3.161 \(\int (a g+b g x) (c i+d i x) (A+B \log (e (\frac{a+b x}{c+d x})^n))^2 \, dx\)

Optimal. Leaf size=372 \[ -\frac{B^2 g i n^2 (b c-a d)^3 \text{PolyLog}\left (2,\frac{d (a+b x)}{b (c+d x)}\right )}{3 b^2 d^2}-\frac{B g i n (b c-a d)^3 \log \left (\frac{b c-a d}{b (c+d x)}\right ) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A+B n\right )}{3 b^2 d^2}-\frac{B g i n (a+b x) (b c-a d)^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{3 b^2 d}+\frac{g i (a+b x)^2 (b c-a d) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )^2}{6 b^2}-\frac{B g i n (a+b x)^2 (b c-a d) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{3 b^2}+\frac{g i (a+b x)^2 (c+d x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )^2}{3 b}-\frac{B^2 g i n^2 (b c-a d)^3 \log (c+d x)}{3 b^2 d^2}+\frac{B^2 g i n^2 x (b c-a d)^2}{3 b d} \]

[Out]

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

________________________________________________________________________________________

Rubi [B]  time = 2.87827, antiderivative size = 1323, normalized size of antiderivative = 3.56, number of steps used = 72, number of rules used = 14, integrand size = 41, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.342, Rules used = {2528, 2523, 12, 2524, 2418, 2390, 2301, 2394, 2393, 2391, 2525, 2486, 31, 72} \[ -\frac{B^2 d g i n^2 \log ^2(a+b x) a^3}{3 b^2}+\frac{2 B d g i n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) a^3}{3 b^2}+\frac{2 B^2 d g i n^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right ) a^3}{3 b^2}+\frac{2 B^2 d g i n^2 \text{PolyLog}\left (2,-\frac{d (a+b x)}{b c-a d}\right ) a^3}{3 b^2}-\frac{B^2 c g i n^2 \log ^2(a+b x) a^2}{b}+\frac{B^2 (b c+a d) g i n^2 \log ^2(a+b x) a^2}{2 b^2}+\frac{B^2 (b c-a d) g i n^2 \log (a+b x) a^2}{3 b^2}+\frac{2 B c g i n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) a^2}{b}-\frac{B (b c+a d) g i n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) a^2}{b^2}+\frac{2 B^2 c g i n^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right ) a^2}{b}-\frac{B^2 (b c+a d) g i n^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right ) a^2}{b^2}+\frac{2 B^2 c g i n^2 \text{PolyLog}\left (2,-\frac{d (a+b x)}{b c-a d}\right ) a^2}{b}-\frac{B^2 (b c+a d) g i n^2 \text{PolyLog}\left (2,-\frac{d (a+b x)}{b c-a d}\right ) a^2}{b^2}+c g i x \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 a-\frac{B^2 c^2 g i n^2 \log ^2(c+d x) a}{d}+\frac{2 B^2 c^2 g i n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x) a}{d}-\frac{2 B c^2 g i n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x) a}{d}+\frac{2 B^2 c^2 g i n^2 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right ) a}{d}+\frac{1}{3} b d g i x^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2+\frac{1}{2} (b c+a d) g i x^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2+\frac{B^2 c^2 (b c+a d) g i n^2 \log ^2(c+d x)}{2 d^2}-\frac{b B^2 c^3 g i n^2 \log ^2(c+d x)}{3 d^2}+\frac{B^2 (b c-a d)^2 g i n^2 x}{3 b d}-\frac{2}{3} A b B \left (\frac{a^2}{b^2}-\frac{c^2}{d^2}\right ) d g i n x-\frac{A B (b c-a d) (b c+a d) g i n x}{b d}-\frac{B^2 (b c-a d) (b c+a d) g i n (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{3 b^2 d}-\frac{1}{3} B (b c-a d) g i n x^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )-\frac{B^2 c^2 (b c-a d) g i n^2 \log (c+d x)}{3 d^2}+\frac{B^2 (b c-a d)^2 (b c+a d) g i n^2 \log (c+d x)}{3 b^2 d^2}-\frac{B^2 c^2 (b c+a d) g i n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{d^2}+\frac{2 b B^2 c^3 g i n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{3 d^2}+\frac{B c^2 (b c+a d) g i n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{d^2}-\frac{2 b B c^3 g i n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{3 d^2}-\frac{B^2 c^2 (b c+a d) g i n^2 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )}{d^2}+\frac{2 b B^2 c^3 g i n^2 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )}{3 d^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2528

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)*(RGx_), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*Log[c*
RFx^p])^n, RGx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, p}, x] && RationalFunctionQ[RFx, x] && RationalF
unctionQ[RGx, x] && IGtQ[n, 0]

Rule 2523

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*Log[c*RFx^p])^n, x] - Dist[b*n*p
, Int[SimplifyIntegrand[(x*(a + b*Log[c*RFx^p])^(n - 1)*D[RFx, x])/RFx, x], x], x] /; FreeQ[{a, b, c, p}, x] &
& RationalFunctionQ[RFx, x] && IGtQ[n, 0]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 2524

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

Rule 2418

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*(RFx_), x_Symbol] :> With[{u = ExpandIntegrand[
(a + b*Log[c*(d + e*x)^n])^p, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, n}, x] && RationalFunct
ionQ[RFx, x] && IntegerQ[p]

Rule 2390

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

Rule 2301

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]

Rule 2394

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

Rule 2393

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Dist[1/g, Subst[Int[(a +
b*Log[1 + (c*e*x)/g])/x, x], x, f + g*x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && EqQ[g
+ c*(e*f - d*g), 0]

Rule 2391

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 2525

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)*((d_.) + (e_.)*(x_))^(m_.), x_Symbol] :> Simp[((d + e*x)^(m
+ 1)*(a + b*Log[c*RFx^p])^n)/(e*(m + 1)), x] - Dist[(b*n*p)/(e*(m + 1)), Int[SimplifyIntegrand[((d + e*x)^(m +
1)*(a + b*Log[c*RFx^p])^(n - 1)*D[RFx, x])/RFx, x], x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && RationalFunc
tionQ[RFx, x] && IGtQ[n, 0] && (EqQ[n, 1] || IntegerQ[m]) && NeQ[m, -1]

Rule 2486

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

Rule 31

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

Rule 72

Int[((e_.) + (f_.)*(x_))^(p_.)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Int[ExpandIntegrand[(
e + f*x)^p/((a + b*x)*(c + d*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IntegerQ[p]

Rubi steps

\begin{align*} \int (161 c+161 d x) (a g+b g x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \, dx &=\int \left (161 a c g \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2+161 (b c+a d) g x \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2+161 b d g x^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2\right ) \, dx\\ &=(161 a c g) \int \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \, dx+(161 b d g) \int x^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \, dx+(161 (b c+a d) g) \int x \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \, dx\\ &=161 a c g x \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2+\frac{161}{2} (b c+a d) g x^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2+\frac{161}{3} b d g x^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2-(322 a B c g n) \int \frac{(b c-a d) x \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{(a+b x) (c+d x)} \, dx-\frac{1}{3} (322 b B d g n) \int \frac{(b c-a d) x^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{(a+b x) (c+d x)} \, dx-(161 B (b c+a d) g n) \int \frac{(b c-a d) x^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{(a+b x) (c+d x)} \, dx\\ &=161 a c g x \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2+\frac{161}{2} (b c+a d) g x^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2+\frac{161}{3} b d g x^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2-(322 a B c (b c-a d) g n) \int \frac{x \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{(a+b x) (c+d x)} \, dx-\frac{1}{3} (322 b B d (b c-a d) g n) \int \frac{x^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{(a+b x) (c+d x)} \, dx-(161 B (b c-a d) (b c+a d) g n) \int \frac{x^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{(a+b x) (c+d x)} \, dx\\ &=161 a c g x \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2+\frac{161}{2} (b c+a d) g x^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2+\frac{161}{3} b d g x^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2-(322 a B c (b c-a d) g n) \int \left (-\frac{a \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d) (a+b x)}+\frac{c \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d) (c+d x)}\right ) \, dx-\frac{1}{3} (322 b B d (b c-a d) g n) \int \left (\frac{(-b c-a d) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^2 d^2}+\frac{x \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b d}-\frac{a^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^2 (b c-a d) (a+b x)}-\frac{c^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{d^2 (-b c+a d) (c+d x)}\right ) \, dx-(161 B (b c-a d) (b c+a d) g n) \int \left (\frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b d}+\frac{a^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b (b c-a d) (a+b x)}+\frac{c^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{d (-b c+a d) (c+d x)}\right ) \, dx\\ &=161 a c g x \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2+\frac{161}{2} (b c+a d) g x^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2+\frac{161}{3} b d g x^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2+\left (322 a^2 B c g n\right ) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{a+b x} \, dx-\left (322 a B c^2 g n\right ) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{c+d x} \, dx-\frac{\left (322 b B c^3 g n\right ) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{c+d x} \, dx}{3 d}+\frac{\left (322 a^3 B d g n\right ) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{a+b x} \, dx}{3 b}-\frac{1}{3} (322 B (b c-a d) g n) \int x \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \, dx-\frac{\left (161 a^2 B (b c+a d) g n\right ) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{a+b x} \, dx}{b}+\frac{\left (161 B c^2 (b c+a d) g n\right ) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{c+d x} \, dx}{d}+\frac{(322 B (b c-a d) (b c+a d) g n) \int \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \, dx}{3 b d}-\frac{(161 B (b c-a d) (b c+a d) g n) \int \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \, dx}{b d}\\ &=-\frac{161 A B (b c-a d) (b c+a d) g n x}{3 b d}-\frac{161}{3} B (b c-a d) g n x^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )+\frac{322 a^2 B c g n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b}+\frac{322 a^3 B d g n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3 b^2}-\frac{161 a^2 B (b c+a d) g n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^2}+161 a c g x \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2+\frac{161}{2} (b c+a d) g x^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2+\frac{161}{3} b d g x^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2-\frac{322 b B c^3 g n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{3 d^2}-\frac{322 a B c^2 g n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{d}+\frac{161 B c^2 (b c+a d) g n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{d^2}+\frac{\left (322 B^2 (b c-a d) (b c+a d) g n\right ) \int \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \, dx}{3 b d}-\frac{\left (161 B^2 (b c-a d) (b c+a d) g n\right ) \int \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \, dx}{b d}-\frac{\left (322 a^2 B^2 c g n^2\right ) \int \frac{(c+d x) \left (-\frac{d (a+b x)}{(c+d x)^2}+\frac{b}{c+d x}\right ) \log (a+b x)}{a+b x} \, dx}{b}+\frac{\left (322 b B^2 c^3 g n^2\right ) \int \frac{(c+d x) \left (-\frac{d (a+b x)}{(c+d x)^2}+\frac{b}{c+d x}\right ) \log (c+d x)}{a+b x} \, dx}{3 d^2}+\frac{\left (322 a B^2 c^2 g n^2\right ) \int \frac{(c+d x) \left (-\frac{d (a+b x)}{(c+d x)^2}+\frac{b}{c+d x}\right ) \log (c+d x)}{a+b x} \, dx}{d}-\frac{\left (322 a^3 B^2 d g n^2\right ) \int \frac{(c+d x) \left (-\frac{d (a+b x)}{(c+d x)^2}+\frac{b}{c+d x}\right ) \log (a+b x)}{a+b x} \, dx}{3 b^2}+\frac{1}{3} \left (161 B^2 (b c-a d) g n^2\right ) \int \frac{(b c-a d) x^2}{(a+b x) (c+d x)} \, dx+\frac{\left (161 a^2 B^2 (b c+a d) g n^2\right ) \int \frac{(c+d x) \left (-\frac{d (a+b x)}{(c+d x)^2}+\frac{b}{c+d x}\right ) \log (a+b x)}{a+b x} \, dx}{b^2}-\frac{\left (161 B^2 c^2 (b c+a d) g n^2\right ) \int \frac{(c+d x) \left (-\frac{d (a+b x)}{(c+d x)^2}+\frac{b}{c+d x}\right ) \log (c+d x)}{a+b x} \, dx}{d^2}\\ &=-\frac{161 A B (b c-a d) (b c+a d) g n x}{3 b d}-\frac{161 B^2 (b c-a d) (b c+a d) g n (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{3 b^2 d}-\frac{161}{3} B (b c-a d) g n x^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )+\frac{322 a^2 B c g n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b}+\frac{322 a^3 B d g n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3 b^2}-\frac{161 a^2 B (b c+a d) g n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^2}+161 a c g x \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2+\frac{161}{2} (b c+a d) g x^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2+\frac{161}{3} b d g x^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2-\frac{322 b B c^3 g n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{3 d^2}-\frac{322 a B c^2 g n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{d}+\frac{161 B c^2 (b c+a d) g n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{d^2}-\frac{\left (322 a^2 B^2 c g n^2\right ) \int \left (\frac{b \log (a+b x)}{a+b x}-\frac{d \log (a+b x)}{c+d x}\right ) \, dx}{b}+\frac{\left (322 b B^2 c^3 g n^2\right ) \int \left (\frac{b \log (c+d x)}{a+b x}-\frac{d \log (c+d x)}{c+d x}\right ) \, dx}{3 d^2}+\frac{\left (322 a B^2 c^2 g n^2\right ) \int \left (\frac{b \log (c+d x)}{a+b x}-\frac{d \log (c+d x)}{c+d x}\right ) \, dx}{d}-\frac{\left (322 a^3 B^2 d g n^2\right ) \int \left (\frac{b \log (a+b x)}{a+b x}-\frac{d \log (a+b x)}{c+d x}\right ) \, dx}{3 b^2}+\frac{1}{3} \left (161 B^2 (b c-a d)^2 g n^2\right ) \int \frac{x^2}{(a+b x) (c+d x)} \, dx+\frac{\left (161 a^2 B^2 (b c+a d) g n^2\right ) \int \left (\frac{b \log (a+b x)}{a+b x}-\frac{d \log (a+b x)}{c+d x}\right ) \, dx}{b^2}-\frac{\left (161 B^2 c^2 (b c+a d) g n^2\right ) \int \left (\frac{b \log (c+d x)}{a+b x}-\frac{d \log (c+d x)}{c+d x}\right ) \, dx}{d^2}-\frac{\left (322 B^2 (b c-a d)^2 (b c+a d) g n^2\right ) \int \frac{1}{c+d x} \, dx}{3 b^2 d}+\frac{\left (161 B^2 (b c-a d)^2 (b c+a d) g n^2\right ) \int \frac{1}{c+d x} \, dx}{b^2 d}\\ &=-\frac{161 A B (b c-a d) (b c+a d) g n x}{3 b d}-\frac{161 B^2 (b c-a d) (b c+a d) g n (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{3 b^2 d}-\frac{161}{3} B (b c-a d) g n x^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )+\frac{322 a^2 B c g n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b}+\frac{322 a^3 B d g n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3 b^2}-\frac{161 a^2 B (b c+a d) g n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^2}+161 a c g x \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2+\frac{161}{2} (b c+a d) g x^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2+\frac{161}{3} b d g x^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2+\frac{161 B^2 (b c-a d)^2 (b c+a d) g n^2 \log (c+d x)}{3 b^2 d^2}-\frac{322 b B c^3 g n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{3 d^2}-\frac{322 a B c^2 g n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{d}+\frac{161 B c^2 (b c+a d) g n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{d^2}-\left (322 a^2 B^2 c g n^2\right ) \int \frac{\log (a+b x)}{a+b x} \, dx-\left (322 a B^2 c^2 g n^2\right ) \int \frac{\log (c+d x)}{c+d x} \, dx+\frac{\left (322 b^2 B^2 c^3 g n^2\right ) \int \frac{\log (c+d x)}{a+b x} \, dx}{3 d^2}+\frac{\left (322 a b B^2 c^2 g n^2\right ) \int \frac{\log (c+d x)}{a+b x} \, dx}{d}-\frac{\left (322 b B^2 c^3 g n^2\right ) \int \frac{\log (c+d x)}{c+d x} \, dx}{3 d}-\frac{\left (322 a^3 B^2 d g n^2\right ) \int \frac{\log (a+b x)}{a+b x} \, dx}{3 b}+\frac{\left (322 a^2 B^2 c d g n^2\right ) \int \frac{\log (a+b x)}{c+d x} \, dx}{b}+\frac{\left (322 a^3 B^2 d^2 g n^2\right ) \int \frac{\log (a+b x)}{c+d x} \, dx}{3 b^2}+\frac{1}{3} \left (161 B^2 (b c-a d)^2 g n^2\right ) \int \left (\frac{1}{b d}+\frac{a^2}{b (b c-a d) (a+b x)}+\frac{c^2}{d (-b c+a d) (c+d x)}\right ) \, dx+\frac{\left (161 a^2 B^2 (b c+a d) g n^2\right ) \int \frac{\log (a+b x)}{a+b x} \, dx}{b}-\frac{\left (161 b B^2 c^2 (b c+a d) g n^2\right ) \int \frac{\log (c+d x)}{a+b x} \, dx}{d^2}+\frac{\left (161 B^2 c^2 (b c+a d) g n^2\right ) \int \frac{\log (c+d x)}{c+d x} \, dx}{d}-\frac{\left (161 a^2 B^2 d (b c+a d) g n^2\right ) \int \frac{\log (a+b x)}{c+d x} \, dx}{b^2}\\ &=-\frac{161 A B (b c-a d) (b c+a d) g n x}{3 b d}+\frac{161 B^2 (b c-a d)^2 g n^2 x}{3 b d}+\frac{161 a^2 B^2 (b c-a d) g n^2 \log (a+b x)}{3 b^2}-\frac{161 B^2 (b c-a d) (b c+a d) g n (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{3 b^2 d}-\frac{161}{3} B (b c-a d) g n x^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )+\frac{322 a^2 B c g n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b}+\frac{322 a^3 B d g n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3 b^2}-\frac{161 a^2 B (b c+a d) g n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^2}+161 a c g x \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2+\frac{161}{2} (b c+a d) g x^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2+\frac{161}{3} b d g x^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2-\frac{161 B^2 c^2 (b c-a d) g n^2 \log (c+d x)}{3 d^2}+\frac{161 B^2 (b c-a d)^2 (b c+a d) g n^2 \log (c+d x)}{3 b^2 d^2}+\frac{322 b B^2 c^3 g n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{3 d^2}+\frac{322 a B^2 c^2 g n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{d}-\frac{161 B^2 c^2 (b c+a d) g n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{d^2}-\frac{322 b B c^3 g n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{3 d^2}-\frac{322 a B c^2 g n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{d}+\frac{161 B c^2 (b c+a d) g n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{d^2}+\frac{322 a^2 B^2 c g n^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b}+\frac{322 a^3 B^2 d g n^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{3 b^2}-\frac{161 a^2 B^2 (b c+a d) g n^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^2}-\left (322 a^2 B^2 c g n^2\right ) \int \frac{\log \left (\frac{b (c+d x)}{b c-a d}\right )}{a+b x} \, dx-\frac{\left (322 a^2 B^2 c g n^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a+b x\right )}{b}-\left (322 a B^2 c^2 g n^2\right ) \int \frac{\log \left (\frac{d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx-\frac{\left (322 b B^2 c^3 g n^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,c+d x\right )}{3 d^2}-\frac{\left (322 a B^2 c^2 g n^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,c+d x\right )}{d}-\frac{\left (322 b B^2 c^3 g n^2\right ) \int \frac{\log \left (\frac{d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{3 d}-\frac{\left (322 a^3 B^2 d g n^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a+b x\right )}{3 b^2}-\frac{\left (322 a^3 B^2 d g n^2\right ) \int \frac{\log \left (\frac{b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{3 b}+\frac{\left (161 a^2 B^2 (b c+a d) g n^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a+b x\right )}{b^2}+\frac{\left (161 a^2 B^2 (b c+a d) g n^2\right ) \int \frac{\log \left (\frac{b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{b}+\frac{\left (161 B^2 c^2 (b c+a d) g n^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,c+d x\right )}{d^2}+\frac{\left (161 B^2 c^2 (b c+a d) g n^2\right ) \int \frac{\log \left (\frac{d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{d}\\ &=-\frac{161 A B (b c-a d) (b c+a d) g n x}{3 b d}+\frac{161 B^2 (b c-a d)^2 g n^2 x}{3 b d}+\frac{161 a^2 B^2 (b c-a d) g n^2 \log (a+b x)}{3 b^2}-\frac{161 a^2 B^2 c g n^2 \log ^2(a+b x)}{b}-\frac{161 a^3 B^2 d g n^2 \log ^2(a+b x)}{3 b^2}+\frac{161 a^2 B^2 (b c+a d) g n^2 \log ^2(a+b x)}{2 b^2}-\frac{161 B^2 (b c-a d) (b c+a d) g n (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{3 b^2 d}-\frac{161}{3} B (b c-a d) g n x^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )+\frac{322 a^2 B c g n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b}+\frac{322 a^3 B d g n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3 b^2}-\frac{161 a^2 B (b c+a d) g n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^2}+161 a c g x \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2+\frac{161}{2} (b c+a d) g x^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2+\frac{161}{3} b d g x^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2-\frac{161 B^2 c^2 (b c-a d) g n^2 \log (c+d x)}{3 d^2}+\frac{161 B^2 (b c-a d)^2 (b c+a d) g n^2 \log (c+d x)}{3 b^2 d^2}+\frac{322 b B^2 c^3 g n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{3 d^2}+\frac{322 a B^2 c^2 g n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{d}-\frac{161 B^2 c^2 (b c+a d) g n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{d^2}-\frac{322 b B c^3 g n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{3 d^2}-\frac{322 a B c^2 g n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{d}+\frac{161 B c^2 (b c+a d) g n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{d^2}-\frac{161 b B^2 c^3 g n^2 \log ^2(c+d x)}{3 d^2}-\frac{161 a B^2 c^2 g n^2 \log ^2(c+d x)}{d}+\frac{161 B^2 c^2 (b c+a d) g n^2 \log ^2(c+d x)}{2 d^2}+\frac{322 a^2 B^2 c g n^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b}+\frac{322 a^3 B^2 d g n^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{3 b^2}-\frac{161 a^2 B^2 (b c+a d) g n^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^2}-\frac{\left (322 a^2 B^2 c g n^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{b}-\frac{\left (322 b B^2 c^3 g n^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{3 d^2}-\frac{\left (322 a B^2 c^2 g n^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{d}-\frac{\left (322 a^3 B^2 d g n^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{3 b^2}+\frac{\left (161 a^2 B^2 (b c+a d) g n^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{b^2}+\frac{\left (161 B^2 c^2 (b c+a d) g n^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{d^2}\\ &=-\frac{161 A B (b c-a d) (b c+a d) g n x}{3 b d}+\frac{161 B^2 (b c-a d)^2 g n^2 x}{3 b d}+\frac{161 a^2 B^2 (b c-a d) g n^2 \log (a+b x)}{3 b^2}-\frac{161 a^2 B^2 c g n^2 \log ^2(a+b x)}{b}-\frac{161 a^3 B^2 d g n^2 \log ^2(a+b x)}{3 b^2}+\frac{161 a^2 B^2 (b c+a d) g n^2 \log ^2(a+b x)}{2 b^2}-\frac{161 B^2 (b c-a d) (b c+a d) g n (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{3 b^2 d}-\frac{161}{3} B (b c-a d) g n x^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )+\frac{322 a^2 B c g n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b}+\frac{322 a^3 B d g n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3 b^2}-\frac{161 a^2 B (b c+a d) g n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^2}+161 a c g x \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2+\frac{161}{2} (b c+a d) g x^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2+\frac{161}{3} b d g x^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2-\frac{161 B^2 c^2 (b c-a d) g n^2 \log (c+d x)}{3 d^2}+\frac{161 B^2 (b c-a d)^2 (b c+a d) g n^2 \log (c+d x)}{3 b^2 d^2}+\frac{322 b B^2 c^3 g n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{3 d^2}+\frac{322 a B^2 c^2 g n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{d}-\frac{161 B^2 c^2 (b c+a d) g n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{d^2}-\frac{322 b B c^3 g n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{3 d^2}-\frac{322 a B c^2 g n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{d}+\frac{161 B c^2 (b c+a d) g n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{d^2}-\frac{161 b B^2 c^3 g n^2 \log ^2(c+d x)}{3 d^2}-\frac{161 a B^2 c^2 g n^2 \log ^2(c+d x)}{d}+\frac{161 B^2 c^2 (b c+a d) g n^2 \log ^2(c+d x)}{2 d^2}+\frac{322 a^2 B^2 c g n^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b}+\frac{322 a^3 B^2 d g n^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{3 b^2}-\frac{161 a^2 B^2 (b c+a d) g n^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^2}+\frac{322 a^2 B^2 c g n^2 \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{b}+\frac{322 a^3 B^2 d g n^2 \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{3 b^2}-\frac{161 a^2 B^2 (b c+a d) g n^2 \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{b^2}+\frac{322 b B^2 c^3 g n^2 \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{3 d^2}+\frac{322 a B^2 c^2 g n^2 \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{d}-\frac{161 B^2 c^2 (b c+a d) g n^2 \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{d^2}\\ \end{align*}

Mathematica [B]  time = 0.72591, size = 937, normalized size = 2.52 \[ \frac{g i \left (2 b^3 B n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x) c^3-b^3 B^2 n^2 \left (\left (2 \log \left (\frac{d (a+b x)}{a d-b c}\right )-\log (c+d x)\right ) \log (c+d x)+2 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )\right ) c^3-6 a b^2 B d n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x) c^2+3 a b^2 B^2 d n^2 \left (\left (2 \log \left (\frac{d (a+b x)}{a d-b c}\right )-\log (c+d x)\right ) \log (c+d x)+2 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )\right ) c^2+6 a b^2 d^2 x \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 c-6 A b^2 B d (b c-a d) n x c-6 b B^2 d (b c-a d) n (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) c+6 a^2 b B d^2 n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) c+6 b B^2 (b c-a d)^2 n^2 \log (c+d x) c-3 a^2 b B^2 d^2 n^2 \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{PolyLog}\left (2,\frac{d (a+b x)}{a d-b c}\right )\right ) c+2 b^3 d^3 x^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2+3 b^2 d^2 (b c+a d) x^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2+6 a A b B d^2 (a d-b c) n x+4 A b B d (b c-a d) (b c+a d) n x+6 a B^2 d^2 (a d-b c) n (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+4 B^2 d (b c-a d) (b c+a d) n (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )-2 b^2 B d^2 (b c-a d) n x^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )-2 a^3 B d^3 n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )+6 a B^2 d (b c-a d)^2 n^2 \log (c+d x)-4 B^2 (b c-a d)^2 (b c+a d) n^2 \log (c+d x)+2 B^2 (b c-a d) n^2 \left (a^2 d^2 \log (a+b x)-b \left (b \log (c+d x) c^2+d (a d-b c) x\right )\right )+a^3 B^2 d^3 n^2 \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{PolyLog}\left (2,\frac{d (a+b x)}{a d-b c}\right )\right )\right )}{6 b^2 d^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [F]  time = 0.335, size = 0, normalized size = 0. \begin{align*} \int \left ( bgx+ag \right ) \left ( dix+ci \right ) \left ( A+B\ln \left ( e \left ({\frac{bx+a}{dx+c}} \right ) ^{n} \right ) \right ) ^{2}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [B]  time = 3.62435, size = 2082, normalized size = 5.6 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (A^{2} b d g i x^{2} + A^{2} a c g i +{\left (A^{2} b c + A^{2} a d\right )} g i x +{\left (B^{2} b d g i x^{2} + B^{2} a c g i +{\left (B^{2} b c + B^{2} a d\right )} g i x\right )} \log \left (e \left (\frac{b x + a}{d x + c}\right )^{n}\right )^{2} + 2 \,{\left (A B b d g i x^{2} + A B a c g i +{\left (A B b c + A B a d\right )} g i x\right )} \log \left (e \left (\frac{b x + a}{d x + c}\right )^{n}\right ), x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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