3.1.93 \(\int \frac {(a g+b g x)^2 (A+B \log (\frac {e (a+b x)}{c+d x}))^2}{(c i+d i x)^2} \, dx\) [93]

Optimal. Leaf size=469 \[ -\frac {2 A B (b c-a d) g^2 (a+b x)}{d^2 i^2 (c+d x)}+\frac {2 B^2 (b c-a d) g^2 (a+b x)}{d^2 i^2 (c+d x)}-\frac {2 B^2 (b c-a d) g^2 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{d^2 i^2 (c+d x)}+\frac {2 b B (b c-a d) g^2 \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{d^3 i^2}+\frac {b g^2 (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{d^2 i^2}+\frac {(b c-a d) g^2 (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{d^2 i^2 (c+d x)}+\frac {2 b (b c-a d) g^2 \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{d^3 i^2}+\frac {2 b B^2 (b c-a d) g^2 \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right )}{d^3 i^2}+\frac {4 b B (b c-a d) g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right )}{d^3 i^2}-\frac {4 b B^2 (b c-a d) g^2 \text {Li}_3\left (\frac {d (a+b x)}{b (c+d x)}\right )}{d^3 i^2} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.31, antiderivative size = 469, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {2562, 2395, 2333, 2332, 2355, 2354, 2438, 2421, 6724} \begin {gather*} \frac {4 b B g^2 (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^3 i^2}+\frac {2 b B^2 g^2 (b c-a d) \text {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{d^3 i^2}-\frac {4 b B^2 g^2 (b c-a d) \text {PolyLog}\left (3,\frac {d (a+b x)}{b (c+d x)}\right )}{d^3 i^2}+\frac {2 b g^2 (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^3 i^2}+\frac {2 b B g^2 (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^3 i^2}+\frac {b g^2 (a+b x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{d^2 i^2}+\frac {g^2 (a+b x) (b c-a d) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{d^2 i^2 (c+d x)}-\frac {2 A B g^2 (a+b x) (b c-a d)}{d^2 i^2 (c+d x)}-\frac {2 B^2 g^2 (a+b x) (b c-a d) \log \left (\frac {e (a+b x)}{c+d x}\right )}{d^2 i^2 (c+d x)}+\frac {2 B^2 g^2 (a+b x) (b c-a d)}{d^2 i^2 (c+d x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2332

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rule 2333

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

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)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(93 c+93 d x)^2} \, dx &=\int \left (\frac {b^2 g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{8649 d^2}+\frac {(-b c+a d)^2 g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{8649 d^2 (c+d x)^2}-\frac {2 b (b c-a d) g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{8649 d^2 (c+d x)}\right ) \, dx\\ &=\frac {\left (b^2 g^2\right ) \int \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \, dx}{8649 d^2}-\frac {\left (2 b (b c-a d) g^2\right ) \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{c+d x} \, dx}{8649 d^2}+\frac {\left ((b c-a d)^2 g^2\right ) \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(c+d x)^2} \, dx}{8649 d^2}\\ &=\frac {b^2 g^2 x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{8649 d^2}-\frac {(b c-a d)^2 g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{8649 d^3 (c+d x)}-\frac {2 b (b c-a d) g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{8649 d^3}-\frac {\left (2 b^2 B g^2\right ) \int \frac {(b c-a d) x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a+b x) (c+d x)} \, dx}{8649 d^2}+\frac {\left (4 b B (b c-a d) g^2\right ) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{e (a+b x)} \, dx}{8649 d^3}+\frac {\left (2 B (b c-a d)^2 g^2\right ) \int \frac {(b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a+b x) (c+d x)^2} \, dx}{8649 d^3}\\ &=\frac {b^2 g^2 x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{8649 d^2}-\frac {(b c-a d)^2 g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{8649 d^3 (c+d x)}-\frac {2 b (b c-a d) g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{8649 d^3}-\frac {\left (2 b^2 B (b c-a d) g^2\right ) \int \frac {x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a+b x) (c+d x)} \, dx}{8649 d^2}+\frac {\left (2 B (b c-a d)^3 g^2\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x) (c+d x)^2} \, dx}{8649 d^3}+\frac {\left (4 b B (b c-a d) g^2\right ) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{a+b x} \, dx}{8649 d^3 e}\\ &=\frac {b^2 g^2 x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{8649 d^2}-\frac {(b c-a d)^2 g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{8649 d^3 (c+d x)}-\frac {2 b (b c-a d) g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{8649 d^3}-\frac {\left (2 b^2 B (b c-a d) g^2\right ) \int \left (-\frac {a \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d) (a+b x)}+\frac {c \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d) (c+d x)}\right ) \, dx}{8649 d^2}+\frac {\left (2 B (b c-a d)^3 g^2\right ) \int \left (\frac {b^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^2 (a+b x)}-\frac {d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d) (c+d x)^2}-\frac {b d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^2 (c+d x)}\right ) \, dx}{8649 d^3}+\frac {\left (4 b B (b c-a d) g^2\right ) \int \frac {(b c-a d) e \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{(a+b x) (c+d x)} \, dx}{8649 d^3 e}\\ &=\frac {b^2 g^2 x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{8649 d^2}-\frac {(b c-a d)^2 g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{8649 d^3 (c+d x)}-\frac {2 b (b c-a d) g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{8649 d^3}+\frac {\left (2 a b^2 B g^2\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{8649 d^2}-\frac {\left (2 b^2 B c g^2\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{c+d x} \, dx}{8649 d^2}+\frac {\left (2 b^2 B (b c-a d) g^2\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{8649 d^3}-\frac {\left (2 b B (b c-a d) g^2\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{c+d x} \, dx}{8649 d^2}+\frac {\left (4 b B (b c-a d)^2 g^2\right ) \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{(a+b x) (c+d x)} \, dx}{8649 d^3}-\frac {\left (2 B (b c-a d)^2 g^2\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(c+d x)^2} \, dx}{8649 d^2}\\ &=\frac {2 B (b c-a d)^2 g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{8649 d^3 (c+d x)}+\frac {2 a b B g^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{8649 d^2}+\frac {2 b B (b c-a d) g^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{8649 d^3}+\frac {b^2 g^2 x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{8649 d^2}-\frac {(b c-a d)^2 g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{8649 d^3 (c+d x)}-\frac {2 b^2 B c g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{8649 d^3}-\frac {2 b B (b c-a d) g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{8649 d^3}-\frac {2 b (b c-a d) g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{8649 d^3}+\frac {\left (2 b^2 B^2 c g^2\right ) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (c+d x)}{e (a+b x)} \, dx}{8649 d^3}-\frac {\left (2 a b B^2 g^2\right ) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (a+b x)}{e (a+b x)} \, dx}{8649 d^2}-\frac {\left (2 b B^2 (b c-a d) g^2\right ) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (a+b x)}{e (a+b x)} \, dx}{8649 d^3}+\frac {\left (2 b B^2 (b c-a d) g^2\right ) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (c+d x)}{e (a+b x)} \, dx}{8649 d^3}+\frac {\left (4 b B (b c-a d)^2 g^2\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}{8649 d^3}-\frac {\left (2 B^2 (b c-a d)^2 g^2\right ) \int \frac {b c-a d}{(a+b x) (c+d x)^2} \, dx}{8649 d^3}\\ &=\frac {2 B (b c-a d)^2 g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{8649 d^3 (c+d x)}+\frac {2 a b B g^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{8649 d^2}+\frac {2 b B (b c-a d) g^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{8649 d^3}+\frac {b^2 g^2 x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{8649 d^2}-\frac {(b c-a d)^2 g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{8649 d^3 (c+d x)}-\frac {2 b^2 B c g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{8649 d^3}-\frac {2 b B (b c-a d) g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{8649 d^3}-\frac {2 b (b c-a d) g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{8649 d^3}+\frac {\left (4 b^2 B (b c-a d) g^2\right ) \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{a+b x} \, dx}{8649 d^3}-\frac {\left (4 b B (b c-a d) g^2\right ) \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{c+d x} \, dx}{8649 d^2}-\frac {\left (2 B^2 (b c-a d)^3 g^2\right ) \int \frac {1}{(a+b x) (c+d x)^2} \, dx}{8649 d^3}+\frac {\left (2 b^2 B^2 c g^2\right ) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (c+d x)}{a+b x} \, dx}{8649 d^3 e}-\frac {\left (2 a b B^2 g^2\right ) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (a+b x)}{a+b x} \, dx}{8649 d^2 e}-\frac {\left (2 b B^2 (b c-a d) g^2\right ) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (a+b x)}{a+b x} \, dx}{8649 d^3 e}+\frac {\left (2 b B^2 (b c-a d) g^2\right ) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (c+d x)}{a+b x} \, dx}{8649 d^3 e}\\ &=\frac {2 B (b c-a d)^2 g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{8649 d^3 (c+d x)}+\frac {2 a b B g^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{8649 d^2}+\frac {2 b B (b c-a d) g^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{8649 d^3}+\frac {b^2 g^2 x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{8649 d^2}-\frac {(b c-a d)^2 g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{8649 d^3 (c+d x)}-\frac {2 b^2 B c g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{8649 d^3}-\frac {2 b B (b c-a d) g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{8649 d^3}-\frac {2 b (b c-a d) g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{8649 d^3}+\frac {\left (4 b^2 B (b c-a d) g^2\right ) \int \left (\frac {A \log (c+d x)}{a+b x}+\frac {B \log \left (\frac {e (a+b x)}{c+d x}\right ) \log (c+d x)}{a+b x}\right ) \, dx}{8649 d^3}-\frac {\left (4 b B (b c-a d) g^2\right ) \int \left (\frac {A \log (c+d x)}{c+d x}+\frac {B \log \left (\frac {e (a+b x)}{c+d x}\right ) \log (c+d x)}{c+d x}\right ) \, dx}{8649 d^2}-\frac {\left (2 B^2 (b c-a d)^3 g^2\right ) \int \left (\frac {b^2}{(b c-a d)^2 (a+b x)}-\frac {d}{(b c-a d) (c+d x)^2}-\frac {b d}{(b c-a d)^2 (c+d x)}\right ) \, dx}{8649 d^3}+\frac {\left (2 b^2 B^2 c g^2\right ) \int \left (\frac {b e \log (c+d x)}{a+b x}-\frac {d e \log (c+d x)}{c+d x}\right ) \, dx}{8649 d^3 e}-\frac {\left (2 a b B^2 g^2\right ) \int \left (\frac {b e \log (a+b x)}{a+b x}-\frac {d e \log (a+b x)}{c+d x}\right ) \, dx}{8649 d^2 e}-\frac {\left (2 b B^2 (b c-a d) g^2\right ) \int \left (\frac {b e \log (a+b x)}{a+b x}-\frac {d e \log (a+b x)}{c+d x}\right ) \, dx}{8649 d^3 e}+\frac {\left (2 b B^2 (b c-a d) g^2\right ) \int \left (\frac {b e \log (c+d x)}{a+b x}-\frac {d e \log (c+d x)}{c+d x}\right ) \, dx}{8649 d^3 e}\\ &=-\frac {2 B^2 (b c-a d)^2 g^2}{8649 d^3 (c+d x)}-\frac {2 b B^2 (b c-a d) g^2 \log (a+b x)}{8649 d^3}+\frac {2 B (b c-a d)^2 g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{8649 d^3 (c+d x)}+\frac {2 a b B g^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{8649 d^2}+\frac {2 b B (b c-a d) g^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{8649 d^3}+\frac {b^2 g^2 x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{8649 d^2}-\frac {(b c-a d)^2 g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{8649 d^3 (c+d x)}+\frac {2 b B^2 (b c-a d) g^2 \log (c+d x)}{8649 d^3}-\frac {2 b^2 B c g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{8649 d^3}-\frac {2 b B (b c-a d) g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{8649 d^3}-\frac {2 b (b c-a d) g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{8649 d^3}+\frac {\left (2 b^3 B^2 c g^2\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{8649 d^3}-\frac {\left (2 a b^2 B^2 g^2\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{8649 d^2}-\frac {\left (2 b^2 B^2 c g^2\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{8649 d^2}+\frac {\left (2 a b B^2 g^2\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{8649 d}+\frac {\left (4 A b^2 B (b c-a d) g^2\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{8649 d^3}-\frac {\left (2 b^2 B^2 (b c-a d) g^2\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{8649 d^3}+\frac {\left (2 b^2 B^2 (b c-a d) g^2\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{8649 d^3}+\frac {\left (4 b^2 B^2 (b c-a d) g^2\right ) \int \frac {\log \left (\frac {e (a+b x)}{c+d x}\right ) \log (c+d x)}{a+b x} \, dx}{8649 d^3}-\frac {\left (4 A b B (b c-a d) g^2\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{8649 d^2}+\frac {\left (2 b B^2 (b c-a d) g^2\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{8649 d^2}-\frac {\left (2 b B^2 (b c-a d) g^2\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{8649 d^2}-\frac {\left (4 b B^2 (b c-a d) g^2\right ) \int \frac {\log \left (\frac {e (a+b x)}{c+d x}\right ) \log (c+d x)}{c+d x} \, dx}{8649 d^2}\\ &=-\frac {2 B^2 (b c-a d)^2 g^2}{8649 d^3 (c+d x)}-\frac {2 b B^2 (b c-a d) g^2 \log (a+b x)}{8649 d^3}+\frac {2 B (b c-a d)^2 g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{8649 d^3 (c+d x)}+\frac {2 a b B g^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{8649 d^2}+\frac {2 b B (b c-a d) g^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{8649 d^3}+\frac {b^2 g^2 x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{8649 d^2}-\frac {(b c-a d)^2 g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{8649 d^3 (c+d x)}+\frac {2 b B^2 (b c-a d) g^2 \log (c+d x)}{8649 d^3}+\frac {2 b^2 B^2 c g^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{8649 d^3}+\frac {4 A b B (b c-a d) g^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{8649 d^3}+\frac {2 b B^2 (b c-a d) g^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{8649 d^3}-\frac {2 b^2 B c g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{8649 d^3}-\frac {2 b B (b c-a d) g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{8649 d^3}-\frac {2 b (b c-a d) g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{8649 d^3}-\frac {2 b B^2 (b c-a d) g^2 \log \left (\frac {e (a+b x)}{c+d x}\right ) \log ^2(c+d x)}{8649 d^3}+\frac {2 a b B^2 g^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{8649 d^2}+\frac {2 b B^2 (b c-a d) g^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{8649 d^3}-\frac {\left (2 b^2 B^2 c g^2\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{8649 d^3}-\frac {\left (2 a b B^2 g^2\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{8649 d^2}-\frac {\left (2 a b^2 B^2 g^2\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{8649 d^2}-\frac {\left (2 b^2 B^2 c g^2\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{8649 d^2}-\frac {\left (4 A b B (b c-a d) g^2\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{8649 d^3}-\frac {\left (2 b B^2 (b c-a d) g^2\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{8649 d^3}-\frac {\left (2 b B^2 (b c-a d) g^2\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{8649 d^3}+\frac {\left (2 b^2 B^2 (b c-a d) g^2\right ) \int \frac {\log ^2(c+d x)}{a+b x} \, dx}{8649 d^3}-\frac {\left (2 b^2 B^2 (b c-a d) g^2\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{8649 d^3}+\frac {\left (4 b^2 B^2 (b c-a d) g^2\right ) \int \frac {\log (a+b x) \log (c+d x)}{a+b x} \, dx}{8649 d^3}+\frac {\left (4 b^2 B^2 (b c-a d) g^2\right ) \int \frac {\log \left (\frac {1}{c+d x}\right ) \log (c+d x)}{a+b x} \, dx}{8649 d^3}-\frac {\left (4 A b B (b c-a d) g^2\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{8649 d^2}-\frac {\left (2 b B^2 (b c-a d) g^2\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{8649 d^2}-\frac {\left (2 b B^2 (b c-a d) g^2\right ) \int \frac {\log ^2(c+d x)}{c+d x} \, dx}{8649 d^2}+\frac {\left (4 b^2 B^2 (b c-a d) g^2 \left (-\log (a+b x)-\log \left (\frac {1}{c+d x}\right )+\log \left (\frac {e (a+b x)}{c+d x}\right )\right )\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{8649 d^3}\\ &=-\frac {2 B^2 (b c-a d)^2 g^2}{8649 d^3 (c+d x)}-\frac {2 b B^2 (b c-a d) g^2 \log (a+b x)}{8649 d^3}-\frac {a b B^2 g^2 \log ^2(a+b x)}{8649 d^2}-\frac {b B^2 (b c-a d) g^2 \log ^2(a+b x)}{8649 d^3}+\frac {2 B (b c-a d)^2 g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{8649 d^3 (c+d x)}+\frac {2 a b B g^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{8649 d^2}+\frac {2 b B (b c-a d) g^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{8649 d^3}+\frac {b^2 g^2 x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{8649 d^2}-\frac {(b c-a d)^2 g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{8649 d^3 (c+d x)}+\frac {2 b B^2 (b c-a d) g^2 \log (c+d x)}{8649 d^3}+\frac {2 b^2 B^2 c g^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{8649 d^3}+\frac {4 A b B (b c-a d) g^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{8649 d^3}+\frac {2 b B^2 (b c-a d) g^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{8649 d^3}-\frac {4 b B^2 (b c-a d) g^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \left (\log (a+b x)+\log \left (\frac {1}{c+d x}\right )-\log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{8649 d^3}-\frac {2 b^2 B c g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{8649 d^3}-\frac {2 b B (b c-a d) g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{8649 d^3}-\frac {2 b (b c-a d) g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{8649 d^3}-\frac {b^2 B^2 c g^2 \log ^2(c+d x)}{8649 d^3}-\frac {2 A b B (b c-a d) g^2 \log ^2(c+d x)}{8649 d^3}-\frac {b B^2 (b c-a d) g^2 \log ^2(c+d x)}{8649 d^3}+\frac {2 b B^2 (b c-a d) g^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(c+d x)}{8649 d^3}-\frac {2 b B^2 (b c-a d) g^2 \log \left (\frac {e (a+b x)}{c+d x}\right ) \log ^2(c+d x)}{8649 d^3}+\frac {2 a b B^2 g^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{8649 d^2}+\frac {2 b B^2 (b c-a d) g^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{8649 d^3}-\frac {\left (2 b^2 B^2 c g^2\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{8649 d^3}-\frac {\left (2 a b B^2 g^2\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{8649 d^2}-\frac {\left (4 A b B (b c-a d) g^2\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{8649 d^3}-\frac {\left (2 b B^2 (b c-a d) g^2\right ) \text {Subst}\left (\int \frac {\log ^2(x)}{x} \, dx,x,c+d x\right )}{8649 d^3}-\frac {\left (2 b B^2 (b c-a d) g^2\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{8649 d^3}-\frac {\left (2 b B^2 (b c-a d) g^2\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{8649 d^3}+\frac {\left (4 b B^2 (b c-a d) g^2\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 )}{8649 d^3}+\frac {\left (4 b B^2 (b c-a d) g^2\right ) \text {Subst}\left (\int \frac {\log \left (\frac {1}{-\frac {-b c+a d}{b}+\frac {d x}{b}}\right ) \log \left (-\frac {-b c+a d}{b}+\frac {d x}{b}\right )}{x} \, dx,x,a+b x\right )}{8649 d^3}-\frac {\left (4 b B^2 (b c-a d) g^2\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right ) \log (c+d x)}{c+d x} \, dx}{8649 d^2}-\frac {\left (4 b B^2 (b c-a d) g^2 \left (-\log (a+b x)-\log \left (\frac {1}{c+d x}\right )+\log \left (\frac {e (a+b x)}{c+d x}\right )\right )\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{8649 d^2}\\ &=-\frac {2 B^2 (b c-a d)^2 g^2}{8649 d^3 (c+d x)}-\frac {2 b B^2 (b c-a d) g^2 \log (a+b x)}{8649 d^3}-\frac {a b B^2 g^2 \log ^2(a+b x)}{8649 d^2}-\frac {b B^2 (b c-a d) g^2 \log ^2(a+b x)}{8649 d^3}+\frac {2 B (b c-a d)^2 g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{8649 d^3 (c+d x)}+\frac {2 a b B g^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{8649 d^2}+\frac {2 b B (b c-a d) g^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{8649 d^3}+\frac {b^2 g^2 x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{8649 d^2}-\frac {(b c-a d)^2 g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{8649 d^3 (c+d x)}+\frac {2 b B^2 (b c-a d) g^2 \log (c+d x)}{8649 d^3}+\frac {2 b B^2 (b c-a d) g^2 \log ^2(a+b x) \log (c+d x)}{8649 d^3}+\frac {2 b^2 B^2 c g^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{8649 d^3}+\frac {4 A b B (b c-a d) g^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{8649 d^3}+\frac {2 b B^2 (b c-a d) g^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{8649 d^3}+\frac {4 b B^2 (b c-a d) g^2 \log (a+b x) \log \left (\frac {1}{c+d x}\right ) \log (c+d x)}{8649 d^3}-\frac {4 b B^2 (b c-a d) g^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \left (\log (a+b x)+\log \left (\frac {1}{c+d x}\right )-\log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{8649 d^3}-\frac {2 b^2 B c g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{8649 d^3}-\frac {2 b B (b c-a d) g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{8649 d^3}-\frac {2 b (b c-a d) g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{8649 d^3}-\frac {b^2 B^2 c g^2 \log ^2(c+d x)}{8649 d^3}-\frac {2 A b B (b c-a d) g^2 \log ^2(c+d x)}{8649 d^3}-\frac {b B^2 (b c-a d) g^2 \log ^2(c+d x)}{8649 d^3}+\frac {2 b B^2 (b c-a d) g^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(c+d x)}{8649 d^3}-\frac {2 b B^2 (b c-a d) g^2 \log \left (\frac {e (a+b x)}{c+d x}\right ) \log ^2(c+d x)}{8649 d^3}+\frac {2 a b B^2 g^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{8649 d^2}+\frac {2 b B^2 (b c-a d) g^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{8649 d^3}+\frac {2 a b B^2 g^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{8649 d^2}+\frac {2 b B^2 (b c-a d) g^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{8649 d^3}+\frac {2 b^2 B^2 c g^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{8649 d^3}+\frac {4 A b B (b c-a d) g^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{8649 d^3}+\frac {2 b B^2 (b c-a d) g^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{8649 d^3}-\frac {\left (2 b B^2 (b c-a d) g^2\right ) \text {Subst}\left (\int x^2 \, dx,x,\log (c+d x)\right )}{8649 d^3}-\frac {\left (4 b B^2 (b c-a d) g^2\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 )}{8649 d^3}-\frac {\left (2 B^2 (b c-a d) g^2\right ) \text {Subst}\left (\int \frac {\log ^2(x)}{\frac {b c-a d}{b}+\frac {d x}{b}} \, dx,x,a+b x\right )}{8649 d^2}-\frac {\left (4 B^2 (b c-a d) g^2\right ) \text {Subst}\left (\int \frac {\log (x) \log \left (\frac {1}{-\frac {-b c+a d}{b}+\frac {d x}{b}}\right )}{-\frac {-b c+a d}{b}+\frac {d x}{b}} \, dx,x,a+b x\right )}{8649 d^2}+\frac {\left (4 B^2 (b c-a d) g^2\right ) \text {Subst}\left (\int \frac {\log (x) \log \left (-\frac {-b c+a d}{b}+\frac {d x}{b}\right )}{-\frac {-b c+a d}{b}+\frac {d x}{b}} \, dx,x,a+b x\right )}{8649 d^2}-\frac {\left (4 b B^2 (b c-a d) g^2 \left (-\log (a+b x)-\log \left (\frac {1}{c+d x}\right )+\log \left (\frac {e (a+b x)}{c+d x}\right )\right )\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{8649 d^3}\\ &=-\frac {2 B^2 (b c-a d)^2 g^2}{8649 d^3 (c+d x)}-\frac {2 b B^2 (b c-a d) g^2 \log (a+b x)}{8649 d^3}-\frac {a b B^2 g^2 \log ^2(a+b x)}{8649 d^2}-\frac {b B^2 (b c-a d) g^2 \log ^2(a+b x)}{8649 d^3}+\frac {2 B (b c-a d)^2 g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{8649 d^3 (c+d x)}+\frac {2 a b B g^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{8649 d^2}+\frac {2 b B (b c-a d) g^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{8649 d^3}+\frac {b^2 g^2 x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{8649 d^2}-\frac {(b c-a d)^2 g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{8649 d^3 (c+d x)}+\frac {2 b B^2 (b c-a d) g^2 \log (c+d x)}{8649 d^3}+\frac {2 b B^2 (b c-a d) g^2 \log ^2(a+b x) \log (c+d x)}{8649 d^3}+\frac {2 b^2 B^2 c g^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{8649 d^3}+\frac {4 A b B (b c-a d) g^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{8649 d^3}+\frac {2 b B^2 (b c-a d) g^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{8649 d^3}+\frac {4 b B^2 (b c-a d) g^2 \log (a+b x) \log \left (\frac {1}{c+d x}\right ) \log (c+d x)}{8649 d^3}-\frac {4 b B^2 (b c-a d) g^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \left (\log (a+b x)+\log \left (\frac {1}{c+d x}\right )-\log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{8649 d^3}-\frac {2 b^2 B c g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{8649 d^3}-\frac {2 b B (b c-a d) g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{8649 d^3}-\frac {2 b (b c-a d) g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{8649 d^3}-\frac {b^2 B^2 c g^2 \log ^2(c+d x)}{8649 d^3}-\frac {2 A b B (b c-a d) g^2 \log ^2(c+d x)}{8649 d^3}-\frac {b B^2 (b c-a d) g^2 \log ^2(c+d x)}{8649 d^3}+\frac {2 b B^2 (b c-a d) g^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(c+d x)}{8649 d^3}-\frac {2 b B^2 (b c-a d) g^2 \log \left (\frac {e (a+b x)}{c+d x}\right ) \log ^2(c+d x)}{8649 d^3}-\frac {2 b B^2 (b c-a d) g^2 \log ^3(c+d x)}{25947 d^3}+\frac {2 a b B^2 g^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{8649 d^2}+\frac {2 b B^2 (b c-a d) g^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{8649 d^3}-\frac {2 b B^2 (b c-a d) g^2 \log ^2(a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{8649 d^3}+\frac {2 a b B^2 g^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{8649 d^2}+\frac {2 b B^2 (b c-a d) g^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{8649 d^3}+\frac {2 b^2 B^2 c g^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{8649 d^3}+\frac {4 A b B (b c-a d) g^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{8649 d^3}+\frac {2 b B^2 (b c-a d) g^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{8649 d^3}-\frac {4 b B^2 (b c-a d) g^2 \left (\log (a+b x)+\log \left (\frac {1}{c+d x}\right )-\log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{8649 d^3}+\frac {4 b B^2 (b c-a d) g^2 \log (c+d x) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{8649 d^3}-\frac {\left (4 b B^2 (b c-a d) g^2\right ) \text {Subst}\left (\int \frac {\log \left (\frac {1}{x}\right ) \log \left (\frac {-b c+a d}{d}+\frac {b x}{d}\right )}{x} \, dx,x,c+d x\right )}{8649 d^3}+\frac {\left (4 b B^2 (b c-a d) g^2\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 )}{8649 d^3}+\frac {\left (4 b B^2 (b c-a d) g^2\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 )}{8649 d^3}-\frac {\left (4 b B^2 (b c-a d) g^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{8649 d^3}\\ &=-\frac {2 B^2 (b c-a d)^2 g^2}{8649 d^3 (c+d x)}-\frac {2 b B^2 (b c-a d) g^2 \log (a+b x)}{8649 d^3}-\frac {a b B^2 g^2 \log ^2(a+b x)}{8649 d^2}-\frac {b B^2 (b c-a d) g^2 \log ^2(a+b x)}{8649 d^3}+\frac {2 b B^2 (b c-a d) g^2 \log (a+b x) \log ^2\left (\frac {1}{c+d x}\right )}{8649 d^3}+\frac {2 B (b c-a d)^2 g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{8649 d^3 (c+d x)}+\frac {2 a b B g^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{8649 d^2}+\frac {2 b B (b c-a d) g^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{8649 d^3}+\frac {b^2 g^2 x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{8649 d^2}-\frac {(b c-a d)^2 g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{8649 d^3 (c+d x)}+\frac {2 b B^2 (b c-a d) g^2 \log (c+d x)}{8649 d^3}+\frac {2 b B^2 (b c-a d) g^2 \log ^2(a+b x) \log (c+d x)}{8649 d^3}+\frac {2 b^2 B^2 c g^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{8649 d^3}+\frac {4 A b B (b c-a d) g^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{8649 d^3}+\frac {2 b B^2 (b c-a d) g^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{8649 d^3}+\frac {4 b B^2 (b c-a d) g^2 \log (a+b x) \log \left (\frac {1}{c+d x}\right ) \log (c+d x)}{8649 d^3}-\frac {4 b B^2 (b c-a d) g^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \left (\log (a+b x)+\log \left (\frac {1}{c+d x}\right )-\log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{8649 d^3}-\frac {2 b^2 B c g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{8649 d^3}-\frac {2 b B (b c-a d) g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{8649 d^3}-\frac {2 b (b c-a d) g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{8649 d^3}-\frac {b^2 B^2 c g^2 \log ^2(c+d x)}{8649 d^3}-\frac {2 A b B (b c-a d) g^2 \log ^2(c+d x)}{8649 d^3}-\frac {b B^2 (b c-a d) g^2 \log ^2(c+d x)}{8649 d^3}+\frac {2 b B^2 (b c-a d) g^2 \log (a+b x) \log ^2(c+d x)}{8649 d^3}+\frac {2 b B^2 (b c-a d) g^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(c+d x)}{8649 d^3}-\frac {2 b B^2 (b c-a d) g^2 \log \left (\frac {e (a+b x)}{c+d x}\right ) \log ^2(c+d x)}{8649 d^3}-\frac {2 b B^2 (b c-a d) g^2 \log ^3(c+d x)}{25947 d^3}+\frac {2 a b B^2 g^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{8649 d^2}+\frac {2 b B^2 (b c-a d) g^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{8649 d^3}-\frac {2 b B^2 (b c-a d) g^2 \log ^2(a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{8649 d^3}+\frac {2 a b B^2 g^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{8649 d^2}+\frac {2 b B^2 (b c-a d) g^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{8649 d^3}-\frac {4 b B^2 (b c-a d) g^2 \log (a+b x) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{8649 d^3}+\frac {2 b^2 B^2 c g^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{8649 d^3}+\frac {4 A b B (b c-a d) g^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{8649 d^3}+\frac {2 b B^2 (b c-a d) g^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{8649 d^3}-\frac {4 b B^2 (b c-a d) g^2 \left (\log (a+b x)+\log \left (\frac {1}{c+d x}\right )-\log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{8649 d^3}+\frac {4 b B^2 (b c-a d) g^2 \log (c+d x) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{8649 d^3}-\frac {4 b B^2 (b c-a d) g^2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{8649 d^3}-\frac {\left (2 b^2 B^2 (b c-a d) g^2\right ) \text {Subst}\left (\int \frac {\log ^2\left (\frac {1}{x}\right )}{\frac {-b c+a d}{d}+\frac {b x}{d}} \, dx,x,c+d x\right )}{8649 d^4}-\frac {\left (2 b^2 B^2 (b c-a d) g^2\right ) \text {Subst}\left (\int \frac {\log ^2(x)}{\frac {-b c+a d}{d}+\frac {b x}{d}} \, dx,x,c+d x\right )}{8649 d^4}+\frac {\left (4 b B^2 (b c-a d) g^2\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 )}{8649 d^3}\\ &=-\frac {2 B^2 (b c-a d)^2 g^2}{8649 d^3 (c+d x)}-\frac {2 b B^2 (b c-a d) g^2 \log (a+b x)}{8649 d^3}-\frac {a b B^2 g^2 \log ^2(a+b x)}{8649 d^2}-\frac {b B^2 (b c-a d) g^2 \log ^2(a+b x)}{8649 d^3}+\frac {2 b B^2 (b c-a d) g^2 \log (a+b x) \log ^2\left (\frac {1}{c+d x}\right )}{8649 d^3}-\frac {2 b B^2 (b c-a d) g^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2\left (\frac {1}{c+d x}\right )}{8649 d^3}+\frac {2 B (b c-a d)^2 g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{8649 d^3 (c+d x)}+\frac {2 a b B g^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{8649 d^2}+\frac {2 b B (b c-a d) g^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{8649 d^3}+\frac {b^2 g^2 x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{8649 d^2}-\frac {(b c-a d)^2 g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{8649 d^3 (c+d x)}+\frac {2 b B^2 (b c-a d) g^2 \log (c+d x)}{8649 d^3}+\frac {2 b B^2 (b c-a d) g^2 \log ^2(a+b x) \log (c+d x)}{8649 d^3}+\frac {2 b^2 B^2 c g^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{8649 d^3}+\frac {4 A b B (b c-a d) g^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{8649 d^3}+\frac {2 b B^2 (b c-a d) g^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{8649 d^3}+\frac {4 b B^2 (b c-a d) g^2 \log (a+b x) \log \left (\frac {1}{c+d x}\right ) \log (c+d x)}{8649 d^3}-\frac {4 b B^2 (b c-a d) g^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \left (\log (a+b x)+\log \left (\frac {1}{c+d x}\right )-\log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{8649 d^3}-\frac {2 b^2 B c g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{8649 d^3}-\frac {2 b B (b c-a d) g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{8649 d^3}-\frac {2 b (b c-a d) g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{8649 d^3}-\frac {b^2 B^2 c g^2 \log ^2(c+d x)}{8649 d^3}-\frac {2 A b B (b c-a d) g^2 \log ^2(c+d x)}{8649 d^3}-\frac {b B^2 (b c-a d) g^2 \log ^2(c+d x)}{8649 d^3}+\frac {2 b B^2 (b c-a d) g^2 \log (a+b x) \log ^2(c+d x)}{8649 d^3}-\frac {2 b B^2 (b c-a d) g^2 \log \left (\frac {e (a+b x)}{c+d x}\right ) \log ^2(c+d x)}{8649 d^3}-\frac {2 b B^2 (b c-a d) g^2 \log ^3(c+d x)}{25947 d^3}+\frac {2 a b B^2 g^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{8649 d^2}+\frac {2 b B^2 (b c-a d) g^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{8649 d^3}-\frac {2 b B^2 (b c-a d) g^2 \log ^2(a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{8649 d^3}+\frac {2 a b B^2 g^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{8649 d^2}+\frac {2 b B^2 (b c-a d) g^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{8649 d^3}-\frac {4 b B^2 (b c-a d) g^2 \log (a+b x) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{8649 d^3}+\frac {2 b^2 B^2 c g^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{8649 d^3}+\frac {4 A b B (b c-a d) g^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{8649 d^3}+\frac {2 b B^2 (b c-a d) g^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{8649 d^3}-\frac {4 b B^2 (b c-a d) g^2 \left (\log (a+b x)+\log \left (\frac {1}{c+d x}\right )-\log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{8649 d^3}+\frac {4 b B^2 (b c-a d) g^2 \log (c+d x) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{8649 d^3}+\frac {4 b B^2 (b c-a d) g^2 \text {Li}_3\left (-\frac {d (a+b x)}{b c-a d}\right )}{8649 d^3}-\frac {4 b B^2 (b c-a d) g^2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{8649 d^3}-\frac {\left (4 b B^2 (b c-a d) g^2\right ) \text {Subst}\left (\int \frac {\log \left (\frac {1}{x}\right ) \log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{8649 d^3}+\frac {\left (4 b B^2 (b c-a d) g^2\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 )}{8649 d^3}\\ &=-\frac {2 B^2 (b c-a d)^2 g^2}{8649 d^3 (c+d x)}-\frac {2 b B^2 (b c-a d) g^2 \log (a+b x)}{8649 d^3}-\frac {a b B^2 g^2 \log ^2(a+b x)}{8649 d^2}-\frac {b B^2 (b c-a d) g^2 \log ^2(a+b x)}{8649 d^3}+\frac {2 b B^2 (b c-a d) g^2 \log (a+b x) \log ^2\left (\frac {1}{c+d x}\right )}{8649 d^3}-\frac {2 b B^2 (b c-a d) g^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2\left (\frac {1}{c+d x}\right )}{8649 d^3}+\frac {2 B (b c-a d)^2 g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{8649 d^3 (c+d x)}+\frac {2 a b B g^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{8649 d^2}+\frac {2 b B (b c-a d) g^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{8649 d^3}+\frac {b^2 g^2 x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{8649 d^2}-\frac {(b c-a d)^2 g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{8649 d^3 (c+d x)}+\frac {2 b B^2 (b c-a d) g^2 \log (c+d x)}{8649 d^3}+\frac {2 b B^2 (b c-a d) g^2 \log ^2(a+b x) \log (c+d x)}{8649 d^3}+\frac {2 b^2 B^2 c g^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{8649 d^3}+\frac {4 A b B (b c-a d) g^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{8649 d^3}+\frac {2 b B^2 (b c-a d) g^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{8649 d^3}+\frac {4 b B^2 (b c-a d) g^2 \log (a+b x) \log \left (\frac {1}{c+d x}\right ) \log (c+d x)}{8649 d^3}-\frac {4 b B^2 (b c-a d) g^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \left (\log (a+b x)+\log \left (\frac {1}{c+d x}\right )-\log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{8649 d^3}-\frac {2 b^2 B c g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{8649 d^3}-\frac {2 b B (b c-a d) g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{8649 d^3}-\frac {2 b (b c-a d) g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{8649 d^3}-\frac {b^2 B^2 c g^2 \log ^2(c+d x)}{8649 d^3}-\frac {2 A b B (b c-a d) g^2 \log ^2(c+d x)}{8649 d^3}-\frac {b B^2 (b c-a d) g^2 \log ^2(c+d x)}{8649 d^3}+\frac {2 b B^2 (b c-a d) g^2 \log (a+b x) \log ^2(c+d x)}{8649 d^3}-\frac {2 b B^2 (b c-a d) g^2 \log \left (\frac {e (a+b x)}{c+d x}\right ) \log ^2(c+d x)}{8649 d^3}-\frac {2 b B^2 (b c-a d) g^2 \log ^3(c+d x)}{25947 d^3}+\frac {2 a b B^2 g^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{8649 d^2}+\frac {2 b B^2 (b c-a d) g^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{8649 d^3}-\frac {2 b B^2 (b c-a d) g^2 \log ^2(a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{8649 d^3}+\frac {2 a b B^2 g^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{8649 d^2}+\frac {2 b B^2 (b c-a d) g^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{8649 d^3}-\frac {4 b B^2 (b c-a d) g^2 \log (a+b x) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{8649 d^3}+\frac {2 b^2 B^2 c g^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{8649 d^3}+\frac {4 A b B (b c-a d) g^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{8649 d^3}+\frac {2 b B^2 (b c-a d) g^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{8649 d^3}+\frac {4 b B^2 (b c-a d) g^2 \log \left (\frac {1}{c+d x}\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{8649 d^3}-\frac {4 b B^2 (b c-a d) g^2 \left (\log (a+b x)+\log \left (\frac {1}{c+d x}\right )-\log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{8649 d^3}+\frac {4 b B^2 (b c-a d) g^2 \text {Li}_3\left (-\frac {d (a+b x)}{b c-a d}\right )}{8649 d^3}-\frac {4 b B^2 (b c-a d) g^2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{8649 d^3}+2 \frac {\left (4 b B^2 (b c-a d) g^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{8649 d^3}\\ &=-\frac {2 B^2 (b c-a d)^2 g^2}{8649 d^3 (c+d x)}-\frac {2 b B^2 (b c-a d) g^2 \log (a+b x)}{8649 d^3}-\frac {a b B^2 g^2 \log ^2(a+b x)}{8649 d^2}-\frac {b B^2 (b c-a d) g^2 \log ^2(a+b x)}{8649 d^3}+\frac {2 b B^2 (b c-a d) g^2 \log (a+b x) \log ^2\left (\frac {1}{c+d x}\right )}{8649 d^3}-\frac {2 b B^2 (b c-a d) g^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2\left (\frac {1}{c+d x}\right )}{8649 d^3}+\frac {2 B (b c-a d)^2 g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{8649 d^3 (c+d x)}+\frac {2 a b B g^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{8649 d^2}+\frac {2 b B (b c-a d) g^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{8649 d^3}+\frac {b^2 g^2 x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{8649 d^2}-\frac {(b c-a d)^2 g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{8649 d^3 (c+d x)}+\frac {2 b B^2 (b c-a d) g^2 \log (c+d x)}{8649 d^3}+\frac {2 b B^2 (b c-a d) g^2 \log ^2(a+b x) \log (c+d x)}{8649 d^3}+\frac {2 b^2 B^2 c g^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{8649 d^3}+\frac {4 A b B (b c-a d) g^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{8649 d^3}+\frac {2 b B^2 (b c-a d) g^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{8649 d^3}+\frac {4 b B^2 (b c-a d) g^2 \log (a+b x) \log \left (\frac {1}{c+d x}\right ) \log (c+d x)}{8649 d^3}-\frac {4 b B^2 (b c-a d) g^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \left (\log (a+b x)+\log \left (\frac {1}{c+d x}\right )-\log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{8649 d^3}-\frac {2 b^2 B c g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{8649 d^3}-\frac {2 b B (b c-a d) g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{8649 d^3}-\frac {2 b (b c-a d) g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{8649 d^3}-\frac {b^2 B^2 c g^2 \log ^2(c+d x)}{8649 d^3}-\frac {2 A b B (b c-a d) g^2 \log ^2(c+d x)}{8649 d^3}-\frac {b B^2 (b c-a d) g^2 \log ^2(c+d x)}{8649 d^3}+\frac {2 b B^2 (b c-a d) g^2 \log (a+b x) \log ^2(c+d x)}{8649 d^3}-\frac {2 b B^2 (b c-a d) g^2 \log \left (\frac {e (a+b x)}{c+d x}\right ) \log ^2(c+d x)}{8649 d^3}-\frac {2 b B^2 (b c-a d) g^2 \log ^3(c+d x)}{25947 d^3}+\frac {2 a b B^2 g^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{8649 d^2}+\frac {2 b B^2 (b c-a d) g^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{8649 d^3}-\frac {2 b B^2 (b c-a d) g^2 \log ^2(a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{8649 d^3}+\frac {2 a b B^2 g^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{8649 d^2}+\frac {2 b B^2 (b c-a d) g^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{8649 d^3}-\frac {4 b B^2 (b c-a d) g^2 \log (a+b x) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{8649 d^3}+\frac {2 b^2 B^2 c g^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{8649 d^3}+\frac {4 A b B (b c-a d) g^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{8649 d^3}+\frac {2 b B^2 (b c-a d) g^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{8649 d^3}+\frac {4 b B^2 (b c-a d) g^2 \log \left (\frac {1}{c+d x}\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{8649 d^3}-\frac {4 b B^2 (b c-a d) g^2 \left (\log (a+b x)+\log \left (\frac {1}{c+d x}\right )-\log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{8649 d^3}+\frac {4 b B^2 (b c-a d) g^2 \text {Li}_3\left (-\frac {d (a+b x)}{b c-a d}\right )}{8649 d^3}+\frac {4 b B^2 (b c-a d) g^2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{8649 d^3}\\ \end {align*}

________________________________________________________________________________________

Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(1969\) vs. \(2(469)=938\).
time = 2.40, size = 1969, normalized size = 4.20 \begin {gather*} \frac {g^2 \left (A^2 b^2 d x-\frac {A^2 (b c-a d)^2}{c+d x}+2 A^2 b (-b c+a d) \log (c+d x)+\frac {2 a^2 A B d^2 \left (b c-a d+b (c+d x) \log \left (\frac {a}{b}+x\right )+(-b c+a d) \log \left (\frac {e (a+b x)}{c+d x}\right )-b c \log \left (\frac {b (c+d x)}{b c-a d}\right )-b d x \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )}{(b c-a d) (c+d x)}+2 a A b B d \left (-\log ^2\left (\frac {c}{d}+x\right )+2 \log \left (\frac {c}{d}+x\right ) \log (c+d x)+2 \left (-\frac {c}{c+d x}+\frac {b c \log (a+b x)}{-b c+a d}+\frac {b c \log (c+d x)}{b c-a d}-\log \left (\frac {a}{b}+x\right ) \log (c+d x)+\log \left (\frac {e (a+b x)}{c+d x}\right ) \left (\frac {c}{c+d x}+\log (c+d x)\right )+\log \left (\frac {a}{b}+x\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )+2 \text {Li}_2\left (\frac {d (a+b x)}{-b c+a d}\right )\right )+2 A b^2 B \left (d \left (\frac {a}{b}+x\right ) \left (-1+\log \left (\frac {a}{b}+x\right )\right )-\frac {c^2 \log \left (\frac {a}{b}+x\right )}{c+d x}-(c+d x) \left (-1+\log \left (\frac {c}{d}+x\right )\right )+c \log ^2\left (\frac {c}{d}+x\right )+\frac {c^2 \left (1+\log \left (\frac {c}{d}+x\right )\right )}{c+d x}+\frac {b c^2 (\log (a+b x)-\log (c+d x))}{b c-a d}+\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 ) \left (d x-\frac {c^2}{c+d x}-2 c \log (c+d x)\right )-2 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 )-\frac {a^2 B^2 d^2 \left (2 b c-2 a d+2 b (c+d x) \log (a+b x)-2 (b c-a d) \log \left (\frac {e (a+b x)}{c+d x}\right )-2 b (c+d x) \log (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )+(b c-a d) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )-2 b (c+d x) \log (c+d x)-2 b (c+d x) \log \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+d x) \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 \text {Li}_2\left (\frac {d (a+b x)}{-b c+a d}\right )\right )+b (c+d x) \left (\log \left (\frac {b c-a d}{b c+b d x}\right ) \left (2 \log \left (\frac {d (a+b x)}{-b c+a d}\right )+\log \left (\frac {b c-a d}{b c+b d x}\right )\right )-2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )\right )\right )}{(b c-a d) (c+d x)}+b^2 B^2 \left (\frac {d (a+b x) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{b}-\frac {c^2 \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{c+d x}+2 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 )-\frac {c^2 \left (2 b c-2 a d+2 b (c+d x) \log (a+b x)-2 (b c-a d) \log \left (\frac {e (a+b x)}{c+d x}\right )-2 b (c+d x) \log (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )-2 b (c+d x) \log (c+d x)-2 b (c+d x) \log \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+d x) \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 \text {Li}_2\left (\frac {d (a+b x)}{-b c+a d}\right )\right )+b (c+d x) \left (\log \left (\frac {b c-a d}{b c+b d x}\right ) \left (2 \log \left (\frac {d (a+b x)}{-b c+a d}\right )+\log \left (\frac {b c-a d}{b c+b d x}\right )\right )-2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )\right )\right )}{(b c-a d) (c+d x)}-\frac {(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 )}{b}+4 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 )+2 a b B^2 d \left (\frac {c \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{c+d x}-\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 )+\frac {c \left (2 b c-2 a d+2 b (c+d x) \log (a+b x)-2 (b c-a d) \log \left (\frac {e (a+b x)}{c+d x}\right )-2 b (c+d x) \log (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )-2 b (c+d x) \log (c+d x)-2 b (c+d x) \log \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+d x) \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 \text {Li}_2\left (\frac {d (a+b x)}{-b c+a d}\right )\right )+b (c+d x) \left (\log \left (\frac {b c-a d}{b c+b d x}\right ) \left (2 \log \left (\frac {d (a+b x)}{-b c+a d}\right )+\log \left (\frac {b c-a d}{b c+b d x}\right )\right )-2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )\right )\right )}{(b c-a d) (c+d x)}+2 \text {Li}_3\left (\frac {d (a+b x)}{b (c+d x)}\right )\right )\right )}{d^3 i^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [F]
time = 0.26, size = 0, normalized size = 0.00 \[\int \frac {\left (b g x +a g \right )^{2} \left (A +B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )\right )^{2}}{\left (d i x +c i \right )^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________