\right ) \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (84 c+84 d x)}{a+b x} \, dx}{42 d^4}+\frac {\left (B (b c-a d)^3 g^3\right ) \int \frac {\left (-A-B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (84 c+84 d x)}{c+d x} \, dx}{42 d^3}-\frac {\left (B^2 (b c-a d)^3 g^3\right ) \int \frac {1}{c+d x} \, dx}{126 d^3}-\frac {\left (B^2 (b c-a d)^3 g^3\right ) \int \frac {1}{c+d x} \, dx}{84 d^3}+\frac {\left (b B^2 c (b c-a d)^2 g^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}{42 d^4 e}-\frac {\left (a B^2 (b c-a d)^2 g^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}{42 d^3 e}+\frac {\left (B^2 (b c-a d)^3 g^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}{126 d^4 e}+\frac {\left (B^2 (b c-a d)^3 g^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}{84 d^4 e}\\ &=\frac {5 A b B (b c-a d)^2 g^3 x}{252 d^3}+\frac {5 B^2 (b c-a d)^2 g^3 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{252 d^3}-\frac {B (b c-a d) g^3 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{252 d^2}+\frac {a B (b c-a d)^2 g^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{42 d^3}+\frac {b (b c-a d)^2 g^3 x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{84 d^3}-\frac {(b c-a d) g^3 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{168 d^2}+\frac {g^3 (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{252 d}-\frac {5 B^2 (b c-a d)^3 g^3 \log (c+d x)}{252 d^4}-\frac {b B c (b c-a d)^2 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{42 d^4}-\frac {5 B (b c-a d)^3 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{252 d^4}-\frac {(b c-a d)^3 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (84 c+84 d x)}{84 d^4}+\frac {\left (B^2 (b c-a d)^2 g^3\right ) \int \left (\frac {b}{d}+\frac {-b c+a d}{d (c+d x)}\right ) \, dx}{252 d^2}+\frac {\left (b B (b c-a d)^3 g^3\right ) \int \left (\frac {A \log (84 c+84 d x)}{a+b x}+\frac {B \log \left (\frac {e (a+b x)}{c+d x}\right ) \log (84 c+84 d x)}{a+b x}\right ) \, dx}{42 d^4}+\frac {\left (B (b c-a d)^3 g^3\right ) \int \left (\frac {A \log (84 c+84 d x)}{-c-d x}+\frac {B \log \left (\frac {e (a+b x)}{c+d x}\right ) \log (84 c+84 d x)}{-c-d x}\right ) \, dx}{42 d^3}+\frac {\left (b B^2 c (b c-a d)^2 g^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}{42 d^4 e}-\frac {\left (a B^2 (b c-a d)^2 g^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}{42 d^3 e}+\frac {\left (B^2 (b c-a d)^3 g^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}{126 d^4 e}+\frac {\left (B^2 (b c-a d)^3 g^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}{84 d^4 e}\\ &=\frac {5 A b B (b c-a d)^2 g^3 x}{252 d^3}+\frac {b B^2 (b c-a d)^2 g^3 x}{252 d^3}+\frac {5 B^2 (b c-a d)^2 g^3 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{252 d^3}-\frac {B (b c-a d) g^3 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{252 d^2}+\frac {a B (b c-a d)^2 g^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{42 d^3}+\frac {b (b c-a d)^2 g^3 x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{84 d^3}-\frac {(b c-a d) g^3 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{168 d^2}+\frac {g^3 (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{252 d}-\frac {B^2 (b c-a d)^3 g^3 \log (c+d x)}{42 d^4}-\frac {b B c (b c-a d)^2 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{42 d^4}-\frac {5 B (b c-a d)^3 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{252 d^4}-\frac {(b c-a d)^3 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (84 c+84 d x)}{84 d^4}+\frac {\left (b^2 B^2 c (b c-a d)^2 g^3\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{42 d^4}-\frac {\left (a b B^2 (b c-a d)^2 g^3\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{42 d^3}-\frac {\left (b B^2 c (b c-a d)^2 g^3\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{42 d^3}+\frac {\left (a B^2 (b c-a d)^2 g^3\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{42 d^2}+\frac {\left (A b B (b c-a d)^3 g^3\right ) \int \frac {\log (84 c+84 d x)}{a+b x} \, dx}{42 d^4}+\frac {\left (b B^2 (b c-a d)^3 g^3\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{126 d^4}+\frac {\left (b B^2 (b c-a d)^3 g^3\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{84 d^4}+\frac {\left (b B^2 (b c-a d)^3 g^3\right ) \int \frac {\log \left (\frac {e (a+b x)}{c+d x}\right ) \log (84 c+84 d x)}{a+b x} \, dx}{42 d^4}+\frac {\left (A B (b c-a d)^3 g^3\right ) \int \frac {\log (84 c+84 d x)}{-c-d x} \, dx}{42 d^3}-\frac {\left (B^2 (b c-a d)^3 g^3\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{126 d^3}-\frac {\left (B^2 (b c-a d)^3 g^3\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{84 d^3}+\frac {\left (B^2 (b c-a d)^3 g^3\right ) \int \frac {\log \left (\frac {e (a+b x)}{c+d x}\right ) \log (84 c+84 d x)}{-c-d x} \, dx}{42 d^3}\\ &=\frac {5 A b B (b c-a d)^2 g^3 x}{252 d^3}+\frac {b B^2 (b c-a d)^2 g^3 x}{252 d^3}+\frac {5 B^2 (b c-a d)^2 g^3 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{252 d^3}-\frac {B (b c-a d) g^3 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{252 d^2}+\frac {a B (b c-a d)^2 g^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{42 d^3}+\frac {b (b c-a d)^2 g^3 x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{84 d^3}-\frac {(b c-a d) g^3 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{168 d^2}+\frac {g^3 (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{252 d}-\frac {B^2 (b c-a d)^3 g^3 \log (c+d x)}{42 d^4}+\frac {b B^2 c (b c-a d)^2 g^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{42 d^4}+\frac {5 B^2 (b c-a d)^3 g^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{252 d^4}-\frac {b B c (b c-a d)^2 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{42 d^4}-\frac {5 B (b c-a d)^3 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{252 d^4}+\frac {a B^2 (b c-a d)^2 g^3 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{42 d^3}+\frac {A B (b c-a d)^3 g^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (84 c+84 d x)}{42 d^4}-\frac {(b c-a d)^3 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (84 c+84 d x)}{84 d^4}-\frac {B^2 (b c-a d)^3 g^3 \log \left (\frac {e (a+b x)}{c+d x}\right ) \log ^2(84 c+84 d x)}{84 d^4}-\frac {\left (b B^2 c (b c-a d)^2 g^3\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{42 d^4}-\frac {\left (a B^2 (b c-a d)^2 g^3\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{42 d^3}-\frac {\left (a b B^2 (b c-a d)^2 g^3\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{42 d^3}-\frac {\left (b B^2 c (b c-a d)^2 g^3\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{42 d^3}+\frac {\left (A B (b c-a d)^3 g^3\right ) \text {Subst}\left (\int -\frac {84 \log (x)}{x} \, dx,x,84 c+84 d x\right )}{3528 d^4}-\frac {\left (B^2 (b c-a d)^3 g^3\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{126 d^4}-\frac {\left (B^2 (b c-a d)^3 g^3\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{84 d^4}+\frac {\left (b B^2 (b c-a d)^3 g^3\right ) \int \frac {\log ^2(84 c+84 d x)}{a+b x} \, dx}{84 d^4}+\frac {\left (b B^2 (b c-a d)^3 g^3\right ) \int \frac {\log (a+b x) \log (84 c+84 d x)}{a+b x} \, dx}{42 d^4}+\frac {\left (b B^2 (b c-a d)^3 g^3\right ) \int \frac {\log \left (\frac {1}{c+d x}\right ) \log (84 c+84 d x)}{a+b x} \, dx}{42 d^4}-\frac {\left (2 A B (b c-a d)^3 g^3\right ) \int \frac {\log \left (\frac {84 d (a+b x)}{-84 b c+84 a d}\right )}{84 c+84 d x} \, dx}{d^3}-\frac {\left (B^2 (b c-a d)^3 g^3\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{126 d^3}-\frac {\left (B^2 (b c-a d)^3 g^3\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{84 d^3}-\frac {\left (B^2 (b c-a d)^3 g^3\right ) \int \frac {\log ^2(84 c+84 d x)}{c+d x} \, dx}{84 d^3}+\frac {\left (b B^2 (b c-a d)^3 g^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 (84 c+84 d x)}{a+b x} \, dx}{42 d^4}\\ &=\frac {5 A b B (b c-a d)^2 g^3 x}{252 d^3}+\frac {b B^2 (b c-a d)^2 g^3 x}{252 d^3}-\frac {a B^2 (b c-a d)^2 g^3 \log ^2(a+b x)}{84 d^3}+\frac {5 B^2 (b c-a d)^2 g^3 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{252 d^3}-\frac {B (b c-a d) g^3 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{252 d^2}+\frac {a B (b c-a d)^2 g^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{42 d^3}+\frac {b (b c-a d)^2 g^3 x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{84 d^3}-\frac {(b c-a d) g^3 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{168 d^2}+\frac {g^3 (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{252 d}-\frac {B^2 (b c-a d)^3 g^3 \log (c+d x)}{42 d^4}+\frac {b B^2 c (b c-a d)^2 g^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{42 d^4}+\frac {5 B^2 (b c-a d)^3 g^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{252 d^4}-\frac {b B c (b c-a d)^2 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{42 d^4}-\frac {5 B (b c-a d)^3 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{252 d^4}-\frac {b B^2 c (b c-a d)^2 g^3 \log ^2(c+d x)}{84 d^4}-\frac {5 B^2 (b c-a d)^3 g^3 \log ^2(c+d x)}{504 d^4}+\frac {a B^2 (b c-a d)^2 g^3 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{42 d^3}+\frac {A B (b c-a d)^3 g^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (84 c+84 d x)}{42 d^4}-\frac {B^2 (b c-a d)^3 g^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 (84 c+84 d x)}{42 d^4}-\frac {(b c-a d)^3 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (84 c+84 d x)}{84 d^4}+\frac {B^2 (b c-a d)^3 g^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(84 c+84 d x)}{84 d^4}-\frac {B^2 (b c-a d)^3 g^3 \log \left (\frac {e (a+b x)}{c+d x}\right ) \log ^2(84 c+84 d x)}{84 d^4}-\frac {\left (b B^2 c (b c-a d)^2 g^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 )}{42 d^4}-\frac {\left (a B^2 (b c-a d)^2 g^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 )}{42 d^3}-\frac {\left (A B (b c-a d)^3 g^3\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,84 c+84 d x\right )}{42 d^4}-\frac {\left (A B (b c-a d)^3 g^3\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-84 b c+84 a d}\right )}{x} \, dx,x,84 c+84 d x\right )}{42 d^4}-\frac {\left (B^2 (b c-a d)^3 g^3\right ) \text {Subst}\left (\int \frac {84 \log ^2(x)}{x} \, dx,x,84 c+84 d x\right )}{7056 d^4}-\frac {\left (B^2 (b c-a d)^3 g^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 )}{126 d^4}-\frac {\left (B^2 (b c-a d)^3 g^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 )}{84 d^4}+\frac {\left (B^2 (b c-a d)^3 g^3\right ) \text {Subst}\left (\int \frac {\log (x) \log \left (\frac {84 b c-84 a d}{b}+\frac {84 d x}{b}\right )}{x} \, dx,x,a+b x\right )}{42 d^4}+\frac {\left (B^2 (b c-a d)^3 g^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 {-84 b c+84 a d}{b}+\frac {84 d x}{b}\right )}{x} \, dx,x,a+b x\right )}{42 d^4}-\frac {\left (2 B^2 (b c-a d)^3 g^3\right ) \int \frac {\log \left (\frac {84 d (a+b x)}{-84 b c+84 a d}\right ) \log (84 c+84 d x)}{84 c+84 d x} \, dx}{d^3}-\frac {\left (2 B^2 (b c-a d)^3 g^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 \left (\frac {84 d (a+b x)}{-84 b c+84 a d}\right )}{84 c+84 d x} \, dx}{d^3}\\ &=\frac {5 A b B (b c-a d)^2 g^3 x}{252 d^3}+\frac {b B^2 (b c-a d)^2 g^3 x}{252 d^3}-\frac {a B^2 (b c-a d)^2 g^3 \log ^2(a+b x)}{84 d^3}+\frac {5 B^2 (b c-a d)^2 g^3 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{252 d^3}-\frac {B (b c-a d) g^3 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{252 d^2}+\frac {a B (b c-a d)^2 g^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{42 d^3}+\frac {b (b c-a d)^2 g^3 x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{84 d^3}-\frac {(b c-a d) g^3 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{168 d^2}+\frac {g^3 (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{252 d}-\frac {B^2 (b c-a d)^3 g^3 \log (c+d x)}{42 d^4}+\frac {b B^2 c (b c-a d)^2 g^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{42 d^4}+\frac {5 B^2 (b c-a d)^3 g^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{252 d^4}-\frac {b B c (b c-a d)^2 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{42 d^4}-\frac {5 B (b c-a d)^3 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{252 d^4}-\frac {b B^2 c (b c-a d)^2 g^3 \log ^2(c+d x)}{84 d^4}-\frac {5 B^2 (b c-a d)^3 g^3 \log ^2(c+d x)}{504 d^4}+\frac {B^2 (b c-a d)^3 g^3 \log ^2(a+b x) \log (84 (c+d x))}{84 d^4}+\frac {B^2 (b c-a d)^3 g^3 \log (a+b x) \log \left (\frac {1}{c+d x}\right ) \log (84 (c+d x))}{42 d^4}-\frac {A B (b c-a d)^3 g^3 \log ^2(84 (c+d x))}{84 d^4}+\frac {a B^2 (b c-a d)^2 g^3 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{42 d^3}+\frac {A B (b c-a d)^3 g^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (84 c+84 d x)}{42 d^4}-\frac {B^2 (b c-a d)^3 g^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 (84 c+84 d x)}{42 d^4}-\frac {(b c-a d)^3 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (84 c+84 d x)}{84 d^4}+\frac {B^2 (b c-a d)^3 g^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(84 c+84 d x)}{84 d^4}-\frac {B^2 (b c-a d)^3 g^3 \log \left (\frac {e (a+b x)}{c+d x}\right ) \log ^2(84 c+84 d x)}{84 d^4}+\frac {a B^2 (b c-a d)^2 g^3 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{42 d^3}+\frac {b B^2 c (b c-a d)^2 g^3 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{42 d^4}+\frac {A B (b c-a d)^3 g^3 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{42 d^4}+\frac {5 B^2 (b c-a d)^3 g^3 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{252 d^4}-\frac {\left (B^2 (b c-a d)^3 g^3\right ) \text {Subst}\left (\int \frac {\log ^2(x)}{x} \, dx,x,84 c+84 d x\right )}{84 d^4}-\frac {\left (B^2 (b c-a d)^3 g^3\right ) \text {Subst}\left (\int \frac {\log (x) \log \left (\frac {84 d \left (\frac {-84 b c+84 a d}{84 d}+\frac {b x}{84 d}\right )}{-84 b c+84 a d}\right )}{x} \, dx,x,84 c+84 d x\right )}{42 d^4}-\frac {\left (B^2 (b c-a d)^3 g^3\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 )}{42 b d^3}-\frac {\left (B^2 (b c-a d)^3 g^3\right ) \text {Subst}\left (\int \frac {\log ^2(x)}{\frac {84 b c-84 a d}{b}+\frac {84 d x}{b}} \, dx,x,a+b x\right )}{b d^3}+\frac {\left (2 B^2 (b c-a d)^3 g^3\right ) \text {Subst}\left (\int \frac {\log (x) \log \left (-\frac {-84 b c+84 a d}{b}+\frac {84 d x}{b}\right )}{-\frac {-84 b c+84 a d}{b}+\frac {84 d x}{b}} \, dx,x,a+b x\right )}{b d^3}-\frac {\left (B^2 (b c-a d)^3 g^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}{-84 b c+84 a d}\right )}{x} \, dx,x,84 c+84 d x\right )}{42 d^4}\\ &=\frac {5 A b B (b c-a d)^2 g^3 x}{252 d^3}+\frac {b B^2 (b c-a d)^2 g^3 x}{252 d^3}-\frac {a B^2 (b c-a d)^2 g^3 \log ^2(a+b x)}{84 d^3}+\frac {5 B^2 (b c-a d)^2 g^3 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{252 d^3}-\frac {B (b c-a d) g^3 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{252 d^2}+\frac {a B (b c-a d)^2 g^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{42 d^3}+\frac {b (b c-a d)^2 g^3 x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{84 d^3}-\frac {(b c-a d) g^3 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{168 d^2}+\frac {g^3 (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{252 d}-\frac {B^2 (b c-a d)^3 g^3 \log (c+d x)}{42 d^4}+\frac {b B^2 c (b c-a d)^2 g^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{42 d^4}+\frac {5 B^2 (b c-a d)^3 g^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{252 d^4}-\frac {b B c (b c-a d)^2 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{42 d^4}-\frac {5 B (b c-a d)^3 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{252 d^4}-\frac {b B^2 c (b c-a d)^2 g^3 \log ^2(c+d x)}{84 d^4}-\frac {5 B^2 (b c-a d)^3 g^3 \log ^2(c+d x)}{504 d^4}+\frac {B^2 (b c-a d)^3 g^3 \log ^2(a+b x) \log (84 (c+d x))}{84 d^4}+\frac {B^2 (b c-a d)^3 g^3 \log (a+b x) \log \left (\frac {1}{c+d x}\right ) \log (84 (c+d x))}{42 d^4}-\frac {A B (b c-a d)^3 g^3 \log ^2(84 (c+d x))}{84 d^4}+\frac {a B^2 (b c-a d)^2 g^3 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{42 d^3}-\frac {B^2 (b c-a d)^3 g^3 \log ^2(a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{84 d^4}+\frac {A B (b c-a d)^3 g^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (84 c+84 d x)}{42 d^4}-\frac {B^2 (b c-a d)^3 g^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 (84 c+84 d x)}{42 d^4}-\frac {(b c-a d)^3 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (84 c+84 d x)}{84 d^4}+\frac {B^2 (b c-a d)^3 g^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(84 c+84 d x)}{84 d^4}-\frac {B^2 (b c-a d)^3 g^3 \log \left (\frac {e (a+b x)}{c+d x}\right ) \log ^2(84 c+84 d x)}{84 d^4}+\frac {a B^2 (b c-a d)^2 g^3 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{42 d^3}+\frac {b B^2 c (b c-a d)^2 g^3 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{42 d^4}+\frac {A B (b c-a d)^3 g^3 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{42 d^4}+\frac {5 B^2 (b c-a d)^3 g^3 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{252 d^4}-\frac {B^2 (b c-a d)^3 g^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 )}{42 d^4}+\frac {B^2 (b c-a d)^3 g^3 \log (84 (c+d x)) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{42 d^4}-\frac {\left (B^2 (b c-a d)^3 g^3\right ) \text {Subst}\left (\int x^2 \, dx,x,\log (84 (c+d x))\right )}{84 d^4}+\frac {\left (B^2 (b c-a d)^3 g^3\right ) \text {Subst}\left (\int \frac {\log (x) \log \left (\frac {-84 b c+84 a d}{84 d}+\frac {b x}{84 d}\right )}{x} \, dx,x,84 (c+d x)\right )}{42 d^4}-\frac {\left (B^2 (b c-a d)^3 g^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 )}{42 d^4}+\frac {\left (B^2 (b c-a d)^3 g^3\right ) \text {Subst}\left (\int \frac {\log (x) \log \left (1+\frac {84 d x}{84 b c-84 a d}\right )}{x} \, dx,x,a+b x\right )}{42 d^4}-\frac {\left (B^2 (b c-a d)^3 g^3\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {b x}{-84 b c+84 a d}\right )}{x} \, dx,x,84 c+84 d x\right )}{42 d^4}\\ &=\frac {5 A b B (b c-a d)^2 g^3 x}{252 d^3}+\frac {b B^2 (b c-a d)^2 g^3 x}{252 d^3}-\frac {a B^2 (b c-a d)^2 g^3 \log ^2(a+b x)}{84 d^3}+\frac {B^2 (b c-a d)^3 g^3 \log (a+b x) \log ^2\left (\frac {1}{c+d x}\right )}{84 d^4}+\frac {5 B^2 (b c-a d)^2 g^3 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{252 d^3}-\frac {B (b c-a d) g^3 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{252 d^2}+\frac {a B (b c-a d)^2 g^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{42 d^3}+\frac {b (b c-a d)^2 g^3 x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{84 d^3}-\frac {(b c-a d) g^3 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{168 d^2}+\frac {g^3 (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{252 d}-\frac {B^2 (b c-a d)^3 g^3 \log (c+d x)}{42 d^4}+\frac {b B^2 c (b c-a d)^2 g^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{42 d^4}+\frac {5 B^2 (b c-a d)^3 g^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{252 d^4}-\frac {b B c (b c-a d)^2 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{42 d^4}-\frac {5 B (b c-a d)^3 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{252 d^4}-\frac {b B^2 c (b c-a d)^2 g^3 \log ^2(c+d x)}{84 d^4}-\frac {5 B^2 (b c-a d)^3 g^3 \log ^2(c+d x)}{504 d^4}+\frac {B^2 (b c-a d)^3 g^3 \log ^2(a+b x) \log (84 (c+d x))}{84 d^4}+\frac {B^2 (b c-a d)^3 g^3 \log (a+b x) \log \left (\frac {1}{c+d x}\right ) \log (84 (c+d x))}{42 d^4}-\frac {A B (b c-a d)^3 g^3 \log ^2(84 (c+d x))}{84 d^4}+\frac {B^2 (b c-a d)^3 g^3 \log (a+b x) \log ^2(84 (c+d x))}{84 d^4}-\frac {B^2 (b c-a d)^3 g^3 \log ^3(84 (c+d x))}{252 d^4}+\frac {a B^2 (b c-a d)^2 g^3 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{42 d^3}-\frac {B^2 (b c-a d)^3 g^3 \log ^2(a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{84 d^4}+\frac {A B (b c-a d)^3 g^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (84 c+84 d x)}{42 d^4}-\frac {B^2 (b c-a d)^3 g^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 (84 c+84 d x)}{42 d^4}-\frac {(b c-a d)^3 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (84 c+84 d x)}{84 d^4}+\frac {B^2 (b c-a d)^3 g^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(84 c+84 d x)}{84 d^4}-\frac {B^2 (b c-a d)^3 g^3 \log \left (\frac {e (a+b x)}{c+d x}\right ) \log ^2(84 c+84 d x)}{84 d^4}+\frac {a B^2 (b c-a d)^2 g^3 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{42 d^3}-\frac {B^2 (b c-a d)^3 g^3 \log (a+b x) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{42 d^4}+\frac {b B^2 c (b c-a d)^2 g^3 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{42 d^4}+\frac {A B (b c-a d)^3 g^3 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{42 d^4}+\frac {5 B^2 (b c-a d)^3 g^3 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{252 d^4}-\frac {B^2 (b c-a d)^3 g^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 )}{42 d^4}+\frac {B^2 (b c-a d)^3 g^3 \log (84 (c+d x)) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{42 d^4}-\frac {B^2 (b c-a d)^3 g^3 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{42 d^4}-\frac {\left (b B^2 (b c-a d)^3 g^3\right ) \text {Subst}\left (\int \frac {\log ^2(x)}{\frac {-84 b c+84 a d}{84 d}+\frac {b x}{84 d}} \, dx,x,84 (c+d x)\right )}{7056 d^5}-\frac {\left (b B^2 (b c-a d)^3 g^3\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 )}{84 d^5}+\frac {\left (B^2 (b c-a d)^3 g^3\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {84 d x}{84 b c-84 a d}\right )}{x} \, dx,x,a+b x\right )}{42 d^4}\\ &=\frac {5 A b B (b c-a d)^2 g^3 x}{252 d^3}+\frac {b B^2 (b c-a d)^2 g^3 x}{252 d^3}-\frac {a B^2 (b c-a d)^2 g^3 \log ^2(a+b x)}{84 d^3}+\frac {B^2 (b c-a d)^3 g^3 \log (a+b x) \log ^2\left (\frac {1}{c+d x}\right )}{84 d^4}-\frac {B^2 (b c-a d)^3 g^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2\left (\frac {1}{c+d x}\right )}{84 d^4}+\frac {5 B^2 (b c-a d)^2 g^3 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{252 d^3}-\frac {B (b c-a d) g^3 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{252 d^2}+\frac {a B (b c-a d)^2 g^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{42 d^3}+\frac {b (b c-a d)^2 g^3 x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{84 d^3}-\frac {(b c-a d) g^3 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{168 d^2}+\frac {g^3 (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{252 d}-\frac {B^2 (b c-a d)^3 g^3 \log (c+d x)}{42 d^4}+\frac {b B^2 c (b c-a d)^2 g^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{42 d^4}+\frac {5 B^2 (b c-a d)^3 g^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{252 d^4}-\frac {b B c (b c-a d)^2 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{42 d^4}-\frac {5 B (b c-a d)^3 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{252 d^4}-\frac {b B^2 c (b c-a d)^2 g^3 \log ^2(c+d x)}{84 d^4}-\frac {5 B^2 (b c-a d)^3 g^3 \log ^2(c+d x)}{504 d^4}+\frac {B^2 (b c-a d)^3 g^3 \log ^2(a+b x) \log (84 (c+d x))}{84 d^4}+\frac {B^2 (b c-a d)^3 g^3 \log (a+b x) \log \left (\frac {1}{c+d x}\right ) \log (84 (c+d x))}{42 d^4}-\frac {A B (b c-a d)^3 g^3 \log ^2(84 (c+d x))}{84 d^4}+\frac {B^2 (b c-a d)^3 g^3 \log (a+b x) \log ^2(84 (c+d x))}{84 d^4}-\frac {B^2 (b c-a d)^3 g^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(84 (c+d x))}{84 d^4}-\frac {B^2 (b c-a d)^3 g^3 \log ^3(84 (c+d x))}{252 d^4}+\frac {a B^2 (b c-a d)^2 g^3 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{42 d^3}-\frac {B^2 (b c-a d)^3 g^3 \log ^2(a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{84 d^4}+\frac {A B (b c-a d)^3 g^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (84 c+84 d x)}{42 d^4}-\frac {B^2 (b c-a d)^3 g^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 (84 c+84 d x)}{42 d^4}-\frac {(b c-a d)^3 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (84 c+84 d x)}{84 d^4}+\frac {B^2 (b c-a d)^3 g^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(84 c+84 d x)}{84 d^4}-\frac {B^2 (b c-a d)^3 g^3 \log \left (\frac {e (a+b x)}{c+d x}\right ) \log ^2(84 c+84 d x)}{84 d^4}+\frac {a B^2 (b c-a d)^2 g^3 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{42 d^3}-\frac {B^2 (b c-a d)^3 g^3 \log (a+b x) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{42 d^4}+\frac {b B^2 c (b c-a d)^2 g^3 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{42 d^4}+\frac {A B (b c-a d)^3 g^3 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{42 d^4}+\frac {5 B^2 (b c-a d)^3 g^3 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{252 d^4}-\frac {B^2 (b c-a d)^3 g^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 )}{42 d^4}+\frac {B^2 (b c-a d)^3 g^3 \log (84 (c+d x)) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{42 d^4}+\frac {B^2 (b c-a d)^3 g^3 \text {Li}_3\left (-\frac {d (a+b x)}{b c-a d}\right )}{42 d^4}-\frac {B^2 (b c-a d)^3 g^3 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{42 d^4}-\frac {\left (B^2 (b c-a d)^3 g^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 )}{42 d^4}+\frac {\left (B^2 (b c-a d)^3 g^3\right ) \text {Subst}\left (\int \frac {\log (x) \log \left (1+\frac {b x}{-84 b c+84 a d}\right )}{x} \, dx,x,84 (c+d x)\right )}{42 d^4}\\ &=\frac {5 A b B (b c-a d)^2 g^3 x}{252 d^3}+\frac {b B^2 (b c-a d)^2 g^3 x}{252 d^3}-\frac {a B^2 (b c-a d)^2 g^3 \log ^2(a+b x)}{84 d^3}+\frac {B^2 (b c-a d)^3 g^3 \log (a+b x) \log ^2\left (\frac {1}{c+d x}\right )}{84 d^4}-\frac {B^2 (b c-a d)^3 g^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2\left (\frac {1}{c+d x}\right )}{84 d^4}+\frac {5 B^2 (b c-a d)^2 g^3 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{252 d^3}-\frac {B (b c-a d) g^3 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{252 d^2}+\frac {a B (b c-a d)^2 g^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{42 d^3}+\frac {b (b c-a d)^2 g^3 x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{84 d^3}-\frac {(b c-a d) g^3 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{168 d^2}+\frac {g^3 (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{252 d}-\frac {B^2 (b c-a d)^3 g^3 \log (c+d x)}{42 d^4}+\frac {b B^2 c (b c-a d)^2 g^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{42 d^4}+\frac {5 B^2 (b c-a d)^3 g^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{252 d^4}-\frac {b B c (b c-a d)^2 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{42 d^4}-\frac {5 B (b c-a d)^3 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{252 d^4}-\frac {b B^2 c (b c-a d)^2 g^3 \log ^2(c+d x)}{84 d^4}-\frac {5 B^2 (b c-a d)^3 g^3 \log ^2(c+d x)}{504 d^4}+\frac {B^2 (b c-a d)^3 g^3 \log ^2(a+b x) \log (84 (c+d x))}{84 d^4}+\frac {B^2 (b c-a d)^3 g^3 \log (a+b x) \log \left (\frac {1}{c+d x}\right ) \log (84 (c+d x))}{42 d^4}-\frac {A B (b c-a d)^3 g^3 \log ^2(84 (c+d x))}{84 d^4}+\frac {B^2 (b c-a d)^3 g^3 \log (a+b x) \log ^2(84 (c+d x))}{84 d^4}-\frac {B^2 (b c-a d)^3 g^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(84 (c+d x))}{84 d^4}-\frac {B^2 (b c-a d)^3 g^3 \log ^3(84 (c+d x))}{252 d^4}+\frac {a B^2 (b c-a d)^2 g^3 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{42 d^3}-\frac {B^2 (b c-a d)^3 g^3 \log ^2(a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{84 d^4}+\frac {A B (b c-a d)^3 g^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (84 c+84 d x)}{42 d^4}-\frac {B^2 (b c-a d)^3 g^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 (84 c+84 d x)}{42 d^4}-\frac {(b c-a d)^3 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (84 c+84 d x)}{84 d^4}+\frac {B^2 (b c-a d)^3 g^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(84 c+84 d x)}{84 d^4}-\frac {B^2 (b c-a d)^3 g^3 \log \left (\frac {e (a+b x)}{c+d x}\right ) \log ^2(84 c+84 d x)}{84 d^4}+\frac {a B^2 (b c-a d)^2 g^3 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{42 d^3}-\frac {B^2 (b c-a d)^3 g^3 \log (a+b x) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{42 d^4}+\frac {b B^2 c (b c-a d)^2 g^3 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{42 d^4}+\frac {A B (b c-a d)^3 g^3 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{42 d^4}+\frac {5 B^2 (b c-a d)^3 g^3 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{252 d^4}+\frac {B^2 (b c-a d)^3 g^3 \log \left (\frac {1}{c+d x}\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{42 d^4}-\frac {B^2 (b c-a d)^3 g^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 )}{42 d^4}+\frac {B^2 (b c-a d)^3 g^3 \text {Li}_3\left (-\frac {d (a+b x)}{b c-a d}\right )}{42 d^4}-\frac {B^2 (b c-a d)^3 g^3 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{42 d^4}+\frac {\left (B^2 (b c-a d)^3 g^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 )}{42 d^4}+\frac {\left (B^2 (b c-a d)^3 g^3\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {b x}{-84 b c+84 a d}\right )}{x} \, dx,x,84 (c+d x)\right )}{42 d^4}\\ &=\frac {5 A b B (b c-a d)^2 g^3 x}{252 d^3}+\frac {b B^2 (b c-a d)^2 g^3 x}{252 d^3}-\frac {a B^2 (b c-a d)^2 g^3 \log ^2(a+b x)}{84 d^3}+\frac {B^2 (b c-a d)^3 g^3 \log (a+b x) \log ^2\left (\frac {1}{c+d x}\right )}{84 d^4}-\frac {B^2 (b c-a d)^3 g^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2\left (\frac {1}{c+d x}\right )}{84 d^4}+\frac {5 B^2 (b c-a d)^2 g^3 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{252 d^3}-\frac {B (b c-a d) g^3 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{252 d^2}+\frac {a B (b c-a d)^2 g^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{42 d^3}+\frac {b (b c-a d)^2 g^3 x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{84 d^3}-\frac {(b c-a d) g^3 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{168 d^2}+\frac {g^3 (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{252 d}-\frac {B^2 (b c-a d)^3 g^3 \log (c+d x)}{42 d^4}+\frac {b B^2 c (b c-a d)^2 g^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{42 d^4}+\frac {5 B^2 (b c-a d)^3 g^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{252 d^4}-\frac {b B c (b c-a d)^2 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{42 d^4}-\frac {5 B (b c-a d)^3 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{252 d^4}-\frac {b B^2 c (b c-a d)^2 g^3 \log ^2(c+d x)}{84 d^4}-\frac {5 B^2 (b c-a d)^3 g^3 \log ^2(c+d x)}{504 d^4}+\frac {B^2 (b c-a d)^3 g^3 \log ^2(a+b x) \log (84 (c+d x))}{84 d^4}+\frac {B^2 (b c-a d)^3 g^3 \log (a+b x) \log \left (\frac {1}{c+d x}\right ) \log (84 (c+d x))}{42 d^4}-\frac {A B (b c-a d)^3 g^3 \log ^2(84 (c+d x))}{84 d^4}+\frac {B^2 (b c-a d)^3 g^3 \log (a+b x) \log ^2(84 (c+d x))}{84 d^4}-\frac {B^2 (b c-a d)^3 g^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(84 (c+d x))}{84 d^4}-\frac {B^2 (b c-a d)^3 g^3 \log ^3(84 (c+d x))}{252 d^4}+\frac {a B^2 (b c-a d)^2 g^3 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{42 d^3}-\frac {B^2 (b c-a d)^3 g^3 \log ^2(a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{84 d^4}+\frac {A B (b c-a d)^3 g^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (84 c+84 d x)}{42 d^4}-\frac {B^2 (b c-a d)^3 g^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 (84 c+84 d x)}{42 d^4}-\frac {(b c-a d)^3 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (84 c+84 d x)}{84 d^4}+\frac {B^2 (b c-a d)^3 g^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(84 c+84 d x)}{84 d^4}-\frac {B^2 (b c-a d)^3 g^3 \log \left (\frac {e (a+b x)}{c+d x}\right ) \log ^2(84 c+84 d x)}{84 d^4}+\frac {a B^2 (b c-a d)^2 g^3 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{42 d^3}-\frac {B^2 (b c-a d)^3 g^3 \log (a+b x) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{42 d^4}+\frac {b B^2 c (b c-a d)^2 g^3 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{42 d^4}+\frac {A B (b c-a d)^3 g^3 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{42 d^4}+\frac {5 B^2 (b c-a d)^3 g^3 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{252 d^4}+\frac {B^2 (b c-a d)^3 g^3 \log \left (\frac {1}{c+d x}\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{42 d^4}-\frac {B^2 (b c-a d)^3 g^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 )}{42 d^4}+\frac {B^2 (b c-a d)^3 g^3 \text {Li}_3\left (-\frac {d (a+b x)}{b c-a d}\right )}{42 d^4}+\frac {B^2 (b c-a d)^3 g^3 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{42 d^4}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(4802\) vs. \(2(718)=1436\).
time = 1.19, size = 4802, normalized size = 6.69 \begin {gather*} \text {Result too large to show} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(g^3*(12*A*b^3*B*c^3 - 48*a*A*b^2*B*c^2*d - 22*a*b^2*B^2*c^2*d + 72*a^2*A*b*B*c*d^2 + 54*a^2*b*B^2*c*d^2 - 36*
a^3*A*B*d^3 - 36*a^3*B^2*d^3 + 6*A^2*b^3*c^2*d*x + 10*A*b^3*B*c^2*d*x + 2*b^3*B^2*c^2*d*x - 18*a*A^2*b^2*c*d^2
*x - 24*a*A*b^2*B*c*d^2*x - 4*a*b^2*B^2*c*d^2*x + 18*a^2*A^2*b*d^3*x + 14*a^2*A*b*B*d^3*x + 2*a^2*b*B^2*d^3*x
- 3*A^2*b^3*c*d^2*x^2 - 2*A*b^3*B*c*d^2*x^2 + 9*a*A^2*b^2*d^3*x^2 + 2*a*A*b^2*B*d^3*x^2 + 2*A^2*b^3*d^3*x^3 -
12*b^3*B^2*c^3*Log[a/b + x] + 12*a*A*b^2*B*c^2*d*Log[a/b + x] + 58*a*b^2*B^2*c^2*d*Log[a/b + x] - 36*a^2*A*b*B
*c*d^2*Log[a/b + x] - 99*a^2*b*B^2*c*d^2*Log[a/b + x] + 36*a^3*A*B*d^3*Log[a/b + x] + 63*a^3*B^2*d^3*Log[a/b +
x] - 6*a*b^2*B^2*c^2*d*Log[a/b + x]^2 + 21*a^2*b*B^2*c*d^2*Log[a/b + x]^2 - 25*a^3*B^2*d^3*Log[a/b + x]^2 - 1
2*A*b^3*B*c^3*Log[c/d + x] - 9*b^3*B^2*c^3*Log[c/d + x] + 36*a*A*b^2*B*c^2*d*Log[c/d + x] + 15*a*b^2*B^2*c^2*d
*Log[c/d + x] - 36*a^2*A*b*B*c*d^2*Log[c/d + x] + 36*a^2*b*B^2*c*d^2*Log[c/d + x] - 36*a^3*B^2*d^3*Log[c/d + x
] + 12*b^3*B^2*c^3*Log[a/b + x]*Log[c/d + x] - 36*a*b^2*B^2*c^2*d*Log[a/b + x]*Log[c/d + x] + 30*a^2*b*B^2*c*d
^2*Log[a/b + x]*Log[c/d + x] + 14*a^3*B^2*d^3*Log[a/b + x]*Log[c/d + x] + 6*A*b^3*B*c^3*Log[c/d + x]^2 - b^3*B
^2*c^3*Log[c/d + x]^2 - 18*a*A*b^2*B*c^2*d*Log[c/d + x]^2 + 9*a*b^2*B^2*c^2*d*Log[c/d + x]^2 + 18*a^2*A*b*B*c*
d^2*Log[c/d + x]^2 - 18*a^2*b*B^2*c*d^2*Log[c/d + x]^2 - 6*a^3*A*B*d^3*Log[c/d + x]^2 + 4*b^3*B^2*c^3*Log[c/d
+ x]^3 - 12*a*b^2*B^2*c^2*d*Log[c/d + x]^3 + 12*a^2*b*B^2*c*d^2*Log[c/d + x]^3 + 6*a^2*A*b*B*c*d^2*Log[a + b*x
] + 5*a^2*b*B^2*c*d^2*Log[a + b*x] - 14*a^3*A*B*d^3*Log[a + b*x] - 15*a^3*B^2*d^3*Log[a + b*x] - 6*a^2*b*B^2*c
*d^2*Log[a/b + x]*Log[a + b*x] + 14*a^3*B^2*d^3*Log[a/b + x]*Log[a + b*x] + 6*a^2*b*B^2*c*d^2*Log[c/d + x]*Log
[a + b*x] - 14*a^3*B^2*d^3*Log[c/d + x]*Log[a + b*x] - 6*b^3*B^2*c^3*Log[c/d + x]^2*Log[(d*(a + b*x))/(-(b*c)
+ a*d)] + 18*a*b^2*B^2*c^2*d*Log[c/d + x]^2*Log[(d*(a + b*x))/(-(b*c) + a*d)] - 18*a^2*b*B^2*c*d^2*Log[c/d + x
]^2*Log[(d*(a + b*x))/(-(b*c) + a*d)] + 12*b^3*B^2*c^3*Log[(e*(a + b*x))/(c + d*x)] - 48*a*b^2*B^2*c^2*d*Log[(
e*(a + b*x))/(c + d*x)] + 72*a^2*b*B^2*c*d^2*Log[(e*(a + b*x))/(c + d*x)] - 36*a^3*B^2*d^3*Log[(e*(a + b*x))/(
c + d*x)] + 12*A*b^3*B*c^2*d*x*Log[(e*(a + b*x))/(c + d*x)] + 10*b^3*B^2*c^2*d*x*Log[(e*(a + b*x))/(c + d*x)]
- 36*a*A*b^2*B*c*d^2*x*Log[(e*(a + b*x))/(c + d*x)] - 24*a*b^2*B^2*c*d^2*x*Log[(e*(a + b*x))/(c + d*x)] + 36*a
^2*A*b*B*d^3*x*Log[(e*(a + b*x))/(c + d*x)] + 14*a^2*b*B^2*d^3*x*Log[(e*(a + b*x))/(c + d*x)] - 6*A*b^3*B*c*d^
2*x^2*Log[(e*(a + b*x))/(c + d*x)] - 2*b^3*B^2*c*d^2*x^2*Log[(e*(a + b*x))/(c + d*x)] + 18*a*A*b^2*B*d^3*x^2*L
og[(e*(a + b*x))/(c + d*x)] + 2*a*b^2*B^2*d^3*x^2*Log[(e*(a + b*x))/(c + d*x)] + 4*A*b^3*B*d^3*x^3*Log[(e*(a +
b*x))/(c + d*x)] + 12*a*b^2*B^2*c^2*d*Log[a/b + x]*Log[(e*(a + b*x))/(c + d*x)] - 36*a^2*b*B^2*c*d^2*Log[a/b
+ x]*Log[(e*(a + b*x))/(c + d*x)] + 36*a^3*B^2*d^3*Log[a/b + x]*Log[(e*(a + b*x))/(c + d*x)] - 12*b^3*B^2*c^3*
Log[c/d + x]*Log[(e*(a + b*x))/(c + d*x)] + 36*a*b^2*B^2*c^2*d*Log[c/d + x]*Log[(e*(a + b*x))/(c + d*x)] - 36*
a^2*b*B^2*c*d^2*Log[c/d + x]*Log[(e*(a + b*x))/(c + d*x)] + 6*b^3*B^2*c^3*Log[c/d + x]^2*Log[(e*(a + b*x))/(c
+ d*x)] - 18*a*b^2*B^2*c^2*d*Log[c/d + x]^2*Log[(e*(a + b*x))/(c + d*x)] + 18*a^2*b*B^2*c*d^2*Log[c/d + x]^2*L
og[(e*(a + b*x))/(c + d*x)] + 6*a^2*b*B^2*c*d^2*Log[a + b*x]*Log[(e*(a + b*x))/(c + d*x)] - 14*a^3*B^2*d^3*Log
[a + b*x]*Log[(e*(a + b*x))/(c + d*x)] + 6*b^3*B^2*c^2*d*x*Log[(e*(a + b*x))/(c + d*x)]^2 - 18*a*b^2*B^2*c*d^2
*x*Log[(e*(a + b*x))/(c + d*x)]^2 + 18*a^2*b*B^2*d^3*x*Log[(e*(a + b*x))/(c + d*x)]^2 - 3*b^3*B^2*c*d^2*x^2*Lo
g[(e*(a + b*x))/(c + d*x)]^2 + 9*a*b^2*B^2*d^3*x^2*Log[(e*(a + b*x))/(c + d*x)]^2 + 2*b^3*B^2*d^3*x^3*Log[(e*(
a + b*x))/(c + d*x)]^2 - 6*A^2*b^3*c^3*Log[c + d*x] - 10*A*b^3*B*c^3*Log[c + d*x] + 9*b^3*B^2*c^3*Log[c + d*x]
+ 18*a*A^2*b^2*c^2*d*Log[c + d*x] + 18*a*A*b^2*B*c^2*d*Log[c + d*x] - 37*a*b^2*B^2*c^2*d*Log[c + d*x] - 18*a^
2*A^2*b*c*d^2*Log[c + d*x] + 22*a^2*b*B^2*c*d^2*Log[c + d*x] + 6*a^3*A^2*d^3*Log[c + d*x] + 12*A*b^3*B*c^3*Log
[a/b + x]*Log[c + d*x] + 10*b^3*B^2*c^3*Log[a/b + x]*Log[c + d*x] - 36*a*A*b^2*B*c^2*d*Log[a/b + x]*Log[c + d*
x] - 18*a*b^2*B^2*c^2*d*Log[a/b + x]*Log[c + d*x] + 36*a^2*A*b*B*c*d^2*Log[a/b + x]*Log[c + d*x] - 12*a^3*A*B*
d^3*Log[a/b + x]*Log[c + d*x] - 6*b^3*B^2*c^3*Log[a/b + x]^2*Log[c + d*x] + 18*a*b^2*B^2*c^2*d*Log[a/b + x]^2*
Log[c + d*x] - 18*a^2*b*B^2*c*d^2*Log[a/b + x]^2*Log[c + d*x] - 12*A*b^3*B*c^3*Log[c/d + x]*Log[c + d*x] - 10*
b^3*B^2*c^3*Log[c/d + x]*Log[c + d*x] + 36*a*A*b^2*B*c^2*d*Log[c/d + x]*Log[c + d*x] + 18*a*b^2*B^2*c^2*d*Log[
c/d + x]*Log[c + d*x] - 36*a^2*A*b*B*c*d^2*Log[c/d + x]*Log[c + d*x] + 12*a^3*A*B*d^3*Log[c/d + x]*Log[c + d*x
] + 12*b^3*B^2*c^3*Log[a/b + x]*Log[c/d + x]*Log[c + d*x] - 36*a*b^2*B^2*c^2*d*Log[a/b + x]*Log[c/d + x]*Log[c
+ d*x] + 36*a^2*b*B^2*c*d^2*Log[a/b + x]*Log[c/d + x]*Log[c + d*x] - 6*b^3*B^2*c^3*Log[c/d + x]^2*Log[c + d*x
] + 18*a*b^2*B^2*c^2*d*Log[c/d + x]^2*Log[c + d...

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Maple [F]
time = 0.36, size = 0, normalized size = 0.00 \[\int \frac {\left (b g x +a g \right )^{3} \left (A +B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )\right )^{2}}{d i x +c i}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*g*x+a*g)^3*(A+B*log(e*(b*x+a)/(d*x+c)))^2/(d*i*x+c*i),x, algorithm="maxima")

[Out]

3*A^2*a^2*b*g^3*(-I*x/d + I*c*log(d*x + c)/d^2) - 1/6*A^2*b^3*g^3*(-6*I*c^3*log(d*x + c)/d^4 + I*(2*d^2*x^3 -
3*c*d*x^2 + 6*c^2*x)/d^3) - 3/2*A^2*a*b^2*g^3*(2*I*c^2*log(d*x + c)/d^3 + I*(d*x^2 - 2*c*x)/d^2) - I*A^2*a^3*g
^3*log(I*d*x + I*c)/d + 1/6*(2*(I*b^3*c^3*g^3 - 3*I*a*b^2*c^2*d*g^3 + 3*I*a^2*b*c*d^2*g^3 - I*a^3*d^3*g^3)*B^2
*log(d*x + c)^3 - (2*I*B^2*b^3*d^3*g^3*x^3 - 3*(I*b^3*c*d^2*g^3 - 3*I*a*b^2*d^3*g^3)*B^2*x^2 - 6*(-I*b^3*c^2*d
*g^3 + 3*I*a*b^2*c*d^2*g^3 - 3*I*a^2*b*d^3*g^3)*B^2*x)*log(d*x + c)^2)/d^4 + integrate(1/3*(-6*I*A*B*a^3*d^2*g
^3 - 3*I*B^2*a^3*d^2*g^3 - 3*(2*I*A*B*b^3*d^2*g^3 + I*B^2*b^3*d^2*g^3)*x^3 - 9*(2*I*A*B*a*b^2*d^2*g^3 + I*B^2*
a*b^2*d^2*g^3)*x^2 - 3*(I*B^2*b^3*d^2*g^3*x^3 + 3*I*B^2*a*b^2*d^2*g^3*x^2 + 3*I*B^2*a^2*b*d^2*g^3*x + I*B^2*a^
3*d^2*g^3)*log(b*x + a)^2 - 9*(2*I*A*B*a^2*b*d^2*g^3 + I*B^2*a^2*b*d^2*g^3)*x - 6*(I*A*B*a^3*d^2*g^3 + I*B^2*a
^3*d^2*g^3 + (I*A*B*b^3*d^2*g^3 + I*B^2*b^3*d^2*g^3)*x^3 + 3*(I*A*B*a*b^2*d^2*g^3 + I*B^2*a*b^2*d^2*g^3)*x^2 +
3*(I*A*B*a^2*b*d^2*g^3 + I*B^2*a^2*b*d^2*g^3)*x)*log(b*x + a) + (6*I*A*B*a^3*d^2*g^3 + 6*I*B^2*a^3*d^2*g^3 -
2*(-3*I*A*B*b^3*d^2*g^3 - 4*I*B^2*b^3*d^2*g^3)*x^3 - 3*(-6*I*A*B*a*b^2*d^2*g^3 + (I*b^3*c*d*g^3 - 9*I*a*b^2*d^
2*g^3)*B^2)*x^2 - 6*(-3*I*A*B*a^2*b*d^2*g^3 + (-I*b^3*c^2*g^3 + 3*I*a*b^2*c*d*g^3 - 6*I*a^2*b*d^2*g^3)*B^2)*x
- 6*(-I*B^2*b^3*d^2*g^3*x^3 - 3*I*B^2*a*b^2*d^2*g^3*x^2 - 3*I*B^2*a^2*b*d^2*g^3*x - I*B^2*a^3*d^2*g^3)*log(b*x
+ a))*log(d*x + c))/(d^3*x + c*d^2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*g*x+a*g)^3*(A+B*log(e*(b*x+a)/(d*x+c)))^2/(d*i*x+c*i),x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*g*x+a*g)**3*(A+B*ln(e*(b*x+a)/(d*x+c)))**2/(d*i*x+c*i),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*g*x+a*g)^3*(A+B*log(e*(b*x+a)/(d*x+c)))^2/(d*i*x+c*i),x, algorithm="giac")

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (a\,g+b\,g\,x\right )}^3\,{\left (A+B\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )\right )}^2}{c\,i+d\,i\,x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a*g + b*g*x)^3*(A + B*log((e*(a + b*x))/(c + d*x)))^2)/(c*i + d*i*x),x)

[Out]

int(((a*g + b*g*x)^3*(A + B*log((e*(a + b*x))/(c + d*x)))^2)/(c*i + d*i*x), x)

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