3.1.86 \(\int \frac {(a g+b g x) (A+B \log (\frac {e (a+b x)}{c+d x}))^2}{c i+d i x} \, dx\) [86]

Optimal. Leaf size=283 \[ \frac {2 B (b c-a d) g \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{d^2 i}+\frac {g (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{d i}+\frac {(b c-a d) g \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{d^2 i}+\frac {2 B^2 (b c-a d) g \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right )}{d^2 i}+\frac {2 B (b c-a d) g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right )}{d^2 i}-\frac {2 B^2 (b c-a d) g \text {Li}_3\left (\frac {d (a+b x)}{b (c+d x)}\right )}{d^2 i} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.22, antiderivative size = 283, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.175, Rules used = {2562, 2395, 2355, 2354, 2438, 2421, 6724} \begin {gather*} \frac {2 B g (b c-a d) \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^2 i}+\frac {2 B^2 g (b c-a d) \text {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{d^2 i}-\frac {2 B^2 g (b c-a d) \text {PolyLog}\left (3,\frac {d (a+b x)}{b (c+d x)}\right )}{d^2 i}+\frac {2 B g (b c-a d) \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^2 i}+\frac {g (b c-a d) \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^2 i}+\frac {g (a+b x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{d i} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 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) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{86 c+86 d x} \, dx &=\int \left (\frac {b g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{86 d}+\frac {(-b c+a d) g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{86 d (c+d x)}\right ) \, dx\\ &=\frac {(b g) \int \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \, dx}{86 d}-\frac {((b c-a d) g) \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{c+d x} \, dx}{86 d}\\ &=\frac {b g x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{86 d}-\frac {(b c-a d) g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{86 d^2}-\frac {(b B g) \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}{43 d}+\frac {(B (b c-a d) g) \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}{43 d^2}\\ &=\frac {b g x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{86 d}-\frac {(b c-a d) g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{86 d^2}-\frac {(b B (b c-a d) g) \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}{43 d}+\frac {(B (b c-a d) g) \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}{43 d^2 e}\\ &=\frac {b g x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{86 d}-\frac {(b c-a d) g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{86 d^2}-\frac {(b B (b c-a d) g) \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}{43 d}+\frac {(B (b c-a d) g) \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}{43 d^2 e}\\ &=\frac {b g x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{86 d}-\frac {(b c-a d) g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{86 d^2}+\frac {(a b B g) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{43 d}-\frac {(b B c g) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{c+d x} \, dx}{43 d}+\frac {\left (B (b c-a d)^2 g\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}{43 d^2}\\ &=\frac {a B g \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{43 d}+\frac {b g x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{86 d}-\frac {b B c g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{43 d^2}-\frac {(b c-a d) g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{86 d^2}+\frac {\left (b B^2 c g\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}{43 d^2}-\frac {\left (a B^2 g\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}{43 d}+\frac {\left (B (b c-a d)^2 g\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}{43 d^2}\\ &=\frac {a B g \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{43 d}+\frac {b g x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{86 d}-\frac {b B c g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{43 d^2}-\frac {(b c-a d) g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{86 d^2}+\frac {(b B (b c-a d) g) \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}{43 d^2}-\frac {(B (b c-a d) g) \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}{43 d}+\frac {\left (b B^2 c g\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}{43 d^2 e}-\frac {\left (a B^2 g\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}{43 d e}\\ &=\frac {a B g \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{43 d}+\frac {b g x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{86 d}-\frac {b B c g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{43 d^2}-\frac {(b c-a d) g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{86 d^2}+\frac {(b B (b c-a d) g) \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}{43 d^2}-\frac {(B (b c-a d) g) \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}{43 d}+\frac {\left (b B^2 c g\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}{43 d^2 e}-\frac {\left (a B^2 g\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}{43 d e}\\ &=\frac {a B g \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{43 d}+\frac {b g x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{86 d}-\frac {b B c g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{43 d^2}-\frac {(b c-a d) g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{86 d^2}+\frac {1}{43} \left (a B^2 g\right ) \int \frac {\log (a+b x)}{c+d x} \, dx+\frac {\left (b^2 B^2 c g\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{43 d^2}-\frac {\left (a b B^2 g\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{43 d}-\frac {\left (b B^2 c g\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{43 d}+\frac {(A b B (b c-a d) g) \int \frac {\log (c+d x)}{a+b x} \, dx}{43 d^2}+\frac {\left (b B^2 (b c-a d) g\right ) \int \frac {\log \left (\frac {e (a+b x)}{c+d x}\right ) \log (c+d x)}{a+b x} \, dx}{43 d^2}-\frac {(A B (b c-a d) g) \int \frac {\log (c+d x)}{c+d x} \, dx}{43 d}-\frac {\left (B^2 (b c-a d) g\right ) \int \frac {\log \left (\frac {e (a+b x)}{c+d x}\right ) \log (c+d x)}{c+d x} \, dx}{43 d}\\ &=\frac {a B g \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{43 d}+\frac {b g x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{86 d}+\frac {b B^2 c g \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{43 d^2}+\frac {A B (b c-a d) g \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{43 d^2}-\frac {b B c g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{43 d^2}-\frac {(b c-a d) g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{86 d^2}-\frac {B^2 (b c-a d) g \log \left (\frac {e (a+b x)}{c+d x}\right ) \log ^2(c+d x)}{86 d^2}+\frac {a B^2 g \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{43 d}-\frac {\left (b B^2 c g\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{43 d^2}-\frac {\left (a B^2 g\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{43 d}-\frac {\left (a b B^2 g\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{43 d}-\frac {\left (b B^2 c g\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{43 d}-\frac {(A B (b c-a d) g) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{43 d^2}+\frac {\left (b B^2 (b c-a d) g\right ) \int \frac {\log ^2(c+d x)}{a+b x} \, dx}{86 d^2}+\frac {\left (b B^2 (b c-a d) g\right ) \int \frac {\log (a+b x) \log (c+d x)}{a+b x} \, dx}{43 d^2}+\frac {\left (b B^2 (b c-a d) g\right ) \int \frac {\log \left (\frac {1}{c+d x}\right ) \log (c+d x)}{a+b x} \, dx}{43 d^2}-\frac {(A B (b c-a d) g) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{43 d}-\frac {\left (B^2 (b c-a d) g\right ) \int \frac {\log ^2(c+d x)}{c+d x} \, dx}{86 d}+\frac {\left (b B^2 (b c-a d) g \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}{43 d^2}\\ &=-\frac {a B^2 g \log ^2(a+b x)}{86 d}+\frac {a B g \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{43 d}+\frac {b g x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{86 d}+\frac {b B^2 c g \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{43 d^2}+\frac {A B (b c-a d) g \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{43 d^2}-\frac {B^2 (b c-a d) g \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)}{43 d^2}-\frac {b B c g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{43 d^2}-\frac {(b c-a d) g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{86 d^2}-\frac {b B^2 c g \log ^2(c+d x)}{86 d^2}-\frac {A B (b c-a d) g \log ^2(c+d x)}{86 d^2}+\frac {B^2 (b c-a d) g \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(c+d x)}{86 d^2}-\frac {B^2 (b c-a d) g \log \left (\frac {e (a+b x)}{c+d x}\right ) \log ^2(c+d x)}{86 d^2}+\frac {a B^2 g \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{43 d}-\frac {\left (b B^2 c g\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{43 d^2}-\frac {\left (a B^2 g\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{43 d}-\frac {(A B (b c-a d) g) \text {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{43 d^2}-\frac {\left (B^2 (b c-a d) g\right ) \text {Subst}\left (\int \frac {\log ^2(x)}{x} \, dx,x,c+d x\right )}{86 d^2}+\frac {\left (B^2 (b c-a d) g\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 )}{43 d^2}+\frac {\left (B^2 (b c-a d) g\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 )}{43 d^2}-\frac {\left (B^2 (b c-a d) g\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right ) \log (c+d x)}{c+d x} \, dx}{43 d}-\frac {\left (B^2 (b c-a d) g \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}{43 d}\\ &=-\frac {a B^2 g \log ^2(a+b x)}{86 d}+\frac {a B g \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{43 d}+\frac {b g x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{86 d}+\frac {B^2 (b c-a d) g \log ^2(a+b x) \log (c+d x)}{86 d^2}+\frac {b B^2 c g \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{43 d^2}+\frac {A B (b c-a d) g \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{43 d^2}+\frac {B^2 (b c-a d) g \log (a+b x) \log \left (\frac {1}{c+d x}\right ) \log (c+d x)}{43 d^2}-\frac {B^2 (b c-a d) g \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)}{43 d^2}-\frac {b B c g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{43 d^2}-\frac {(b c-a d) g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{86 d^2}-\frac {b B^2 c g \log ^2(c+d x)}{86 d^2}-\frac {A B (b c-a d) g \log ^2(c+d x)}{86 d^2}+\frac {B^2 (b c-a d) g \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(c+d x)}{86 d^2}-\frac {B^2 (b c-a d) g \log \left (\frac {e (a+b x)}{c+d x}\right ) \log ^2(c+d x)}{86 d^2}+\frac {a B^2 g \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{43 d}+\frac {a B^2 g \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{43 d}+\frac {b B^2 c g \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{43 d^2}+\frac {A B (b c-a d) g \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{43 d^2}-\frac {\left (B^2 (b c-a d) g\right ) \text {Subst}\left (\int x^2 \, dx,x,\log (c+d x)\right )}{86 d^2}-\frac {\left (B^2 (b c-a d) g\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 )}{43 d^2}-\frac {\left (B^2 (b c-a d) g\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 )}{86 b d}-\frac {\left (B^2 (b c-a d) g\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 )}{43 b d}+\frac {\left (B^2 (b c-a d) g\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 )}{43 b d}-\frac {\left (B^2 (b c-a d) g \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 )}{43 d^2}\\ &=-\frac {a B^2 g \log ^2(a+b x)}{86 d}+\frac {a B g \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{43 d}+\frac {b g x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{86 d}+\frac {B^2 (b c-a d) g \log ^2(a+b x) \log (c+d x)}{86 d^2}+\frac {b B^2 c g \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{43 d^2}+\frac {A B (b c-a d) g \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{43 d^2}+\frac {B^2 (b c-a d) g \log (a+b x) \log \left (\frac {1}{c+d x}\right ) \log (c+d x)}{43 d^2}-\frac {B^2 (b c-a d) g \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)}{43 d^2}-\frac {b B c g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{43 d^2}-\frac {(b c-a d) g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{86 d^2}-\frac {b B^2 c g \log ^2(c+d x)}{86 d^2}-\frac {A B (b c-a d) g \log ^2(c+d x)}{86 d^2}+\frac {B^2 (b c-a d) g \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(c+d x)}{86 d^2}-\frac {B^2 (b c-a d) g \log \left (\frac {e (a+b x)}{c+d x}\right ) \log ^2(c+d x)}{86 d^2}-\frac {B^2 (b c-a d) g \log ^3(c+d x)}{258 d^2}+\frac {a B^2 g \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{43 d}-\frac {B^2 (b c-a d) g \log ^2(a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{86 d^2}+\frac {a B^2 g \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{43 d}+\frac {b B^2 c g \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{43 d^2}+\frac {A B (b c-a d) g \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{43 d^2}-\frac {B^2 (b c-a d) g \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 )}{43 d^2}+\frac {B^2 (b c-a d) g \log (c+d x) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{43 d^2}-\frac {\left (B^2 (b c-a d) g\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 )}{43 d^2}+\frac {\left (B^2 (b c-a d) g\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 )}{43 d^2}+\frac {\left (B^2 (b c-a d) g\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 )}{43 d^2}-\frac {\left (B^2 (b c-a d) g\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 )}{43 d^2}\\ &=-\frac {a B^2 g \log ^2(a+b x)}{86 d}+\frac {B^2 (b c-a d) g \log (a+b x) \log ^2\left (\frac {1}{c+d x}\right )}{86 d^2}+\frac {a B g \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{43 d}+\frac {b g x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{86 d}+\frac {B^2 (b c-a d) g \log ^2(a+b x) \log (c+d x)}{86 d^2}+\frac {b B^2 c g \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{43 d^2}+\frac {A B (b c-a d) g \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{43 d^2}+\frac {B^2 (b c-a d) g \log (a+b x) \log \left (\frac {1}{c+d x}\right ) \log (c+d x)}{43 d^2}-\frac {B^2 (b c-a d) g \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)}{43 d^2}-\frac {b B c g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{43 d^2}-\frac {(b c-a d) g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{86 d^2}-\frac {b B^2 c g \log ^2(c+d x)}{86 d^2}-\frac {A B (b c-a d) g \log ^2(c+d x)}{86 d^2}+\frac {B^2 (b c-a d) g \log (a+b x) \log ^2(c+d x)}{86 d^2}+\frac {B^2 (b c-a d) g \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(c+d x)}{86 d^2}-\frac {B^2 (b c-a d) g \log \left (\frac {e (a+b x)}{c+d x}\right ) \log ^2(c+d x)}{86 d^2}-\frac {B^2 (b c-a d) g \log ^3(c+d x)}{258 d^2}+\frac {a B^2 g \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{43 d}-\frac {B^2 (b c-a d) g \log ^2(a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{86 d^2}+\frac {a B^2 g \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{43 d}-\frac {B^2 (b c-a d) g \log (a+b x) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{43 d^2}+\frac {b B^2 c g \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{43 d^2}+\frac {A B (b c-a d) g \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{43 d^2}-\frac {B^2 (b c-a d) g \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 )}{43 d^2}+\frac {B^2 (b c-a d) g \log (c+d x) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{43 d^2}-\frac {B^2 (b c-a d) g \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{43 d^2}-\frac {\left (b B^2 (b c-a d) g\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 )}{86 d^3}-\frac {\left (b B^2 (b c-a d) g\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 )}{86 d^3}+\frac {\left (B^2 (b c-a d) g\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 )}{43 d^2}\\ &=-\frac {a B^2 g \log ^2(a+b x)}{86 d}+\frac {B^2 (b c-a d) g \log (a+b x) \log ^2\left (\frac {1}{c+d x}\right )}{86 d^2}-\frac {B^2 (b c-a d) g \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2\left (\frac {1}{c+d x}\right )}{86 d^2}+\frac {a B g \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{43 d}+\frac {b g x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{86 d}+\frac {B^2 (b c-a d) g \log ^2(a+b x) \log (c+d x)}{86 d^2}+\frac {b B^2 c g \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{43 d^2}+\frac {A B (b c-a d) g \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{43 d^2}+\frac {B^2 (b c-a d) g \log (a+b x) \log \left (\frac {1}{c+d x}\right ) \log (c+d x)}{43 d^2}-\frac {B^2 (b c-a d) g \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)}{43 d^2}-\frac {b B c g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{43 d^2}-\frac {(b c-a d) g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{86 d^2}-\frac {b B^2 c g \log ^2(c+d x)}{86 d^2}-\frac {A B (b c-a d) g \log ^2(c+d x)}{86 d^2}+\frac {B^2 (b c-a d) g \log (a+b x) \log ^2(c+d x)}{86 d^2}-\frac {B^2 (b c-a d) g \log \left (\frac {e (a+b x)}{c+d x}\right ) \log ^2(c+d x)}{86 d^2}-\frac {B^2 (b c-a d) g \log ^3(c+d x)}{258 d^2}+\frac {a B^2 g \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{43 d}-\frac {B^2 (b c-a d) g \log ^2(a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{86 d^2}+\frac {a B^2 g \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{43 d}-\frac {B^2 (b c-a d) g \log (a+b x) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{43 d^2}+\frac {b B^2 c g \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{43 d^2}+\frac {A B (b c-a d) g \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{43 d^2}-\frac {B^2 (b c-a d) g \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 )}{43 d^2}+\frac {B^2 (b c-a d) g \log (c+d x) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{43 d^2}+\frac {B^2 (b c-a d) g \text {Li}_3\left (-\frac {d (a+b x)}{b c-a d}\right )}{43 d^2}-\frac {B^2 (b c-a d) g \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{43 d^2}-\frac {\left (B^2 (b c-a d) g\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 )}{43 d^2}+\frac {\left (B^2 (b c-a d) g\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 )}{43 d^2}\\ &=-\frac {a B^2 g \log ^2(a+b x)}{86 d}+\frac {B^2 (b c-a d) g \log (a+b x) \log ^2\left (\frac {1}{c+d x}\right )}{86 d^2}-\frac {B^2 (b c-a d) g \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2\left (\frac {1}{c+d x}\right )}{86 d^2}+\frac {a B g \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{43 d}+\frac {b g x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{86 d}+\frac {B^2 (b c-a d) g \log ^2(a+b x) \log (c+d x)}{86 d^2}+\frac {b B^2 c g \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{43 d^2}+\frac {A B (b c-a d) g \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{43 d^2}+\frac {B^2 (b c-a d) g \log (a+b x) \log \left (\frac {1}{c+d x}\right ) \log (c+d x)}{43 d^2}-\frac {B^2 (b c-a d) g \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)}{43 d^2}-\frac {b B c g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{43 d^2}-\frac {(b c-a d) g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{86 d^2}-\frac {b B^2 c g \log ^2(c+d x)}{86 d^2}-\frac {A B (b c-a d) g \log ^2(c+d x)}{86 d^2}+\frac {B^2 (b c-a d) g \log (a+b x) \log ^2(c+d x)}{86 d^2}-\frac {B^2 (b c-a d) g \log \left (\frac {e (a+b x)}{c+d x}\right ) \log ^2(c+d x)}{86 d^2}-\frac {B^2 (b c-a d) g \log ^3(c+d x)}{258 d^2}+\frac {a B^2 g \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{43 d}-\frac {B^2 (b c-a d) g \log ^2(a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{86 d^2}+\frac {a B^2 g \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{43 d}-\frac {B^2 (b c-a d) g \log (a+b x) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{43 d^2}+\frac {b B^2 c g \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{43 d^2}+\frac {A B (b c-a d) g \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{43 d^2}+\frac {B^2 (b c-a d) g \log \left (\frac {1}{c+d x}\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{43 d^2}-\frac {B^2 (b c-a d) g \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 )}{43 d^2}+\frac {B^2 (b c-a d) g \text {Li}_3\left (-\frac {d (a+b x)}{b c-a d}\right )}{43 d^2}-\frac {B^2 (b c-a d) g \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{43 d^2}+2 \frac {\left (B^2 (b c-a d) g\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 )}{43 d^2}\\ &=-\frac {a B^2 g \log ^2(a+b x)}{86 d}+\frac {B^2 (b c-a d) g \log (a+b x) \log ^2\left (\frac {1}{c+d x}\right )}{86 d^2}-\frac {B^2 (b c-a d) g \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2\left (\frac {1}{c+d x}\right )}{86 d^2}+\frac {a B g \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{43 d}+\frac {b g x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{86 d}+\frac {B^2 (b c-a d) g \log ^2(a+b x) \log (c+d x)}{86 d^2}+\frac {b B^2 c g \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{43 d^2}+\frac {A B (b c-a d) g \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{43 d^2}+\frac {B^2 (b c-a d) g \log (a+b x) \log \left (\frac {1}{c+d x}\right ) \log (c+d x)}{43 d^2}-\frac {B^2 (b c-a d) g \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)}{43 d^2}-\frac {b B c g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{43 d^2}-\frac {(b c-a d) g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{86 d^2}-\frac {b B^2 c g \log ^2(c+d x)}{86 d^2}-\frac {A B (b c-a d) g \log ^2(c+d x)}{86 d^2}+\frac {B^2 (b c-a d) g \log (a+b x) \log ^2(c+d x)}{86 d^2}-\frac {B^2 (b c-a d) g \log \left (\frac {e (a+b x)}{c+d x}\right ) \log ^2(c+d x)}{86 d^2}-\frac {B^2 (b c-a d) g \log ^3(c+d x)}{258 d^2}+\frac {a B^2 g \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{43 d}-\frac {B^2 (b c-a d) g \log ^2(a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{86 d^2}+\frac {a B^2 g \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{43 d}-\frac {B^2 (b c-a d) g \log (a+b x) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{43 d^2}+\frac {b B^2 c g \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{43 d^2}+\frac {A B (b c-a d) g \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{43 d^2}+\frac {B^2 (b c-a d) g \log \left (\frac {1}{c+d x}\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{43 d^2}-\frac {B^2 (b c-a d) g \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 )}{43 d^2}+\frac {B^2 (b c-a d) g \text {Li}_3\left (-\frac {d (a+b x)}{b c-a d}\right )}{43 d^2}+\frac {B^2 (b c-a d) g \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{43 d^2}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(646\) vs. \(2(283)=566\).
time = 0.43, size = 646, normalized size = 2.28 \begin {gather*} -\frac {g \left (-A^2 b d x+A^2 (b c-a d) \log (c+d x)+a A B d \left (\log ^2\left (\frac {c}{d}+x\right )+2 \left (\log \left (\frac {a}{b}+x\right )-\log \left (\frac {c}{d}+x\right )-\log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)-2 \left (\log \left (\frac {a}{b}+x\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )+\text {Li}_2\left (\frac {d (a+b x)}{-b c+a d}\right )\right )\right )+A B \left (-2 d (a+b x) \left (-1+\log \left (\frac {a}{b}+x\right )\right )+2 b (c+d x) \left (-1+\log \left (\frac {c}{d}+x\right )\right )-b c \log ^2\left (\frac {c}{d}+x\right )+2 b \left (\log \left (\frac {a}{b}+x\right )-\log \left (\frac {c}{d}+x\right )-\log \left (\frac {e (a+b x)}{c+d x}\right )\right ) (d x-c \log (c+d x))+2 b c \left (\log \left (\frac {a}{b}+x\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )+\text {Li}_2\left (\frac {d (a+b x)}{-b c+a d}\right )\right )\right )-B^2 \left (d (a+b x) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )+b c \log ^2\left (\frac {e (a+b x)}{c+d x}\right ) \log \left (\frac {b c-a d}{b c+b d x}\right )-(b c-a d) \left (\log \left (\frac {b c-a d}{b c+b d x}\right ) \left (2 \log \left (\frac {d (a+b x)}{-b c+a d}\right )-2 \log \left (\frac {e (a+b x)}{c+d x}\right )+\log \left (\frac {b c-a d}{b c+b d x}\right )\right )-2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )\right )+2 b c \left (\log \left (\frac {e (a+b x)}{c+d x}\right ) \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right )-\text {Li}_3\left (\frac {d (a+b x)}{b (c+d x)}\right )\right )\right )+a B^2 d \left (\log ^2\left (\frac {e (a+b x)}{c+d x}\right ) \log \left (\frac {b c-a d}{b c+b d x}\right )+2 \log \left (\frac {e (a+b x)}{c+d x}\right ) \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right )-2 \text {Li}_3\left (\frac {d (a+b x)}{b (c+d x)}\right )\right )\right )}{d^2 i} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]
time = 0.17, size = 0, normalized size = 0.00 \[\int \frac {\left (b g x +a g \right ) \left (A +B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )\right )^{2}}{d i x +c i}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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