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

Optimal. Leaf size=819 \[ \frac{B g^2 i^2 n \left (2 A+3 B n+2 B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log \left (\frac{b c-a d}{b (c+d x)}\right ) (b c-a d)^5}{30 b^3 d^3}+\frac{B^2 g^2 i^2 n^2 \log \left (\frac{a+b x}{c+d x}\right ) (b c-a d)^5}{30 b^3 d^3}+\frac{B^2 g^2 i^2 n^2 \log (c+d x) (b c-a d)^5}{10 b^3 d^3}+\frac{B^2 g^2 i^2 n^2 \text{PolyLog}\left (2,\frac{d (a+b x)}{b (c+d x)}\right ) (b c-a d)^5}{15 b^3 d^3}-\frac{B^2 g^2 i^2 n^2 x (b c-a d)^4}{10 b^2 d^2}+\frac{B g^2 i^2 n (a+b x) \left (2 A+B n+2 B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) (b c-a d)^4}{30 b^3 d^2}-\frac{B^2 g^2 i^2 n^2 (c+d x)^2 (b c-a d)^3}{20 b d^3}-\frac{B g^2 i^2 n (a+b x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) (b c-a d)^3}{30 b^3 d}-\frac{B g^2 i^2 n (c+d x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) (b c-a d)^3}{5 b d^3}+\frac{B^2 g^2 i^2 n^2 (c+d x)^3 (b c-a d)^2}{30 d^3}+\frac{g^2 i^2 (a+b x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 (b c-a d)^2}{30 b^3}-\frac{B g^2 i^2 n (a+b x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) (b c-a d)^2}{15 b^3}+\frac{4 B g^2 i^2 n (c+d x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) (b c-a d)^2}{15 d^3}+\frac{g^2 i^2 (a+b x)^3 (c+d x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 (b c-a d)}{10 b^2}-\frac{b B g^2 i^2 n (c+d x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) (b c-a d)}{10 d^3}+\frac{g^2 i^2 (a+b x)^3 (c+d x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{5 b} \]

[Out]

-(B^2*(b*c - a*d)^4*g^2*i^2*n^2*x)/(10*b^2*d^2) - (B^2*(b*c - a*d)^3*g^2*i^2*n^2*(c + d*x)^2)/(20*b*d^3) + (B^
2*(b*c - a*d)^2*g^2*i^2*n^2*(c + d*x)^3)/(30*d^3) - (B*(b*c - a*d)^3*g^2*i^2*n*(a + b*x)^2*(A + B*Log[e*((a +
b*x)/(c + d*x))^n]))/(30*b^3*d) - (B*(b*c - a*d)^2*g^2*i^2*n*(a + b*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]
))/(15*b^3) - (B*(b*c - a*d)^3*g^2*i^2*n*(c + d*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(5*b*d^3) + (4*B*
(b*c - a*d)^2*g^2*i^2*n*(c + d*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(15*d^3) - (b*B*(b*c - a*d)*g^2*i^
2*n*(c + d*x)^4*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(10*d^3) + ((b*c - a*d)^2*g^2*i^2*(a + b*x)^3*(A + B*L
og[e*((a + b*x)/(c + d*x))^n])^2)/(30*b^3) + ((b*c - a*d)*g^2*i^2*(a + b*x)^3*(c + d*x)*(A + B*Log[e*((a + b*x
)/(c + d*x))^n])^2)/(10*b^2) + (g^2*i^2*(a + b*x)^3*(c + d*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(5*b
) + (B*(b*c - a*d)^4*g^2*i^2*n*(a + b*x)*(2*A + B*n + 2*B*Log[e*((a + b*x)/(c + d*x))^n]))/(30*b^3*d^2) + (B*(
b*c - a*d)^5*g^2*i^2*n*(2*A + 3*B*n + 2*B*Log[e*((a + b*x)/(c + d*x))^n])*Log[(b*c - a*d)/(b*(c + d*x))])/(30*
b^3*d^3) + (B^2*(b*c - a*d)^5*g^2*i^2*n^2*Log[(a + b*x)/(c + d*x)])/(30*b^3*d^3) + (B^2*(b*c - a*d)^5*g^2*i^2*
n^2*Log[c + d*x])/(10*b^3*d^3) + (B^2*(b*c - a*d)^5*g^2*i^2*n^2*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(15*b
^3*d^3)

________________________________________________________________________________________

Rubi [A]  time = 2.60906, antiderivative size = 714, normalized size of antiderivative = 0.87, number of steps used = 71, number of rules used = 13, integrand size = 45, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.289, Rules used = {2528, 2525, 12, 2486, 31, 43, 2524, 2418, 2394, 2393, 2391, 2390, 2301} \[ \frac{B^2 g^2 i^2 n^2 (b c-a d)^5 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )}{15 b^3 d^3}-\frac{B g^2 i^2 n (b c-a d)^5 \log (c+d x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{15 b^3 d^3}+\frac{d^2 g^2 i^2 (a+b x)^5 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )^2}{5 b^3}+\frac{A B g^2 i^2 n x (b c-a d)^4}{15 b^2 d^2}-\frac{B g^2 i^2 n (a+b x)^2 (b c-a d)^3 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{30 b^3 d}+\frac{g^2 i^2 (a+b x)^3 (b c-a d)^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )^2}{3 b^3}-\frac{B g^2 i^2 n (a+b x)^3 (b c-a d)^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{5 b^3}+\frac{d g^2 i^2 (a+b x)^4 (b c-a d) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )^2}{2 b^3}-\frac{B d g^2 i^2 n (a+b x)^4 (b c-a d) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{10 b^3}+\frac{B^2 g^2 i^2 n (a+b x) (b c-a d)^4 \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{15 b^3 d^2}-\frac{B^2 g^2 i^2 n^2 x (b c-a d)^4}{15 b^2 d^2}-\frac{B^2 g^2 i^2 n^2 (b c-a d)^5 \log ^2(c+d x)}{30 b^3 d^3}+\frac{B^2 g^2 i^2 n^2 (b c-a d)^5 \log (c+d x) \log \left (-\frac{d (a+b x)}{b c-a d}\right )}{15 b^3 d^3}+\frac{B^2 g^2 i^2 n^2 (a+b x)^2 (b c-a d)^3}{20 b^3 d}+\frac{B^2 g^2 i^2 n^2 (a+b x)^3 (b c-a d)^2}{30 b^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(A*B*(b*c - a*d)^4*g^2*i^2*n*x)/(15*b^2*d^2) - (B^2*(b*c - a*d)^4*g^2*i^2*n^2*x)/(15*b^2*d^2) + (B^2*(b*c - a*
d)^3*g^2*i^2*n^2*(a + b*x)^2)/(20*b^3*d) + (B^2*(b*c - a*d)^2*g^2*i^2*n^2*(a + b*x)^3)/(30*b^3) + (B^2*(b*c -
a*d)^4*g^2*i^2*n*(a + b*x)*Log[e*((a + b*x)/(c + d*x))^n])/(15*b^3*d^2) - (B*(b*c - a*d)^3*g^2*i^2*n*(a + b*x)
^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(30*b^3*d) - (B*(b*c - a*d)^2*g^2*i^2*n*(a + b*x)^3*(A + B*Log[e*((
a + b*x)/(c + d*x))^n]))/(5*b^3) - (B*d*(b*c - a*d)*g^2*i^2*n*(a + b*x)^4*(A + B*Log[e*((a + b*x)/(c + d*x))^n
]))/(10*b^3) + ((b*c - a*d)^2*g^2*i^2*(a + b*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(3*b^3) + (d*(b*c
- a*d)*g^2*i^2*(a + b*x)^4*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(2*b^3) + (d^2*g^2*i^2*(a + b*x)^5*(A + B
*Log[e*((a + b*x)/(c + d*x))^n])^2)/(5*b^3) + (B^2*(b*c - a*d)^5*g^2*i^2*n^2*Log[-((d*(a + b*x))/(b*c - a*d))]
*Log[c + d*x])/(15*b^3*d^3) - (B*(b*c - a*d)^5*g^2*i^2*n*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*Log[c + d*x])/
(15*b^3*d^3) - (B^2*(b*c - a*d)^5*g^2*i^2*n^2*Log[c + d*x]^2)/(30*b^3*d^3) + (B^2*(b*c - a*d)^5*g^2*i^2*n^2*Po
lyLog[2, (b*(c + d*x))/(b*c - a*d)])/(15*b^3*d^3)

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 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 12

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

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 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

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 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 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]

Rubi steps

\begin{align*} \int (169 c+169 d x)^2 (a g+b g x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \, dx &=\int \left (\frac{(-b c+a d)^2 g^2 (169 c+169 d x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{d^2}-\frac{2 b (b c-a d) g^2 (169 c+169 d x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{169 d^2}+\frac{b^2 g^2 (169 c+169 d x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{28561 d^2}\right ) \, dx\\ &=\frac{\left (b^2 g^2\right ) \int (169 c+169 d x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \, dx}{28561 d^2}-\frac{\left (2 b (b c-a d) g^2\right ) \int (169 c+169 d x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \, dx}{169 d^2}+\frac{\left ((b c-a d)^2 g^2\right ) \int (169 c+169 d x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \, dx}{d^2}\\ &=\frac{28561 (b c-a d)^2 g^2 (c+d x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{3 d^3}-\frac{28561 b (b c-a d) g^2 (c+d x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{2 d^3}+\frac{28561 b^2 g^2 (c+d x)^5 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{5 d^3}-\frac{\left (2 b^2 B g^2 n\right ) \int \frac{137858491849 (b c-a d) (c+d x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{a+b x} \, dx}{24134045 d^3}+\frac{\left (b B (b c-a d) g^2 n\right ) \int \frac{815730721 (b c-a d) (c+d x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{a+b x} \, dx}{28561 d^3}-\frac{\left (2 B (b c-a d)^2 g^2 n\right ) \int \frac{4826809 (b c-a d) (c+d x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{a+b x} \, dx}{507 d^3}\\ &=\frac{28561 (b c-a d)^2 g^2 (c+d x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{3 d^3}-\frac{28561 b (b c-a d) g^2 (c+d x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{2 d^3}+\frac{28561 b^2 g^2 (c+d x)^5 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{5 d^3}-\frac{\left (57122 b^2 B (b c-a d) g^2 n\right ) \int \frac{(c+d x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{a+b x} \, dx}{5 d^3}+\frac{\left (28561 b B (b c-a d)^2 g^2 n\right ) \int \frac{(c+d x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{a+b x} \, dx}{d^3}-\frac{\left (57122 B (b c-a d)^3 g^2 n\right ) \int \frac{(c+d x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{a+b x} \, dx}{3 d^3}\\ &=\frac{28561 (b c-a d)^2 g^2 (c+d x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{3 d^3}-\frac{28561 b (b c-a d) g^2 (c+d x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{2 d^3}+\frac{28561 b^2 g^2 (c+d x)^5 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{5 d^3}-\frac{\left (57122 b^2 B (b c-a d) g^2 n\right ) \int \left (\frac{d (b c-a d)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^4}+\frac{(b c-a d)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^4 (a+b x)}+\frac{d (b c-a d)^2 (c+d x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^3}+\frac{d (b c-a d) (c+d x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^2}+\frac{d (c+d x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b}\right ) \, dx}{5 d^3}+\frac{\left (28561 b B (b c-a d)^2 g^2 n\right ) \int \left (\frac{d (b c-a d)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^3}+\frac{(b c-a d)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^3 (a+b x)}+\frac{d (b c-a d) (c+d x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^2}+\frac{d (c+d x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b}\right ) \, dx}{d^3}-\frac{\left (57122 B (b c-a d)^3 g^2 n\right ) \int \left (\frac{d (b c-a d) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^2}+\frac{(b c-a d)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^2 (a+b x)}+\frac{d (c+d x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b}\right ) \, dx}{3 d^3}\\ &=\frac{28561 (b c-a d)^2 g^2 (c+d x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{3 d^3}-\frac{28561 b (b c-a d) g^2 (c+d x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{2 d^3}+\frac{28561 b^2 g^2 (c+d x)^5 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{5 d^3}-\frac{\left (57122 b B (b c-a d) g^2 n\right ) \int (c+d x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \, dx}{5 d^2}-\frac{\left (57122 B (b c-a d)^2 g^2 n\right ) \int (c+d x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \, dx}{5 d^2}+\frac{\left (28561 B (b c-a d)^2 g^2 n\right ) \int (c+d x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \, dx}{d^2}-\frac{\left (57122 B (b c-a d)^3 g^2 n\right ) \int (c+d x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \, dx}{5 b d^2}-\frac{\left (57122 B (b c-a d)^3 g^2 n\right ) \int (c+d x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \, dx}{3 b d^2}+\frac{\left (28561 B (b c-a d)^3 g^2 n\right ) \int (c+d x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \, dx}{b d^2}-\frac{\left (57122 B (b c-a d)^4 g^2 n\right ) \int \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \, dx}{5 b^2 d^2}-\frac{\left (57122 B (b c-a d)^4 g^2 n\right ) \int \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \, dx}{3 b^2 d^2}+\frac{\left (28561 B (b c-a d)^4 g^2 n\right ) \int \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \, dx}{b^2 d^2}-\frac{\left (57122 B (b c-a d)^5 g^2 n\right ) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{a+b x} \, dx}{5 b^2 d^3}-\frac{\left (57122 B (b c-a d)^5 g^2 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^2 d^3}+\frac{\left (28561 B (b c-a d)^5 g^2 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^2 d^3}\\ &=-\frac{28561 A B (b c-a d)^4 g^2 n x}{15 b^2 d^2}-\frac{28561 B (b c-a d)^3 g^2 n (c+d x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{30 b d^3}+\frac{28561 B (b c-a d)^2 g^2 n (c+d x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{5 d^3}-\frac{28561 b B (b c-a d) g^2 n (c+d x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{10 d^3}-\frac{28561 B (b c-a d)^5 g^2 n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{15 b^3 d^3}+\frac{28561 (b c-a d)^2 g^2 (c+d x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{3 d^3}-\frac{28561 b (b c-a d) g^2 (c+d x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{2 d^3}+\frac{28561 b^2 g^2 (c+d x)^5 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{5 d^3}-\frac{\left (57122 B^2 (b c-a d)^4 g^2 n\right ) \int \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \, dx}{5 b^2 d^2}-\frac{\left (57122 B^2 (b c-a d)^4 g^2 n\right ) \int \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \, dx}{3 b^2 d^2}+\frac{\left (28561 B^2 (b c-a d)^4 g^2 n\right ) \int \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \, dx}{b^2 d^2}+\frac{\left (28561 b B^2 (b c-a d) g^2 n^2\right ) \int \frac{(b c-a d) (c+d x)^3}{a+b x} \, dx}{10 d^3}+\frac{\left (57122 B^2 (b c-a d)^2 g^2 n^2\right ) \int \frac{(b c-a d) (c+d x)^2}{a+b x} \, dx}{15 d^3}-\frac{\left (28561 B^2 (b c-a d)^2 g^2 n^2\right ) \int \frac{(b c-a d) (c+d x)^2}{a+b x} \, dx}{3 d^3}+\frac{\left (28561 B^2 (b c-a d)^3 g^2 n^2\right ) \int \frac{(b c-a d) (c+d x)}{a+b x} \, dx}{5 b d^3}+\frac{\left (28561 B^2 (b c-a d)^3 g^2 n^2\right ) \int \frac{(b c-a d) (c+d x)}{a+b x} \, dx}{3 b d^3}-\frac{\left (28561 B^2 (b c-a d)^3 g^2 n^2\right ) \int \frac{(b c-a d) (c+d x)}{a+b x} \, dx}{2 b d^3}+\frac{\left (57122 B^2 (b c-a d)^5 g^2 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}{5 b^3 d^3}+\frac{\left (57122 B^2 (b c-a d)^5 g^2 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^3 d^3}-\frac{\left (28561 B^2 (b c-a d)^5 g^2 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^3 d^3}\\ &=-\frac{28561 A B (b c-a d)^4 g^2 n x}{15 b^2 d^2}-\frac{28561 B^2 (b c-a d)^4 g^2 n (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{15 b^3 d^2}-\frac{28561 B (b c-a d)^3 g^2 n (c+d x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{30 b d^3}+\frac{28561 B (b c-a d)^2 g^2 n (c+d x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{5 d^3}-\frac{28561 b B (b c-a d) g^2 n (c+d x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{10 d^3}-\frac{28561 B (b c-a d)^5 g^2 n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{15 b^3 d^3}+\frac{28561 (b c-a d)^2 g^2 (c+d x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{3 d^3}-\frac{28561 b (b c-a d) g^2 (c+d x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{2 d^3}+\frac{28561 b^2 g^2 (c+d x)^5 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{5 d^3}+\frac{\left (28561 b B^2 (b c-a d)^2 g^2 n^2\right ) \int \frac{(c+d x)^3}{a+b x} \, dx}{10 d^3}+\frac{\left (57122 B^2 (b c-a d)^3 g^2 n^2\right ) \int \frac{(c+d x)^2}{a+b x} \, dx}{15 d^3}-\frac{\left (28561 B^2 (b c-a d)^3 g^2 n^2\right ) \int \frac{(c+d x)^2}{a+b x} \, dx}{3 d^3}+\frac{\left (28561 B^2 (b c-a d)^4 g^2 n^2\right ) \int \frac{c+d x}{a+b x} \, dx}{5 b d^3}+\frac{\left (28561 B^2 (b c-a d)^4 g^2 n^2\right ) \int \frac{c+d x}{a+b x} \, dx}{3 b d^3}-\frac{\left (28561 B^2 (b c-a d)^4 g^2 n^2\right ) \int \frac{c+d x}{a+b x} \, dx}{2 b d^3}+\frac{\left (57122 B^2 (b c-a d)^5 g^2 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}{5 b^3 d^3}+\frac{\left (57122 B^2 (b c-a d)^5 g^2 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^3 d^3}-\frac{\left (28561 B^2 (b c-a d)^5 g^2 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^3 d^3}+\frac{\left (57122 B^2 (b c-a d)^5 g^2 n^2\right ) \int \frac{1}{c+d x} \, dx}{5 b^3 d^2}+\frac{\left (57122 B^2 (b c-a d)^5 g^2 n^2\right ) \int \frac{1}{c+d x} \, dx}{3 b^3 d^2}-\frac{\left (28561 B^2 (b c-a d)^5 g^2 n^2\right ) \int \frac{1}{c+d x} \, dx}{b^3 d^2}\\ &=-\frac{28561 A B (b c-a d)^4 g^2 n x}{15 b^2 d^2}-\frac{28561 B^2 (b c-a d)^4 g^2 n (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{15 b^3 d^2}-\frac{28561 B (b c-a d)^3 g^2 n (c+d x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{30 b d^3}+\frac{28561 B (b c-a d)^2 g^2 n (c+d x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{5 d^3}-\frac{28561 b B (b c-a d) g^2 n (c+d x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{10 d^3}-\frac{28561 B (b c-a d)^5 g^2 n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{15 b^3 d^3}+\frac{28561 (b c-a d)^2 g^2 (c+d x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{3 d^3}-\frac{28561 b (b c-a d) g^2 (c+d x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{2 d^3}+\frac{28561 b^2 g^2 (c+d x)^5 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{5 d^3}+\frac{28561 B^2 (b c-a d)^5 g^2 n^2 \log (c+d x)}{15 b^3 d^3}+\frac{\left (28561 b B^2 (b c-a d)^2 g^2 n^2\right ) \int \left (\frac{d (b c-a d)^2}{b^3}+\frac{(b c-a d)^3}{b^3 (a+b x)}+\frac{d (b c-a d) (c+d x)}{b^2}+\frac{d (c+d x)^2}{b}\right ) \, dx}{10 d^3}+\frac{\left (57122 B^2 (b c-a d)^3 g^2 n^2\right ) \int \left (\frac{d (b c-a d)}{b^2}+\frac{(b c-a d)^2}{b^2 (a+b x)}+\frac{d (c+d x)}{b}\right ) \, dx}{15 d^3}-\frac{\left (28561 B^2 (b c-a d)^3 g^2 n^2\right ) \int \left (\frac{d (b c-a d)}{b^2}+\frac{(b c-a d)^2}{b^2 (a+b x)}+\frac{d (c+d x)}{b}\right ) \, dx}{3 d^3}+\frac{\left (28561 B^2 (b c-a d)^4 g^2 n^2\right ) \int \left (\frac{d}{b}+\frac{b c-a d}{b (a+b x)}\right ) \, dx}{5 b d^3}+\frac{\left (28561 B^2 (b c-a d)^4 g^2 n^2\right ) \int \left (\frac{d}{b}+\frac{b c-a d}{b (a+b x)}\right ) \, dx}{3 b d^3}-\frac{\left (28561 B^2 (b c-a d)^4 g^2 n^2\right ) \int \left (\frac{d}{b}+\frac{b c-a d}{b (a+b x)}\right ) \, dx}{2 b d^3}+\frac{\left (57122 B^2 (b c-a d)^5 g^2 n^2\right ) \int \frac{\log (a+b x)}{a+b x} \, dx}{5 b^2 d^3}+\frac{\left (57122 B^2 (b c-a d)^5 g^2 n^2\right ) \int \frac{\log (a+b x)}{a+b x} \, dx}{3 b^2 d^3}-\frac{\left (28561 B^2 (b c-a d)^5 g^2 n^2\right ) \int \frac{\log (a+b x)}{a+b x} \, dx}{b^2 d^3}-\frac{\left (57122 B^2 (b c-a d)^5 g^2 n^2\right ) \int \frac{\log (a+b x)}{c+d x} \, dx}{5 b^3 d^2}-\frac{\left (57122 B^2 (b c-a d)^5 g^2 n^2\right ) \int \frac{\log (a+b x)}{c+d x} \, dx}{3 b^3 d^2}+\frac{\left (28561 B^2 (b c-a d)^5 g^2 n^2\right ) \int \frac{\log (a+b x)}{c+d x} \, dx}{b^3 d^2}\\ &=-\frac{28561 A B (b c-a d)^4 g^2 n x}{15 b^2 d^2}-\frac{28561 B^2 (b c-a d)^4 g^2 n^2 x}{15 b^2 d^2}-\frac{28561 B^2 (b c-a d)^3 g^2 n^2 (c+d x)^2}{20 b d^3}+\frac{28561 B^2 (b c-a d)^2 g^2 n^2 (c+d x)^3}{30 d^3}-\frac{28561 B^2 (b c-a d)^5 g^2 n^2 \log (a+b x)}{15 b^3 d^3}-\frac{28561 B^2 (b c-a d)^4 g^2 n (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{15 b^3 d^2}-\frac{28561 B (b c-a d)^3 g^2 n (c+d x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{30 b d^3}+\frac{28561 B (b c-a d)^2 g^2 n (c+d x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{5 d^3}-\frac{28561 b B (b c-a d) g^2 n (c+d x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{10 d^3}-\frac{28561 B (b c-a d)^5 g^2 n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{15 b^3 d^3}+\frac{28561 (b c-a d)^2 g^2 (c+d x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{3 d^3}-\frac{28561 b (b c-a d) g^2 (c+d x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{2 d^3}+\frac{28561 b^2 g^2 (c+d x)^5 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{5 d^3}+\frac{28561 B^2 (b c-a d)^5 g^2 n^2 \log (c+d x)}{15 b^3 d^3}-\frac{28561 B^2 (b c-a d)^5 g^2 n^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{15 b^3 d^3}+\frac{\left (57122 B^2 (b c-a d)^5 g^2 n^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a+b x\right )}{5 b^3 d^3}+\frac{\left (57122 B^2 (b c-a d)^5 g^2 n^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a+b x\right )}{3 b^3 d^3}-\frac{\left (28561 B^2 (b c-a d)^5 g^2 n^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a+b x\right )}{b^3 d^3}+\frac{\left (57122 B^2 (b c-a d)^5 g^2 n^2\right ) \int \frac{\log \left (\frac{b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{5 b^2 d^3}+\frac{\left (57122 B^2 (b c-a d)^5 g^2 n^2\right ) \int \frac{\log \left (\frac{b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{3 b^2 d^3}-\frac{\left (28561 B^2 (b c-a d)^5 g^2 n^2\right ) \int \frac{\log \left (\frac{b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{b^2 d^3}\\ &=-\frac{28561 A B (b c-a d)^4 g^2 n x}{15 b^2 d^2}-\frac{28561 B^2 (b c-a d)^4 g^2 n^2 x}{15 b^2 d^2}-\frac{28561 B^2 (b c-a d)^3 g^2 n^2 (c+d x)^2}{20 b d^3}+\frac{28561 B^2 (b c-a d)^2 g^2 n^2 (c+d x)^3}{30 d^3}-\frac{28561 B^2 (b c-a d)^5 g^2 n^2 \log (a+b x)}{15 b^3 d^3}+\frac{28561 B^2 (b c-a d)^5 g^2 n^2 \log ^2(a+b x)}{30 b^3 d^3}-\frac{28561 B^2 (b c-a d)^4 g^2 n (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{15 b^3 d^2}-\frac{28561 B (b c-a d)^3 g^2 n (c+d x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{30 b d^3}+\frac{28561 B (b c-a d)^2 g^2 n (c+d x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{5 d^3}-\frac{28561 b B (b c-a d) g^2 n (c+d x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{10 d^3}-\frac{28561 B (b c-a d)^5 g^2 n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{15 b^3 d^3}+\frac{28561 (b c-a d)^2 g^2 (c+d x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{3 d^3}-\frac{28561 b (b c-a d) g^2 (c+d x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{2 d^3}+\frac{28561 b^2 g^2 (c+d x)^5 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{5 d^3}+\frac{28561 B^2 (b c-a d)^5 g^2 n^2 \log (c+d x)}{15 b^3 d^3}-\frac{28561 B^2 (b c-a d)^5 g^2 n^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{15 b^3 d^3}+\frac{\left (57122 B^2 (b c-a d)^5 g^2 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 )}{5 b^3 d^3}+\frac{\left (57122 B^2 (b c-a d)^5 g^2 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^3 d^3}-\frac{\left (28561 B^2 (b c-a d)^5 g^2 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^3 d^3}\\ &=-\frac{28561 A B (b c-a d)^4 g^2 n x}{15 b^2 d^2}-\frac{28561 B^2 (b c-a d)^4 g^2 n^2 x}{15 b^2 d^2}-\frac{28561 B^2 (b c-a d)^3 g^2 n^2 (c+d x)^2}{20 b d^3}+\frac{28561 B^2 (b c-a d)^2 g^2 n^2 (c+d x)^3}{30 d^3}-\frac{28561 B^2 (b c-a d)^5 g^2 n^2 \log (a+b x)}{15 b^3 d^3}+\frac{28561 B^2 (b c-a d)^5 g^2 n^2 \log ^2(a+b x)}{30 b^3 d^3}-\frac{28561 B^2 (b c-a d)^4 g^2 n (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{15 b^3 d^2}-\frac{28561 B (b c-a d)^3 g^2 n (c+d x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{30 b d^3}+\frac{28561 B (b c-a d)^2 g^2 n (c+d x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{5 d^3}-\frac{28561 b B (b c-a d) g^2 n (c+d x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{10 d^3}-\frac{28561 B (b c-a d)^5 g^2 n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{15 b^3 d^3}+\frac{28561 (b c-a d)^2 g^2 (c+d x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{3 d^3}-\frac{28561 b (b c-a d) g^2 (c+d x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{2 d^3}+\frac{28561 b^2 g^2 (c+d x)^5 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{5 d^3}+\frac{28561 B^2 (b c-a d)^5 g^2 n^2 \log (c+d x)}{15 b^3 d^3}-\frac{28561 B^2 (b c-a d)^5 g^2 n^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{15 b^3 d^3}-\frac{28561 B^2 (b c-a d)^5 g^2 n^2 \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{15 b^3 d^3}\\ \end{align*}

Mathematica [A]  time = 1.02239, size = 1254, normalized size = 1.53 \[ \frac{g^2 i^2 \left (12 d^5 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 (a+b x)^5+30 d^4 (b c-a d) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 (a+b x)^4+20 d^3 (b c-a d)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 (a+b x)^3+20 B (b c-a d)^3 n \left (-2 B n \log (c+d x) (b c-a d)^2-2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x) (b c-a d)^2+B n \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 ) (b c-a d)^2+2 A b d x (b c-a d)+2 B d (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) (b c-a d)+B n (b d x+(a d-b c) \log (c+d x)) (b c-a d)-d^2 (a+b x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )\right )-10 B (b c-a d)^2 n \left (-6 B n \log (c+d x) (b c-a d)^3-6 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x) (b c-a d)^3+3 B n \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 ) (b c-a d)^3+6 A b d x (b c-a d)^2+6 B d (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) (b c-a d)^2+3 B n (b d x+(a d-b c) \log (c+d x)) (b c-a d)^2+B n \left (-2 \log (c+d x) (b c-a d)^2+2 b d x (b c-a d)-d^2 (a+b x)^2\right ) (b c-a d)+2 d^3 (a+b x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )+3 d^2 (a d-b c) (a+b x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )\right )+B (b c-a d) n \left (-24 B n \log (c+d x) (b c-a d)^4-24 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x) (b c-a d)^4+12 B n \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 ) (b c-a d)^4+24 A b d x (b c-a d)^3+24 B d (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) (b c-a d)^3+12 B n (b d x+(a d-b c) \log (c+d x)) (b c-a d)^3-12 d^2 (a+b x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) (b c-a d)^2+4 B n \left (-2 \log (c+d x) (b c-a d)^2+2 b d x (b c-a d)-d^2 (a+b x)^2\right ) (b c-a d)^2+8 d^3 (a+b x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) (b c-a d)+B n \left (-6 \log (c+d x) (b c-a d)^3+6 b d x (b c-a d)^2+2 d^3 (a+b x)^3+3 d^2 (a d-b c) (a+b x)^2\right ) (b c-a d)-6 d^4 (a+b x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )\right )\right )}{60 b^3 d^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(g^2*i^2*(20*d^3*(b*c - a*d)^2*(a + b*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2 + 30*d^4*(b*c - a*d)*(a +
b*x)^4*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2 + 12*d^5*(a + b*x)^5*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2
+ 20*B*(b*c - a*d)^3*n*(2*A*b*d*(b*c - a*d)*x + 2*B*d*(b*c - a*d)*(a + b*x)*Log[e*((a + b*x)/(c + d*x))^n] - d
^2*(a + b*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) - 2*B*(b*c - a*d)^2*n*Log[c + d*x] - 2*(b*c - a*d)^2*(A
+ B*Log[e*((a + b*x)/(c + d*x))^n])*Log[c + d*x] + B*(b*c - a*d)*n*(b*d*x + (-(b*c) + a*d)*Log[c + d*x]) + B*(
b*c - a*d)^2*n*((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)])) - 10*B*(b*c - a*d)^2*n*(6*A*b*d*(b*c - a*d)^2*x + 6*B*d*(b*c - a*d)^2*(a + b*x)*Log[e*((a + b*
x)/(c + d*x))^n] + 3*d^2*(-(b*c) + a*d)*(a + b*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) + 2*d^3*(a + b*x)^3
*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) - 6*B*(b*c - a*d)^3*n*Log[c + d*x] - 6*(b*c - a*d)^3*(A + B*Log[e*((a
+ b*x)/(c + d*x))^n])*Log[c + d*x] + B*(b*c - a*d)*n*(2*b*d*(b*c - a*d)*x - d^2*(a + b*x)^2 - 2*(b*c - a*d)^2*
Log[c + d*x]) + 3*B*(b*c - a*d)^2*n*(b*d*x + (-(b*c) + a*d)*Log[c + d*x]) + 3*B*(b*c - a*d)^3*n*((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)])) + B*(b*c - a*
d)*n*(24*A*b*d*(b*c - a*d)^3*x + 24*B*d*(b*c - a*d)^3*(a + b*x)*Log[e*((a + b*x)/(c + d*x))^n] - 12*d^2*(b*c -
a*d)^2*(a + b*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) + 8*d^3*(b*c - a*d)*(a + b*x)^3*(A + B*Log[e*((a +
b*x)/(c + d*x))^n]) - 6*d^4*(a + b*x)^4*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) - 24*B*(b*c - a*d)^4*n*Log[c +
d*x] - 24*(b*c - a*d)^4*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*Log[c + d*x] + 4*B*(b*c - a*d)^2*n*(2*b*d*(b*c
- a*d)*x - d^2*(a + b*x)^2 - 2*(b*c - a*d)^2*Log[c + d*x]) + B*(b*c - a*d)*n*(6*b*d*(b*c - a*d)^2*x + 3*d^2*(-
(b*c) + a*d)*(a + b*x)^2 + 2*d^3*(a + b*x)^3 - 6*(b*c - a*d)^3*Log[c + d*x]) + 12*B*(b*c - a*d)^3*n*(b*d*x + (
-(b*c) + a*d)*Log[c + d*x]) + 12*B*(b*c - a*d)^4*n*((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)]))))/(60*b^3*d^3)

________________________________________________________________________________________

Maple [F]  time = 0.691, size = 0, normalized size = 0. \begin{align*} \int \left ( bgx+ag \right ) ^{2} \left ( dix+ci \right ) ^{2} \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)^2*(d*i*x+c*i)^2*(A+B*ln(e*((b*x+a)/(d*x+c))^n))^2,x)

[Out]

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

________________________________________________________________________________________

Maxima [B]  time = 3.92027, size = 5733, normalized size = 7. \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)^2*(d*i*x+c*i)^2*(A+B*log(e*((b*x+a)/(d*x+c))^n))^2,x, algorithm="maxima")

[Out]

2/5*A*B*b^2*d^2*g^2*i^2*x^5*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 1/5*A^2*b^2*d^2*g^2*i^2*x^5 + A*B*b^2*c*d
*g^2*i^2*x^4*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + A*B*a*b*d^2*g^2*i^2*x^4*log(e*(b*x/(d*x + c) + a/(d*x +
c))^n) + 1/2*A^2*b^2*c*d*g^2*i^2*x^4 + 1/2*A^2*a*b*d^2*g^2*i^2*x^4 + 2/3*A*B*b^2*c^2*g^2*i^2*x^3*log(e*(b*x/(d
*x + c) + a/(d*x + c))^n) + 8/3*A*B*a*b*c*d*g^2*i^2*x^3*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 2/3*A*B*a^2*d
^2*g^2*i^2*x^3*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 1/3*A^2*b^2*c^2*g^2*i^2*x^3 + 4/3*A^2*a*b*c*d*g^2*i^2*
x^3 + 1/3*A^2*a^2*d^2*g^2*i^2*x^3 + 2*A*B*a*b*c^2*g^2*i^2*x^2*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 2*A*B*a
^2*c*d*g^2*i^2*x^2*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + A^2*a*b*c^2*g^2*i^2*x^2 + A^2*a^2*c*d*g^2*i^2*x^2
+ 1/30*A*B*b^2*d^2*g^2*i^2*n*(12*a^5*log(b*x + a)/b^5 - 12*c^5*log(d*x + c)/d^5 - (3*(b^4*c*d^3 - a*b^3*d^4)*x
^4 - 4*(b^4*c^2*d^2 - a^2*b^2*d^4)*x^3 + 6*(b^4*c^3*d - a^3*b*d^4)*x^2 - 12*(b^4*c^4 - a^4*d^4)*x)/(b^4*d^4))
- 1/6*A*B*b^2*c*d*g^2*i^2*n*(6*a^4*log(b*x + a)/b^4 - 6*c^4*log(d*x + c)/d^4 + (2*(b^3*c*d^2 - a*b^2*d^3)*x^3
- 3*(b^3*c^2*d - a^2*b*d^3)*x^2 + 6*(b^3*c^3 - a^3*d^3)*x)/(b^3*d^3)) - 1/6*A*B*a*b*d^2*g^2*i^2*n*(6*a^4*log(b
*x + a)/b^4 - 6*c^4*log(d*x + c)/d^4 + (2*(b^3*c*d^2 - a*b^2*d^3)*x^3 - 3*(b^3*c^2*d - a^2*b*d^3)*x^2 + 6*(b^3
*c^3 - a^3*d^3)*x)/(b^3*d^3)) + 1/3*A*B*b^2*c^2*g^2*i^2*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)) + 4/3*A*B*a*b*c*d*g^2*i^2*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)) + 1/3*A*B*a^2*d^
2*g^2*i^2*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)) - 2*A*B*a*b*c^2*g^2*i^2*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^2*c*d*g^2*i^2*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^2*c^2*
g^2*i^2*n*(a*log(b*x + a)/b - c*log(d*x + c)/d) + 2*A*B*a^2*c^2*g^2*i^2*x*log(e*(b*x/(d*x + c) + a/(d*x + c))^
n) + A^2*a^2*c^2*g^2*i^2*x - 1/30*(9*a^3*b*c^2*d^3*g^2*i^2*n^2 - 2*a^4*c*d^4*g^2*i^2*n^2 + 2*b^4*c^5*g^2*i^2*n
*log(e) + 2*(g^2*i^2*n^2 - 5*g^2*i^2*n*log(e))*a*b^3*c^4*d - (9*g^2*i^2*n^2 - 20*g^2*i^2*n*log(e))*a^2*b^2*c^3
*d^2)*B^2*log(d*x + c)/(b^2*d^3) - 1/15*(b^5*c^5*g^2*i^2*n^2 - 5*a*b^4*c^4*d*g^2*i^2*n^2 + 10*a^2*b^3*c^3*d^2*
g^2*i^2*n^2 - 10*a^3*b^2*c^2*d^3*g^2*i^2*n^2 + 5*a^4*b*c*d^4*g^2*i^2*n^2 - a^5*d^5*g^2*i^2*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^3*d^3) + 1/60*(12*B^2*b^5*d^5*g
^2*i^2*x^5*log(e)^2 - 6*((g^2*i^2*n*log(e) - 5*g^2*i^2*log(e)^2)*b^5*c*d^4 - (g^2*i^2*n*log(e) + 5*g^2*i^2*log
(e)^2)*a*b^4*d^5)*B^2*x^4 + 2*((g^2*i^2*n^2 - 6*g^2*i^2*n*log(e) + 10*g^2*i^2*log(e)^2)*b^5*c^2*d^3 - 2*(g^2*i
^2*n^2 - 20*g^2*i^2*log(e)^2)*a*b^4*c*d^4 + (g^2*i^2*n^2 + 6*g^2*i^2*n*log(e) + 10*g^2*i^2*log(e)^2)*a^2*b^3*d
^5)*B^2*x^3 + ((3*g^2*i^2*n^2 - 2*g^2*i^2*n*log(e))*b^5*c^3*d^2 - 3*(g^2*i^2*n^2 + 10*g^2*i^2*n*log(e) - 20*g^
2*i^2*log(e)^2)*a*b^4*c^2*d^3 - 3*(g^2*i^2*n^2 - 10*g^2*i^2*n*log(e) - 20*g^2*i^2*log(e)^2)*a^2*b^3*c*d^4 + (3
*g^2*i^2*n^2 + 2*g^2*i^2*n*log(e))*a^3*b^2*d^5)*B^2*x^2 - 2*(10*a^3*b^2*c^2*d^3*g^2*i^2*n^2 - 5*a^4*b*c*d^4*g^
2*i^2*n^2 + a^5*d^5*g^2*i^2*n^2)*B^2*log(b*x + a)^2 + 4*(b^5*c^5*g^2*i^2*n^2 - 5*a*b^4*c^4*d*g^2*i^2*n^2 + 10*
a^2*b^3*c^3*d^2*g^2*i^2*n^2)*B^2*log(b*x + a)*log(d*x + c) - 2*(b^5*c^5*g^2*i^2*n^2 - 5*a*b^4*c^4*d*g^2*i^2*n^
2 + 10*a^2*b^3*c^3*d^2*g^2*i^2*n^2)*B^2*log(d*x + c)^2 - 2*(2*(g^2*i^2*n^2 - g^2*i^2*n*log(e))*b^5*c^4*d - (11
*g^2*i^2*n^2 - 10*g^2*i^2*n*log(e))*a*b^4*c^3*d^2 + 6*(3*g^2*i^2*n^2 - 5*g^2*i^2*log(e)^2)*a^2*b^3*c^2*d^3 - (
11*g^2*i^2*n^2 + 10*g^2*i^2*n*log(e))*a^3*b^2*c*d^4 + 2*(g^2*i^2*n^2 + g^2*i^2*n*log(e))*a^4*b*d^5)*B^2*x + 2*
(2*a*b^4*c^4*d*g^2*i^2*n^2 - 9*a^2*b^3*c^3*d^2*g^2*i^2*n^2 + 2*a^5*d^5*g^2*i^2*n*log(e) + (9*g^2*i^2*n^2 + 20*
g^2*i^2*n*log(e))*a^3*b^2*c^2*d^3 - 2*(g^2*i^2*n^2 + 5*g^2*i^2*n*log(e))*a^4*b*c*d^4)*B^2*log(b*x + a) + 2*(6*
B^2*b^5*d^5*g^2*i^2*x^5 + 30*B^2*a^2*b^3*c^2*d^3*g^2*i^2*x + 15*(b^5*c*d^4*g^2*i^2 + a*b^4*d^5*g^2*i^2)*B^2*x^
4 + 10*(b^5*c^2*d^3*g^2*i^2 + 4*a*b^4*c*d^4*g^2*i^2 + a^2*b^3*d^5*g^2*i^2)*B^2*x^3 + 30*(a*b^4*c^2*d^3*g^2*i^2
+ a^2*b^3*c*d^4*g^2*i^2)*B^2*x^2)*log((b*x + a)^n)^2 + 2*(6*B^2*b^5*d^5*g^2*i^2*x^5 + 30*B^2*a^2*b^3*c^2*d^3*
g^2*i^2*x + 15*(b^5*c*d^4*g^2*i^2 + a*b^4*d^5*g^2*i^2)*B^2*x^4 + 10*(b^5*c^2*d^3*g^2*i^2 + 4*a*b^4*c*d^4*g^2*i
^2 + a^2*b^3*d^5*g^2*i^2)*B^2*x^3 + 30*(a*b^4*c^2*d^3*g^2*i^2 + a^2*b^3*c*d^4*g^2*i^2)*B^2*x^2)*log((d*x + c)^
n)^2 + 2*(12*B^2*b^5*d^5*g^2*i^2*x^5*log(e) - 3*((g^2*i^2*n - 10*g^2*i^2*log(e))*b^5*c*d^4 - (g^2*i^2*n + 10*g
^2*i^2*log(e))*a*b^4*d^5)*B^2*x^4 + 2*(40*a*b^4*c*d^4*g^2*i^2*log(e) - (3*g^2*i^2*n - 10*g^2*i^2*log(e))*b^5*c
^2*d^3 + (3*g^2*i^2*n + 10*g^2*i^2*log(e))*a^2*b^3*d^5)*B^2*x^3 - (b^5*c^3*d^2*g^2*i^2*n - a^3*b^2*d^5*g^2*i^2
*n + 15*(g^2*i^2*n - 4*g^2*i^2*log(e))*a*b^4*c^2*d^3 - 15*(g^2*i^2*n + 4*g^2*i^2*log(e))*a^2*b^3*c*d^4)*B^2*x^
2 + 2*(b^5*c^4*d*g^2*i^2*n - 5*a*b^4*c^3*d^2*g^2*i^2*n + 5*a^3*b^2*c*d^4*g^2*i^2*n - a^4*b*d^5*g^2*i^2*n + 30*
a^2*b^3*c^2*d^3*g^2*i^2*log(e))*B^2*x + 2*(10*a^3*b^2*c^2*d^3*g^2*i^2*n - 5*a^4*b*c*d^4*g^2*i^2*n + a^5*d^5*g^
2*i^2*n)*B^2*log(b*x + a) - 2*(b^5*c^5*g^2*i^2*n - 5*a*b^4*c^4*d*g^2*i^2*n + 10*a^2*b^3*c^3*d^2*g^2*i^2*n)*B^2
*log(d*x + c))*log((b*x + a)^n) - 2*(12*B^2*b^5*d^5*g^2*i^2*x^5*log(e) - 3*((g^2*i^2*n - 10*g^2*i^2*log(e))*b^
5*c*d^4 - (g^2*i^2*n + 10*g^2*i^2*log(e))*a*b^4*d^5)*B^2*x^4 + 2*(40*a*b^4*c*d^4*g^2*i^2*log(e) - (3*g^2*i^2*n
- 10*g^2*i^2*log(e))*b^5*c^2*d^3 + (3*g^2*i^2*n + 10*g^2*i^2*log(e))*a^2*b^3*d^5)*B^2*x^3 - (b^5*c^3*d^2*g^2*
i^2*n - a^3*b^2*d^5*g^2*i^2*n + 15*(g^2*i^2*n - 4*g^2*i^2*log(e))*a*b^4*c^2*d^3 - 15*(g^2*i^2*n + 4*g^2*i^2*lo
g(e))*a^2*b^3*c*d^4)*B^2*x^2 + 2*(b^5*c^4*d*g^2*i^2*n - 5*a*b^4*c^3*d^2*g^2*i^2*n + 5*a^3*b^2*c*d^4*g^2*i^2*n
- a^4*b*d^5*g^2*i^2*n + 30*a^2*b^3*c^2*d^3*g^2*i^2*log(e))*B^2*x + 2*(10*a^3*b^2*c^2*d^3*g^2*i^2*n - 5*a^4*b*c
*d^4*g^2*i^2*n + a^5*d^5*g^2*i^2*n)*B^2*log(b*x + a) - 2*(b^5*c^5*g^2*i^2*n - 5*a*b^4*c^4*d*g^2*i^2*n + 10*a^2
*b^3*c^3*d^2*g^2*i^2*n)*B^2*log(d*x + c) + 2*(6*B^2*b^5*d^5*g^2*i^2*x^5 + 30*B^2*a^2*b^3*c^2*d^3*g^2*i^2*x + 1
5*(b^5*c*d^4*g^2*i^2 + a*b^4*d^5*g^2*i^2)*B^2*x^4 + 10*(b^5*c^2*d^3*g^2*i^2 + 4*a*b^4*c*d^4*g^2*i^2 + a^2*b^3*
d^5*g^2*i^2)*B^2*x^3 + 30*(a*b^4*c^2*d^3*g^2*i^2 + a^2*b^3*c*d^4*g^2*i^2)*B^2*x^2)*log((b*x + a)^n))*log((d*x
+ c)^n))/(b^3*d^3)

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (A^{2} b^{2} d^{2} g^{2} i^{2} x^{4} + A^{2} a^{2} c^{2} g^{2} i^{2} + 2 \,{\left (A^{2} b^{2} c d + A^{2} a b d^{2}\right )} g^{2} i^{2} x^{3} +{\left (A^{2} b^{2} c^{2} + 4 \, A^{2} a b c d + A^{2} a^{2} d^{2}\right )} g^{2} i^{2} x^{2} + 2 \,{\left (A^{2} a b c^{2} + A^{2} a^{2} c d\right )} g^{2} i^{2} x +{\left (B^{2} b^{2} d^{2} g^{2} i^{2} x^{4} + B^{2} a^{2} c^{2} g^{2} i^{2} + 2 \,{\left (B^{2} b^{2} c d + B^{2} a b d^{2}\right )} g^{2} i^{2} x^{3} +{\left (B^{2} b^{2} c^{2} + 4 \, B^{2} a b c d + B^{2} a^{2} d^{2}\right )} g^{2} i^{2} x^{2} + 2 \,{\left (B^{2} a b c^{2} + B^{2} a^{2} c d\right )} g^{2} i^{2} x\right )} \log \left (e \left (\frac{b x + a}{d x + c}\right )^{n}\right )^{2} + 2 \,{\left (A B b^{2} d^{2} g^{2} i^{2} x^{4} + A B a^{2} c^{2} g^{2} i^{2} + 2 \,{\left (A B b^{2} c d + A B a b d^{2}\right )} g^{2} i^{2} x^{3} +{\left (A B b^{2} c^{2} + 4 \, A B a b c d + A B a^{2} d^{2}\right )} g^{2} i^{2} x^{2} + 2 \,{\left (A B a b c^{2} + A B a^{2} c d\right )} g^{2} i^{2} 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)^2*(d*i*x+c*i)^2*(A+B*log(e*((b*x+a)/(d*x+c))^n))^2,x, algorithm="fricas")

[Out]

integral(A^2*b^2*d^2*g^2*i^2*x^4 + A^2*a^2*c^2*g^2*i^2 + 2*(A^2*b^2*c*d + A^2*a*b*d^2)*g^2*i^2*x^3 + (A^2*b^2*
c^2 + 4*A^2*a*b*c*d + A^2*a^2*d^2)*g^2*i^2*x^2 + 2*(A^2*a*b*c^2 + A^2*a^2*c*d)*g^2*i^2*x + (B^2*b^2*d^2*g^2*i^
2*x^4 + B^2*a^2*c^2*g^2*i^2 + 2*(B^2*b^2*c*d + B^2*a*b*d^2)*g^2*i^2*x^3 + (B^2*b^2*c^2 + 4*B^2*a*b*c*d + B^2*a
^2*d^2)*g^2*i^2*x^2 + 2*(B^2*a*b*c^2 + B^2*a^2*c*d)*g^2*i^2*x)*log(e*((b*x + a)/(d*x + c))^n)^2 + 2*(A*B*b^2*d
^2*g^2*i^2*x^4 + A*B*a^2*c^2*g^2*i^2 + 2*(A*B*b^2*c*d + A*B*a*b*d^2)*g^2*i^2*x^3 + (A*B*b^2*c^2 + 4*A*B*a*b*c*
d + A*B*a^2*d^2)*g^2*i^2*x^2 + 2*(A*B*a*b*c^2 + A*B*a^2*c*d)*g^2*i^2*x)*log(e*((b*x + a)/(d*x + c))^n), x)

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

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

[Out]

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