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

Optimal. Leaf size=766 \[ -\frac{B^2 g^3 i^2 n^2 (b c-a d)^6 \text{PolyLog}\left (2,\frac{d (a+b x)}{b (c+d x)}\right )}{30 b^3 d^4}-\frac{B g^3 i^2 n (b c-a d)^6 \log \left (\frac{b c-a d}{b (c+d x)}\right ) \left (6 B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+6 A+11 B n\right )}{180 b^3 d^4}-\frac{B g^3 i^2 n (a+b x) (b c-a d)^5 \left (6 B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+6 A+5 B n\right )}{180 b^3 d^3}+\frac{B g^3 i^2 n (a+b x)^2 (b c-a d)^4 \left (3 B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+3 A+B n\right )}{180 b^3 d^2}-\frac{B g^3 i^2 n (a+b x)^3 (b c-a d)^3 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{90 b^3 d}+\frac{g^3 i^2 (a+b x)^4 (b c-a d)^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )^2}{60 b^3}-\frac{B g^3 i^2 n (a+b x)^4 (b c-a d)^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{20 b^3}+\frac{g^3 i^2 (a+b x)^4 (c+d x) (b c-a d) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )^2}{15 b^2}-\frac{B g^3 i^2 n (a+b x)^4 (c+d x) (b c-a d) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{15 b^2}+\frac{g^3 i^2 (a+b x)^4 (c+d x)^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )^2}{6 b}+\frac{3 B^2 g^3 i^2 n^2 x (b c-a d)^5}{20 b^2 d^3}-\frac{B^2 g^3 i^2 n^2 (b c-a d)^6 \log (c+d x)}{20 b^3 d^4}+\frac{B^2 g^3 i^2 n^2 (a+b x)^4 (b c-a d)^2}{60 b^3}-\frac{3 B^2 g^3 i^2 n^2 (c+d x)^2 (b c-a d)^4}{40 b d^4}+\frac{B^2 g^3 i^2 n^2 (c+d x)^3 (b c-a d)^3}{60 d^4} \]

[Out]

(3*B^2*(b*c - a*d)^5*g^3*i^2*n^2*x)/(20*b^2*d^3) + (B^2*(b*c - a*d)^2*g^3*i^2*n^2*(a + b*x)^4)/(60*b^3) - (3*B
^2*(b*c - a*d)^4*g^3*i^2*n^2*(c + d*x)^2)/(40*b*d^4) + (B^2*(b*c - a*d)^3*g^3*i^2*n^2*(c + d*x)^3)/(60*d^4) -
(B*(b*c - a*d)^3*g^3*i^2*n*(a + b*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(90*b^3*d) - (B*(b*c - a*d)^2*g
^3*i^2*n*(a + b*x)^4*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(20*b^3) - (B*(b*c - a*d)*g^3*i^2*n*(a + b*x)^4*(
c + d*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(15*b^2) + ((b*c - a*d)^2*g^3*i^2*(a + b*x)^4*(A + B*Log[e*((
a + b*x)/(c + d*x))^n])^2)/(60*b^3) + ((b*c - a*d)*g^3*i^2*(a + b*x)^4*(c + d*x)*(A + B*Log[e*((a + b*x)/(c +
d*x))^n])^2)/(15*b^2) + (g^3*i^2*(a + b*x)^4*(c + d*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(6*b) + (B*
(b*c - a*d)^4*g^3*i^2*n*(a + b*x)^2*(3*A + B*n + 3*B*Log[e*((a + b*x)/(c + d*x))^n]))/(180*b^3*d^2) - (B*(b*c
- a*d)^5*g^3*i^2*n*(a + b*x)*(6*A + 5*B*n + 6*B*Log[e*((a + b*x)/(c + d*x))^n]))/(180*b^3*d^3) - (B*(b*c - a*d
)^6*g^3*i^2*n*(6*A + 11*B*n + 6*B*Log[e*((a + b*x)/(c + d*x))^n])*Log[(b*c - a*d)/(b*(c + d*x))])/(180*b^3*d^4
) - (B^2*(b*c - a*d)^6*g^3*i^2*n^2*Log[c + d*x])/(20*b^3*d^4) - (B^2*(b*c - a*d)^6*g^3*i^2*n^2*PolyLog[2, (d*(
a + b*x))/(b*(c + d*x))])/(30*b^3*d^4)

________________________________________________________________________________________

Rubi [A]  time = 3.34846, antiderivative size = 848, normalized size of antiderivative = 1.11, number of steps used = 83, 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} \[ \frac{B^2 g^3 i^2 n^2 \log ^2(c+d x) (b c-a d)^6}{60 b^3 d^4}+\frac{B^2 g^3 i^2 n^2 \log (c+d x) (b c-a d)^6}{90 b^3 d^4}-\frac{B^2 g^3 i^2 n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x) (b c-a d)^6}{30 b^3 d^4}+\frac{B g^3 i^2 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x) (b c-a d)^6}{30 b^3 d^4}-\frac{B^2 g^3 i^2 n^2 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right ) (b c-a d)^6}{30 b^3 d^4}+\frac{B^2 g^3 i^2 n^2 x (b c-a d)^5}{45 b^2 d^3}-\frac{A B g^3 i^2 n x (b c-a d)^5}{30 b^2 d^3}-\frac{B^2 g^3 i^2 n (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) (b c-a d)^5}{30 b^3 d^3}-\frac{7 B^2 g^3 i^2 n^2 (a+b x)^2 (b c-a d)^4}{360 b^3 d^2}+\frac{B g^3 i^2 n (a+b x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) (b c-a d)^4}{60 b^3 d^2}+\frac{B^2 g^3 i^2 n^2 (a+b x)^3 (b c-a d)^3}{60 b^3 d}-\frac{B g^3 i^2 n (a+b x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) (b c-a d)^3}{90 b^3 d}+\frac{B^2 g^3 i^2 n^2 (a+b x)^4 (b c-a d)^2}{60 b^3}+\frac{g^3 i^2 (a+b x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 (b c-a d)^2}{4 b^3}-\frac{7 B g^3 i^2 n (a+b x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) (b c-a d)^2}{60 b^3}+\frac{2 d g^3 i^2 (a+b x)^5 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 (b c-a d)}{5 b^3}-\frac{B d g^3 i^2 n (a+b x)^5 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) (b c-a d)}{15 b^3}+\frac{d^2 g^3 i^2 (a+b x)^6 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{6 b^3} \]

Antiderivative was successfully verified.

[In]

Int[(a*g + b*g*x)^3*(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)^5*g^3*i^2*n*x)/(30*b^2*d^3) + (B^2*(b*c - a*d)^5*g^3*i^2*n^2*x)/(45*b^2*d^3) - (7*B^2*(b*c -
a*d)^4*g^3*i^2*n^2*(a + b*x)^2)/(360*b^3*d^2) + (B^2*(b*c - a*d)^3*g^3*i^2*n^2*(a + b*x)^3)/(60*b^3*d) + (B^2
*(b*c - a*d)^2*g^3*i^2*n^2*(a + b*x)^4)/(60*b^3) - (B^2*(b*c - a*d)^5*g^3*i^2*n*(a + b*x)*Log[e*((a + b*x)/(c
+ d*x))^n])/(30*b^3*d^3) + (B*(b*c - a*d)^4*g^3*i^2*n*(a + b*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(60*
b^3*d^2) - (B*(b*c - a*d)^3*g^3*i^2*n*(a + b*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(90*b^3*d) - (7*B*(b
*c - a*d)^2*g^3*i^2*n*(a + b*x)^4*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(60*b^3) - (B*d*(b*c - a*d)*g^3*i^2*
n*(a + b*x)^5*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(15*b^3) + ((b*c - a*d)^2*g^3*i^2*(a + b*x)^4*(A + B*Log
[e*((a + b*x)/(c + d*x))^n])^2)/(4*b^3) + (2*d*(b*c - a*d)*g^3*i^2*(a + b*x)^5*(A + B*Log[e*((a + b*x)/(c + d*
x))^n])^2)/(5*b^3) + (d^2*g^3*i^2*(a + b*x)^6*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(6*b^3) + (B^2*(b*c -
a*d)^6*g^3*i^2*n^2*Log[c + d*x])/(90*b^3*d^4) - (B^2*(b*c - a*d)^6*g^3*i^2*n^2*Log[-((d*(a + b*x))/(b*c - a*d)
)]*Log[c + d*x])/(30*b^3*d^4) + (B*(b*c - a*d)^6*g^3*i^2*n*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*Log[c + d*x]
)/(30*b^3*d^4) + (B^2*(b*c - a*d)^6*g^3*i^2*n^2*Log[c + d*x]^2)/(60*b^3*d^4) - (B^2*(b*c - a*d)^6*g^3*i^2*n^2*
PolyLog[2, (b*(c + d*x))/(b*c - a*d)])/(30*b^3*d^4)

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 (168 c+168 d x)^2 (a g+b g x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \, dx &=\int \left (\frac{28224 (b c-a d)^2 (a g+b g x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^2}+\frac{56448 d (b c-a d) (a g+b g x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^2 g}+\frac{28224 d^2 (a g+b g x)^5 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^2 g^2}\right ) \, dx\\ &=\frac{\left (28224 (b c-a d)^2\right ) \int (a g+b g x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \, dx}{b^2}+\frac{\left (28224 d^2\right ) \int (a g+b g x)^5 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \, dx}{b^2 g^2}+\frac{(56448 d (b c-a d)) \int (a g+b g x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \, dx}{b^2 g}\\ &=\frac{7056 (b c-a d)^2 g^3 (a+b x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^3}+\frac{56448 d (b c-a d) g^3 (a+b x)^5 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{5 b^3}+\frac{4704 d^2 g^3 (a+b x)^6 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^3}-\frac{\left (9408 B d^2 n\right ) \int \frac{(b c-a d) g^6 (a+b x)^5 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{c+d x} \, dx}{b^3 g^3}-\frac{(112896 B d (b c-a d) n) \int \frac{(b c-a d) g^5 (a+b x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{c+d x} \, dx}{5 b^3 g^2}-\frac{\left (14112 B (b c-a d)^2 n\right ) \int \frac{(b c-a d) g^4 (a+b x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{c+d x} \, dx}{b^3 g}\\ &=\frac{7056 (b c-a d)^2 g^3 (a+b x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^3}+\frac{56448 d (b c-a d) g^3 (a+b x)^5 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{5 b^3}+\frac{4704 d^2 g^3 (a+b x)^6 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^3}-\frac{\left (9408 B d^2 (b c-a d) g^3 n\right ) \int \frac{(a+b x)^5 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{c+d x} \, dx}{b^3}-\frac{\left (112896 B d (b c-a d)^2 g^3 n\right ) \int \frac{(a+b x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{c+d x} \, dx}{5 b^3}-\frac{\left (14112 B (b c-a d)^3 g^3 n\right ) \int \frac{(a+b x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{c+d x} \, dx}{b^3}\\ &=\frac{7056 (b c-a d)^2 g^3 (a+b x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^3}+\frac{56448 d (b c-a d) g^3 (a+b x)^5 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{5 b^3}+\frac{4704 d^2 g^3 (a+b x)^6 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^3}-\frac{\left (9408 B d^2 (b c-a d) g^3 n\right ) \int \left (\frac{b (b c-a d)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{d^5}-\frac{b (b c-a d)^3 (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{d^4}+\frac{b (b c-a d)^2 (a+b x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{d^3}-\frac{b (b c-a d) (a+b x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{d^2}+\frac{b (a+b x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{d}+\frac{(-b c+a d)^5 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{d^5 (c+d x)}\right ) \, dx}{b^3}-\frac{\left (112896 B d (b c-a d)^2 g^3 n\right ) \int \left (-\frac{b (b c-a d)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{d^4}+\frac{b (b c-a d)^2 (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{d^3}-\frac{b (b c-a d) (a+b x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{d^2}+\frac{b (a+b x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{d}+\frac{(-b c+a d)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{d^4 (c+d x)}\right ) \, dx}{5 b^3}-\frac{\left (14112 B (b c-a d)^3 g^3 n\right ) \int \left (\frac{b (b c-a d)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{d^3}-\frac{b (b c-a d) (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{d^2}+\frac{b (a+b x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{d}+\frac{(-b c+a d)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{d^3 (c+d x)}\right ) \, dx}{b^3}\\ &=\frac{7056 (b c-a d)^2 g^3 (a+b x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^3}+\frac{56448 d (b c-a d) g^3 (a+b x)^5 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{5 b^3}+\frac{4704 d^2 g^3 (a+b x)^6 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^3}-\frac{\left (9408 B d (b c-a d) g^3 n\right ) \int (a+b x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \, dx}{b^2}+\frac{\left (9408 B (b c-a d)^2 g^3 n\right ) \int (a+b x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \, dx}{b^2}-\frac{\left (112896 B (b c-a d)^2 g^3 n\right ) \int (a+b x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \, dx}{5 b^2}-\frac{\left (9408 B (b c-a d)^3 g^3 n\right ) \int (a+b x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \, dx}{b^2 d}-\frac{\left (14112 B (b c-a d)^3 g^3 n\right ) \int (a+b x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \, dx}{b^2 d}+\frac{\left (112896 B (b c-a d)^3 g^3 n\right ) \int (a+b x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \, dx}{5 b^2 d}+\frac{\left (9408 B (b c-a d)^4 g^3 n\right ) \int (a+b x) \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 (14112 B (b c-a d)^4 g^3 n\right ) \int (a+b x) \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 (112896 B (b c-a d)^4 g^3 n\right ) \int (a+b x) \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 (9408 B (b c-a d)^5 g^3 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^3}-\frac{\left (14112 B (b c-a d)^5 g^3 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^3}+\frac{\left (112896 B (b c-a d)^5 g^3 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^3}+\frac{\left (9408 B (b c-a d)^6 g^3 n\right ) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{c+d x} \, dx}{b^3 d^3}+\frac{\left (14112 B (b c-a d)^6 g^3 n\right ) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{c+d x} \, dx}{b^3 d^3}-\frac{\left (112896 B (b c-a d)^6 g^3 n\right ) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{c+d x} \, dx}{5 b^3 d^3}\\ &=-\frac{4704 A B (b c-a d)^5 g^3 n x}{5 b^2 d^3}+\frac{2352 B (b c-a d)^4 g^3 n (a+b x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{5 b^3 d^2}-\frac{1568 B (b c-a d)^3 g^3 n (a+b x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{5 b^3 d}-\frac{16464 B (b c-a d)^2 g^3 n (a+b x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{5 b^3}-\frac{9408 B d (b c-a d) g^3 n (a+b x)^5 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{5 b^3}+\frac{7056 (b c-a d)^2 g^3 (a+b x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^3}+\frac{56448 d (b c-a d) g^3 (a+b x)^5 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{5 b^3}+\frac{4704 d^2 g^3 (a+b x)^6 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^3}+\frac{4704 B (b c-a d)^6 g^3 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{5 b^3 d^4}-\frac{\left (9408 B^2 (b c-a d)^5 g^3 n\right ) \int \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \, dx}{b^2 d^3}-\frac{\left (14112 B^2 (b c-a d)^5 g^3 n\right ) \int \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \, dx}{b^2 d^3}+\frac{\left (112896 B^2 (b c-a d)^5 g^3 n\right ) \int \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \, dx}{5 b^2 d^3}+\frac{\left (9408 B^2 d (b c-a d) g^3 n^2\right ) \int \frac{(b c-a d) (a+b x)^4}{c+d x} \, dx}{5 b^3}-\frac{\left (2352 B^2 (b c-a d)^2 g^3 n^2\right ) \int \frac{(b c-a d) (a+b x)^3}{c+d x} \, dx}{b^3}+\frac{\left (28224 B^2 (b c-a d)^2 g^3 n^2\right ) \int \frac{(b c-a d) (a+b x)^3}{c+d x} \, dx}{5 b^3}+\frac{\left (3136 B^2 (b c-a d)^3 g^3 n^2\right ) \int \frac{(b c-a d) (a+b x)^2}{c+d x} \, dx}{b^3 d}+\frac{\left (4704 B^2 (b c-a d)^3 g^3 n^2\right ) \int \frac{(b c-a d) (a+b x)^2}{c+d x} \, dx}{b^3 d}-\frac{\left (37632 B^2 (b c-a d)^3 g^3 n^2\right ) \int \frac{(b c-a d) (a+b x)^2}{c+d x} \, dx}{5 b^3 d}-\frac{\left (4704 B^2 (b c-a d)^4 g^3 n^2\right ) \int \frac{(b c-a d) (a+b x)}{c+d x} \, dx}{b^3 d^2}-\frac{\left (7056 B^2 (b c-a d)^4 g^3 n^2\right ) \int \frac{(b c-a d) (a+b x)}{c+d x} \, dx}{b^3 d^2}+\frac{\left (56448 B^2 (b c-a d)^4 g^3 n^2\right ) \int \frac{(b c-a d) (a+b x)}{c+d x} \, dx}{5 b^3 d^2}-\frac{\left (9408 B^2 (b c-a d)^6 g^3 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}{b^3 d^4}-\frac{\left (14112 B^2 (b c-a d)^6 g^3 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}{b^3 d^4}+\frac{\left (112896 B^2 (b c-a d)^6 g^3 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}{5 b^3 d^4}\\ &=-\frac{4704 A B (b c-a d)^5 g^3 n x}{5 b^2 d^3}-\frac{4704 B^2 (b c-a d)^5 g^3 n (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{5 b^3 d^3}+\frac{2352 B (b c-a d)^4 g^3 n (a+b x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{5 b^3 d^2}-\frac{1568 B (b c-a d)^3 g^3 n (a+b x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{5 b^3 d}-\frac{16464 B (b c-a d)^2 g^3 n (a+b x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{5 b^3}-\frac{9408 B d (b c-a d) g^3 n (a+b x)^5 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{5 b^3}+\frac{7056 (b c-a d)^2 g^3 (a+b x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^3}+\frac{56448 d (b c-a d) g^3 (a+b x)^5 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{5 b^3}+\frac{4704 d^2 g^3 (a+b x)^6 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^3}+\frac{4704 B (b c-a d)^6 g^3 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{5 b^3 d^4}+\frac{\left (9408 B^2 d (b c-a d)^2 g^3 n^2\right ) \int \frac{(a+b x)^4}{c+d x} \, dx}{5 b^3}-\frac{\left (2352 B^2 (b c-a d)^3 g^3 n^2\right ) \int \frac{(a+b x)^3}{c+d x} \, dx}{b^3}+\frac{\left (28224 B^2 (b c-a d)^3 g^3 n^2\right ) \int \frac{(a+b x)^3}{c+d x} \, dx}{5 b^3}+\frac{\left (3136 B^2 (b c-a d)^4 g^3 n^2\right ) \int \frac{(a+b x)^2}{c+d x} \, dx}{b^3 d}+\frac{\left (4704 B^2 (b c-a d)^4 g^3 n^2\right ) \int \frac{(a+b x)^2}{c+d x} \, dx}{b^3 d}-\frac{\left (37632 B^2 (b c-a d)^4 g^3 n^2\right ) \int \frac{(a+b x)^2}{c+d x} \, dx}{5 b^3 d}-\frac{\left (4704 B^2 (b c-a d)^5 g^3 n^2\right ) \int \frac{a+b x}{c+d x} \, dx}{b^3 d^2}-\frac{\left (7056 B^2 (b c-a d)^5 g^3 n^2\right ) \int \frac{a+b x}{c+d x} \, dx}{b^3 d^2}+\frac{\left (56448 B^2 (b c-a d)^5 g^3 n^2\right ) \int \frac{a+b x}{c+d x} \, dx}{5 b^3 d^2}-\frac{\left (9408 B^2 (b c-a d)^6 g^3 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}{b^3 d^4}-\frac{\left (14112 B^2 (b c-a d)^6 g^3 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}{b^3 d^4}+\frac{\left (112896 B^2 (b c-a d)^6 g^3 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}{5 b^3 d^4}+\frac{\left (9408 B^2 (b c-a d)^6 g^3 n^2\right ) \int \frac{1}{c+d x} \, dx}{b^3 d^3}+\frac{\left (14112 B^2 (b c-a d)^6 g^3 n^2\right ) \int \frac{1}{c+d x} \, dx}{b^3 d^3}-\frac{\left (112896 B^2 (b c-a d)^6 g^3 n^2\right ) \int \frac{1}{c+d x} \, dx}{5 b^3 d^3}\\ &=-\frac{4704 A B (b c-a d)^5 g^3 n x}{5 b^2 d^3}-\frac{4704 B^2 (b c-a d)^5 g^3 n (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{5 b^3 d^3}+\frac{2352 B (b c-a d)^4 g^3 n (a+b x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{5 b^3 d^2}-\frac{1568 B (b c-a d)^3 g^3 n (a+b x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{5 b^3 d}-\frac{16464 B (b c-a d)^2 g^3 n (a+b x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{5 b^3}-\frac{9408 B d (b c-a d) g^3 n (a+b x)^5 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{5 b^3}+\frac{7056 (b c-a d)^2 g^3 (a+b x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^3}+\frac{56448 d (b c-a d) g^3 (a+b x)^5 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{5 b^3}+\frac{4704 d^2 g^3 (a+b x)^6 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^3}+\frac{4704 B^2 (b c-a d)^6 g^3 n^2 \log (c+d x)}{5 b^3 d^4}+\frac{4704 B (b c-a d)^6 g^3 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{5 b^3 d^4}+\frac{\left (9408 B^2 d (b c-a d)^2 g^3 n^2\right ) \int \left (-\frac{b (b c-a d)^3}{d^4}+\frac{b (b c-a d)^2 (a+b x)}{d^3}-\frac{b (b c-a d) (a+b x)^2}{d^2}+\frac{b (a+b x)^3}{d}+\frac{(-b c+a d)^4}{d^4 (c+d x)}\right ) \, dx}{5 b^3}-\frac{\left (2352 B^2 (b c-a d)^3 g^3 n^2\right ) \int \left (\frac{b (b c-a d)^2}{d^3}-\frac{b (b c-a d) (a+b x)}{d^2}+\frac{b (a+b x)^2}{d}+\frac{(-b c+a d)^3}{d^3 (c+d x)}\right ) \, dx}{b^3}+\frac{\left (28224 B^2 (b c-a d)^3 g^3 n^2\right ) \int \left (\frac{b (b c-a d)^2}{d^3}-\frac{b (b c-a d) (a+b x)}{d^2}+\frac{b (a+b x)^2}{d}+\frac{(-b c+a d)^3}{d^3 (c+d x)}\right ) \, dx}{5 b^3}+\frac{\left (3136 B^2 (b c-a d)^4 g^3 n^2\right ) \int \left (-\frac{b (b c-a d)}{d^2}+\frac{b (a+b x)}{d}+\frac{(-b c+a d)^2}{d^2 (c+d x)}\right ) \, dx}{b^3 d}+\frac{\left (4704 B^2 (b c-a d)^4 g^3 n^2\right ) \int \left (-\frac{b (b c-a d)}{d^2}+\frac{b (a+b x)}{d}+\frac{(-b c+a d)^2}{d^2 (c+d x)}\right ) \, dx}{b^3 d}-\frac{\left (37632 B^2 (b c-a d)^4 g^3 n^2\right ) \int \left (-\frac{b (b c-a d)}{d^2}+\frac{b (a+b x)}{d}+\frac{(-b c+a d)^2}{d^2 (c+d x)}\right ) \, dx}{5 b^3 d}-\frac{\left (4704 B^2 (b c-a d)^5 g^3 n^2\right ) \int \left (\frac{b}{d}+\frac{-b c+a d}{d (c+d x)}\right ) \, dx}{b^3 d^2}-\frac{\left (7056 B^2 (b c-a d)^5 g^3 n^2\right ) \int \left (\frac{b}{d}+\frac{-b c+a d}{d (c+d x)}\right ) \, dx}{b^3 d^2}+\frac{\left (56448 B^2 (b c-a d)^5 g^3 n^2\right ) \int \left (\frac{b}{d}+\frac{-b c+a d}{d (c+d x)}\right ) \, dx}{5 b^3 d^2}-\frac{\left (9408 B^2 (b c-a d)^6 g^3 n^2\right ) \int \frac{\log (c+d x)}{a+b x} \, dx}{b^2 d^4}-\frac{\left (14112 B^2 (b c-a d)^6 g^3 n^2\right ) \int \frac{\log (c+d x)}{a+b x} \, dx}{b^2 d^4}+\frac{\left (112896 B^2 (b c-a d)^6 g^3 n^2\right ) \int \frac{\log (c+d x)}{a+b x} \, dx}{5 b^2 d^4}+\frac{\left (9408 B^2 (b c-a d)^6 g^3 n^2\right ) \int \frac{\log (c+d x)}{c+d x} \, dx}{b^3 d^3}+\frac{\left (14112 B^2 (b c-a d)^6 g^3 n^2\right ) \int \frac{\log (c+d x)}{c+d x} \, dx}{b^3 d^3}-\frac{\left (112896 B^2 (b c-a d)^6 g^3 n^2\right ) \int \frac{\log (c+d x)}{c+d x} \, dx}{5 b^3 d^3}\\ &=-\frac{4704 A B (b c-a d)^5 g^3 n x}{5 b^2 d^3}+\frac{3136 B^2 (b c-a d)^5 g^3 n^2 x}{5 b^2 d^3}-\frac{2744 B^2 (b c-a d)^4 g^3 n^2 (a+b x)^2}{5 b^3 d^2}+\frac{2352 B^2 (b c-a d)^3 g^3 n^2 (a+b x)^3}{5 b^3 d}+\frac{2352 B^2 (b c-a d)^2 g^3 n^2 (a+b x)^4}{5 b^3}-\frac{4704 B^2 (b c-a d)^5 g^3 n (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{5 b^3 d^3}+\frac{2352 B (b c-a d)^4 g^3 n (a+b x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{5 b^3 d^2}-\frac{1568 B (b c-a d)^3 g^3 n (a+b x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{5 b^3 d}-\frac{16464 B (b c-a d)^2 g^3 n (a+b x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{5 b^3}-\frac{9408 B d (b c-a d) g^3 n (a+b x)^5 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{5 b^3}+\frac{7056 (b c-a d)^2 g^3 (a+b x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^3}+\frac{56448 d (b c-a d) g^3 (a+b x)^5 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{5 b^3}+\frac{4704 d^2 g^3 (a+b x)^6 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^3}+\frac{1568 B^2 (b c-a d)^6 g^3 n^2 \log (c+d x)}{5 b^3 d^4}-\frac{4704 B^2 (b c-a d)^6 g^3 n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{5 b^3 d^4}+\frac{4704 B (b c-a d)^6 g^3 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{5 b^3 d^4}+\frac{\left (9408 B^2 (b c-a d)^6 g^3 n^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,c+d x\right )}{b^3 d^4}+\frac{\left (14112 B^2 (b c-a d)^6 g^3 n^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,c+d x\right )}{b^3 d^4}-\frac{\left (112896 B^2 (b c-a d)^6 g^3 n^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,c+d x\right )}{5 b^3 d^4}+\frac{\left (9408 B^2 (b c-a d)^6 g^3 n^2\right ) \int \frac{\log \left (\frac{d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{b^3 d^3}+\frac{\left (14112 B^2 (b c-a d)^6 g^3 n^2\right ) \int \frac{\log \left (\frac{d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{b^3 d^3}-\frac{\left (112896 B^2 (b c-a d)^6 g^3 n^2\right ) \int \frac{\log \left (\frac{d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{5 b^3 d^3}\\ &=-\frac{4704 A B (b c-a d)^5 g^3 n x}{5 b^2 d^3}+\frac{3136 B^2 (b c-a d)^5 g^3 n^2 x}{5 b^2 d^3}-\frac{2744 B^2 (b c-a d)^4 g^3 n^2 (a+b x)^2}{5 b^3 d^2}+\frac{2352 B^2 (b c-a d)^3 g^3 n^2 (a+b x)^3}{5 b^3 d}+\frac{2352 B^2 (b c-a d)^2 g^3 n^2 (a+b x)^4}{5 b^3}-\frac{4704 B^2 (b c-a d)^5 g^3 n (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{5 b^3 d^3}+\frac{2352 B (b c-a d)^4 g^3 n (a+b x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{5 b^3 d^2}-\frac{1568 B (b c-a d)^3 g^3 n (a+b x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{5 b^3 d}-\frac{16464 B (b c-a d)^2 g^3 n (a+b x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{5 b^3}-\frac{9408 B d (b c-a d) g^3 n (a+b x)^5 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{5 b^3}+\frac{7056 (b c-a d)^2 g^3 (a+b x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^3}+\frac{56448 d (b c-a d) g^3 (a+b x)^5 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{5 b^3}+\frac{4704 d^2 g^3 (a+b x)^6 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^3}+\frac{1568 B^2 (b c-a d)^6 g^3 n^2 \log (c+d x)}{5 b^3 d^4}-\frac{4704 B^2 (b c-a d)^6 g^3 n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{5 b^3 d^4}+\frac{4704 B (b c-a d)^6 g^3 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{5 b^3 d^4}+\frac{2352 B^2 (b c-a d)^6 g^3 n^2 \log ^2(c+d x)}{5 b^3 d^4}+\frac{\left (9408 B^2 (b c-a d)^6 g^3 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 )}{b^3 d^4}+\frac{\left (14112 B^2 (b c-a d)^6 g^3 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 )}{b^3 d^4}-\frac{\left (112896 B^2 (b c-a d)^6 g^3 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 )}{5 b^3 d^4}\\ &=-\frac{4704 A B (b c-a d)^5 g^3 n x}{5 b^2 d^3}+\frac{3136 B^2 (b c-a d)^5 g^3 n^2 x}{5 b^2 d^3}-\frac{2744 B^2 (b c-a d)^4 g^3 n^2 (a+b x)^2}{5 b^3 d^2}+\frac{2352 B^2 (b c-a d)^3 g^3 n^2 (a+b x)^3}{5 b^3 d}+\frac{2352 B^2 (b c-a d)^2 g^3 n^2 (a+b x)^4}{5 b^3}-\frac{4704 B^2 (b c-a d)^5 g^3 n (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{5 b^3 d^3}+\frac{2352 B (b c-a d)^4 g^3 n (a+b x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{5 b^3 d^2}-\frac{1568 B (b c-a d)^3 g^3 n (a+b x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{5 b^3 d}-\frac{16464 B (b c-a d)^2 g^3 n (a+b x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{5 b^3}-\frac{9408 B d (b c-a d) g^3 n (a+b x)^5 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{5 b^3}+\frac{7056 (b c-a d)^2 g^3 (a+b x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^3}+\frac{56448 d (b c-a d) g^3 (a+b x)^5 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{5 b^3}+\frac{4704 d^2 g^3 (a+b x)^6 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^3}+\frac{1568 B^2 (b c-a d)^6 g^3 n^2 \log (c+d x)}{5 b^3 d^4}-\frac{4704 B^2 (b c-a d)^6 g^3 n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{5 b^3 d^4}+\frac{4704 B (b c-a d)^6 g^3 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{5 b^3 d^4}+\frac{2352 B^2 (b c-a d)^6 g^3 n^2 \log ^2(c+d x)}{5 b^3 d^4}-\frac{4704 B^2 (b c-a d)^6 g^3 n^2 \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{5 b^3 d^4}\\ \end{align*}

Mathematica [B]  time = 1.44315, size = 1634, normalized size = 2.13 \[ \text{result too large to display} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(g^3*i^2*(15*(b*c - a*d)^2*(a + b*x)^4*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2 + 24*d*(b*c - a*d)*(a + b*x)^5
*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2 + 10*d^2*(a + b*x)^6*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2 - (5*B
*(b*c - a*d)^3*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)])))/d^4 + (2*B*(b*c - a*d)^2*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)])))/d^4 - (B*(b*c - a*d)*n*(120*A*b*d*(b*c - a*d)^4*x + 120*B*d*(b*c - a*
d)^4*(a + b*x)*Log[e*((a + b*x)/(c + d*x))^n] + 60*d^2*(-(b*c) + a*d)^3*(a + b*x)^2*(A + B*Log[e*((a + b*x)/(c
+ d*x))^n]) + 40*d^3*(b*c - a*d)^2*(a + b*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) + 30*d^4*(-(b*c) + a*d)
*(a + b*x)^4*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) + 24*d^5*(a + b*x)^5*(A + B*Log[e*((a + b*x)/(c + d*x))^n]
) - 120*B*(b*c - a*d)^5*n*Log[c + d*x] - 120*(b*c - a*d)^5*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*Log[c + d*x]
+ 20*B*(b*c - a*d)^3*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]) + 5*B*(b*c - a*
d)^2*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]) + 2*B*(b*c - a*d)*n*(12*b*d*(b*c - a*d)^3*x - 6*d^2*(b*c - a*d)^2*(a + b*x)^2 + 4*d^3*(b*c - a*d)*(a +
b*x)^3 - 3*d^4*(a + b*x)^4 - 12*(b*c - a*d)^4*Log[c + d*x]) + 60*B*(b*c - a*d)^4*n*(b*d*x + (-(b*c) + a*d)*Log
[c + d*x]) + 60*B*(b*c - a*d)^5*n*((2*Log[(d*(a + b*x))/(-(b*c) + a*d)] - Log[c + d*x])*Log[c + d*x] + 2*PolyL
og[2, (b*(c + d*x))/(b*c - a*d)])))/(6*d^4)))/(60*b^3)

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Maple [F]  time = 0.734, size = 0, normalized size = 0. \begin{align*} \int \left ( bgx+ag \right ) ^{3} \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)^3*(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)^3*(d*i*x+c*i)^2*(A+B*ln(e*((b*x+a)/(d*x+c))^n))^2,x)

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Maxima [B]  time = 4.16418, size = 8035, normalized size = 10.49 \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)^3*(d*i*x+c*i)^2*(A+B*log(e*((b*x+a)/(d*x+c))^n))^2,x, algorithm="maxima")

[Out]

1/3*A*B*b^3*d^2*g^3*i^2*x^6*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 1/6*A^2*b^3*d^2*g^3*i^2*x^6 + 4/5*A*B*b^3
*c*d*g^3*i^2*x^5*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 6/5*A*B*a*b^2*d^2*g^3*i^2*x^5*log(e*(b*x/(d*x + c) +
a/(d*x + c))^n) + 2/5*A^2*b^3*c*d*g^3*i^2*x^5 + 3/5*A^2*a*b^2*d^2*g^3*i^2*x^5 + 1/2*A*B*b^3*c^2*g^3*i^2*x^4*l
og(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 3*A*B*a*b^2*c*d*g^3*i^2*x^4*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 3
/2*A*B*a^2*b*d^2*g^3*i^2*x^4*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 1/4*A^2*b^3*c^2*g^3*i^2*x^4 + 3/2*A^2*a*
b^2*c*d*g^3*i^2*x^4 + 3/4*A^2*a^2*b*d^2*g^3*i^2*x^4 + 2*A*B*a*b^2*c^2*g^3*i^2*x^3*log(e*(b*x/(d*x + c) + a/(d*
x + c))^n) + 4*A*B*a^2*b*c*d*g^3*i^2*x^3*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 2/3*A*B*a^3*d^2*g^3*i^2*x^3*
log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + A^2*a*b^2*c^2*g^3*i^2*x^3 + 2*A^2*a^2*b*c*d*g^3*i^2*x^3 + 1/3*A^2*a^3
*d^2*g^3*i^2*x^3 + 3*A*B*a^2*b*c^2*g^3*i^2*x^2*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 2*A*B*a^3*c*d*g^3*i^2*
x^2*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 3/2*A^2*a^2*b*c^2*g^3*i^2*x^2 + A^2*a^3*c*d*g^3*i^2*x^2 - 1/180*A
*B*b^3*d^2*g^3*i^2*n*(60*a^6*log(b*x + a)/b^6 - 60*c^6*log(d*x + c)/d^6 + (12*(b^5*c*d^4 - a*b^4*d^5)*x^5 - 15
*(b^5*c^2*d^3 - a^2*b^3*d^5)*x^4 + 20*(b^5*c^3*d^2 - a^3*b^2*d^5)*x^3 - 30*(b^5*c^4*d - a^4*b*d^5)*x^2 + 60*(b
^5*c^5 - a^5*d^5)*x)/(b^5*d^5)) + 1/15*A*B*b^3*c*d*g^3*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)) + 1/10*A*B*a*b^2*d^2*g^3*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)) - 1/12*A*B*b^3*c^2*g^3*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)) - 1/2
*A*B*a*b^2*c*d*g^3*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)) - 1/4*A*B*a^2*b*d^2*g^3*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)) + A*B*a*b^2*c^2*g^3*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^2*b*c*d*g^3*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^3*d^2*g
^3*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)) - 3*A*B*a^2*b*c^2*g^3*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^3*c*d*g^3*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^3*c^2*g
^3*i^2*n*(a*log(b*x + a)/b - c*log(d*x + c)/d) + 2*A*B*a^3*c^2*g^3*i^2*x*log(e*(b*x/(d*x + c) + a/(d*x + c))^n
) + A^2*a^3*c^2*g^3*i^2*x - 1/180*(33*a^4*b*c^2*d^4*g^3*i^2*n^2 - 6*a^5*c*d^5*g^3*i^2*n^2 - 2*(g^3*i^2*n^2 + 3
*g^3*i^2*n*log(e))*b^5*c^6 + 6*(g^3*i^2*n^2 + 6*g^3*i^2*n*log(e))*a*b^4*c^5*d + 3*(g^3*i^2*n^2 - 30*g^3*i^2*n*
log(e))*a^2*b^3*c^4*d^2 - 2*(17*g^3*i^2*n^2 - 60*g^3*i^2*n*log(e))*a^3*b^2*c^3*d^3)*B^2*log(d*x + c)/(b^2*d^4)
+ 1/30*(b^6*c^6*g^3*i^2*n^2 - 6*a*b^5*c^5*d*g^3*i^2*n^2 + 15*a^2*b^4*c^4*d^2*g^3*i^2*n^2 - 20*a^3*b^3*c^3*d^3
*g^3*i^2*n^2 + 15*a^4*b^2*c^2*d^4*g^3*i^2*n^2 - 6*a^5*b*c*d^5*g^3*i^2*n^2 + a^6*d^6*g^3*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^4) + 1/360*(60*B^2*b^6*d^6
*g^3*i^2*x^6*log(e)^2 - 24*((g^3*i^2*n*log(e) - 6*g^3*i^2*log(e)^2)*b^6*c*d^5 - (g^3*i^2*n*log(e) + 9*g^3*i^2*
log(e)^2)*a*b^5*d^6)*B^2*x^5 + 6*((g^3*i^2*n^2 - 7*g^3*i^2*n*log(e) + 15*g^3*i^2*log(e)^2)*b^6*c^2*d^4 - 2*(g^
3*i^2*n^2 + 3*g^3*i^2*n*log(e) - 45*g^3*i^2*log(e)^2)*a*b^5*c*d^5 + (g^3*i^2*n^2 + 13*g^3*i^2*n*log(e) + 45*g^
3*i^2*log(e)^2)*a^2*b^4*d^6)*B^2*x^4 + 2*((3*g^3*i^2*n^2 - 2*g^3*i^2*n*log(e))*b^6*c^3*d^3 + 3*(g^3*i^2*n^2 -
26*g^3*i^2*n*log(e) + 60*g^3*i^2*log(e)^2)*a*b^5*c^2*d^4 - 3*(5*g^3*i^2*n^2 - 14*g^3*i^2*n*log(e) - 120*g^3*i^
2*log(e)^2)*a^2*b^4*c*d^5 + (9*g^3*i^2*n^2 + 38*g^3*i^2*n*log(e) + 60*g^3*i^2*log(e)^2)*a^3*b^3*d^6)*B^2*x^3 -
((7*g^3*i^2*n^2 - 6*g^3*i^2*n*log(e))*b^6*c^4*d^2 - 2*(23*g^3*i^2*n^2 - 18*g^3*i^2*n*log(e))*a*b^5*c^3*d^3 +
60*(g^3*i^2*n^2 + 3*g^3*i^2*n*log(e) - 9*g^3*i^2*log(e)^2)*a^2*b^4*c^2*d^4 - 2*(5*g^3*i^2*n^2 + 102*g^3*i^2*n*
log(e) + 180*g^3*i^2*log(e)^2)*a^3*b^3*c*d^5 - (11*g^3*i^2*n^2 + 6*g^3*i^2*n*log(e))*a^4*b^2*d^6)*B^2*x^2 - 6*
(15*a^4*b^2*c^2*d^4*g^3*i^2*n^2 - 6*a^5*b*c*d^5*g^3*i^2*n^2 + a^6*d^6*g^3*i^2*n^2)*B^2*log(b*x + a)^2 - 12*(b^
6*c^6*g^3*i^2*n^2 - 6*a*b^5*c^5*d*g^3*i^2*n^2 + 15*a^2*b^4*c^4*d^2*g^3*i^2*n^2 - 20*a^3*b^3*c^3*d^3*g^3*i^2*n^
2)*B^2*log(b*x + a)*log(d*x + c) + 6*(b^6*c^6*g^3*i^2*n^2 - 6*a*b^5*c^5*d*g^3*i^2*n^2 + 15*a^2*b^4*c^4*d^2*g^3
*i^2*n^2 - 20*a^3*b^3*c^3*d^3*g^3*i^2*n^2)*B^2*log(d*x + c)^2 + 2*(2*(2*g^3*i^2*n^2 - 3*g^3*i^2*n*log(e))*b^6*
c^5*d - 9*(3*g^3*i^2*n^2 - 4*g^3*i^2*n*log(e))*a*b^5*c^4*d^2 + (77*g^3*i^2*n^2 - 90*g^3*i^2*n*log(e))*a^2*b^4*
c^3*d^3 - (97*g^3*i^2*n^2 - 30*g^3*i^2*n*log(e) - 180*g^3*i^2*log(e)^2)*a^3*b^3*c^2*d^4 + 3*(17*g^3*i^2*n^2 +
12*g^3*i^2*n*log(e))*a^4*b^2*c*d^5 - 2*(4*g^3*i^2*n^2 + 3*g^3*i^2*n*log(e))*a^5*b*d^6)*B^2*x - 2*(6*a*b^5*c^5*
d*g^3*i^2*n^2 - 33*a^2*b^4*c^4*d^2*g^3*i^2*n^2 + 74*a^3*b^3*c^3*d^3*g^3*i^2*n^2 - 9*(7*g^3*i^2*n^2 + 10*g^3*i^
2*n*log(e))*a^4*b^2*c^2*d^4 + 18*(g^3*i^2*n^2 + 2*g^3*i^2*n*log(e))*a^5*b*c*d^5 - 2*(g^3*i^2*n^2 + 3*g^3*i^2*n
*log(e))*a^6*d^6)*B^2*log(b*x + a) + 6*(10*B^2*b^6*d^6*g^3*i^2*x^6 + 60*B^2*a^3*b^3*c^2*d^4*g^3*i^2*x + 12*(2*
b^6*c*d^5*g^3*i^2 + 3*a*b^5*d^6*g^3*i^2)*B^2*x^5 + 15*(b^6*c^2*d^4*g^3*i^2 + 6*a*b^5*c*d^5*g^3*i^2 + 3*a^2*b^4
*d^6*g^3*i^2)*B^2*x^4 + 20*(3*a*b^5*c^2*d^4*g^3*i^2 + 6*a^2*b^4*c*d^5*g^3*i^2 + a^3*b^3*d^6*g^3*i^2)*B^2*x^3 +
30*(3*a^2*b^4*c^2*d^4*g^3*i^2 + 2*a^3*b^3*c*d^5*g^3*i^2)*B^2*x^2)*log((b*x + a)^n)^2 + 6*(10*B^2*b^6*d^6*g^3*
i^2*x^6 + 60*B^2*a^3*b^3*c^2*d^4*g^3*i^2*x + 12*(2*b^6*c*d^5*g^3*i^2 + 3*a*b^5*d^6*g^3*i^2)*B^2*x^5 + 15*(b^6*
c^2*d^4*g^3*i^2 + 6*a*b^5*c*d^5*g^3*i^2 + 3*a^2*b^4*d^6*g^3*i^2)*B^2*x^4 + 20*(3*a*b^5*c^2*d^4*g^3*i^2 + 6*a^2
*b^4*c*d^5*g^3*i^2 + a^3*b^3*d^6*g^3*i^2)*B^2*x^3 + 30*(3*a^2*b^4*c^2*d^4*g^3*i^2 + 2*a^3*b^3*c*d^5*g^3*i^2)*B
^2*x^2)*log((d*x + c)^n)^2 + 2*(60*B^2*b^6*d^6*g^3*i^2*x^6*log(e) - 12*((g^3*i^2*n - 12*g^3*i^2*log(e))*b^6*c*
d^5 - (g^3*i^2*n + 18*g^3*i^2*log(e))*a*b^5*d^6)*B^2*x^5 - 3*((7*g^3*i^2*n - 30*g^3*i^2*log(e))*b^6*c^2*d^4 +
6*(g^3*i^2*n - 30*g^3*i^2*log(e))*a*b^5*c*d^5 - (13*g^3*i^2*n + 90*g^3*i^2*log(e))*a^2*b^4*d^6)*B^2*x^4 - 2*(b
^6*c^3*d^3*g^3*i^2*n + 3*(13*g^3*i^2*n - 60*g^3*i^2*log(e))*a*b^5*c^2*d^4 - 3*(7*g^3*i^2*n + 120*g^3*i^2*log(e
))*a^2*b^4*c*d^5 - (19*g^3*i^2*n + 60*g^3*i^2*log(e))*a^3*b^3*d^6)*B^2*x^3 + 3*(b^6*c^4*d^2*g^3*i^2*n - 6*a*b^
5*c^3*d^3*g^3*i^2*n + a^4*b^2*d^6*g^3*i^2*n - 30*(g^3*i^2*n - 6*g^3*i^2*log(e))*a^2*b^4*c^2*d^4 + 2*(17*g^3*i^
2*n + 60*g^3*i^2*log(e))*a^3*b^3*c*d^5)*B^2*x^2 - 6*(b^6*c^5*d*g^3*i^2*n - 6*a*b^5*c^4*d^2*g^3*i^2*n + 15*a^2*
b^4*c^3*d^3*g^3*i^2*n - 6*a^4*b^2*c*d^5*g^3*i^2*n + a^5*b*d^6*g^3*i^2*n - 5*(g^3*i^2*n + 12*g^3*i^2*log(e))*a^
3*b^3*c^2*d^4)*B^2*x + 6*(15*a^4*b^2*c^2*d^4*g^3*i^2*n - 6*a^5*b*c*d^5*g^3*i^2*n + a^6*d^6*g^3*i^2*n)*B^2*log(
b*x + a) + 6*(b^6*c^6*g^3*i^2*n - 6*a*b^5*c^5*d*g^3*i^2*n + 15*a^2*b^4*c^4*d^2*g^3*i^2*n - 20*a^3*b^3*c^3*d^3*
g^3*i^2*n)*B^2*log(d*x + c))*log((b*x + a)^n) - 2*(60*B^2*b^6*d^6*g^3*i^2*x^6*log(e) - 12*((g^3*i^2*n - 12*g^3
*i^2*log(e))*b^6*c*d^5 - (g^3*i^2*n + 18*g^3*i^2*log(e))*a*b^5*d^6)*B^2*x^5 - 3*((7*g^3*i^2*n - 30*g^3*i^2*log
(e))*b^6*c^2*d^4 + 6*(g^3*i^2*n - 30*g^3*i^2*log(e))*a*b^5*c*d^5 - (13*g^3*i^2*n + 90*g^3*i^2*log(e))*a^2*b^4*
d^6)*B^2*x^4 - 2*(b^6*c^3*d^3*g^3*i^2*n + 3*(13*g^3*i^2*n - 60*g^3*i^2*log(e))*a*b^5*c^2*d^4 - 3*(7*g^3*i^2*n
+ 120*g^3*i^2*log(e))*a^2*b^4*c*d^5 - (19*g^3*i^2*n + 60*g^3*i^2*log(e))*a^3*b^3*d^6)*B^2*x^3 + 3*(b^6*c^4*d^2
*g^3*i^2*n - 6*a*b^5*c^3*d^3*g^3*i^2*n + a^4*b^2*d^6*g^3*i^2*n - 30*(g^3*i^2*n - 6*g^3*i^2*log(e))*a^2*b^4*c^2
*d^4 + 2*(17*g^3*i^2*n + 60*g^3*i^2*log(e))*a^3*b^3*c*d^5)*B^2*x^2 - 6*(b^6*c^5*d*g^3*i^2*n - 6*a*b^5*c^4*d^2*
g^3*i^2*n + 15*a^2*b^4*c^3*d^3*g^3*i^2*n - 6*a^4*b^2*c*d^5*g^3*i^2*n + a^5*b*d^6*g^3*i^2*n - 5*(g^3*i^2*n + 12
*g^3*i^2*log(e))*a^3*b^3*c^2*d^4)*B^2*x + 6*(15*a^4*b^2*c^2*d^4*g^3*i^2*n - 6*a^5*b*c*d^5*g^3*i^2*n + a^6*d^6*
g^3*i^2*n)*B^2*log(b*x + a) + 6*(b^6*c^6*g^3*i^2*n - 6*a*b^5*c^5*d*g^3*i^2*n + 15*a^2*b^4*c^4*d^2*g^3*i^2*n -
20*a^3*b^3*c^3*d^3*g^3*i^2*n)*B^2*log(d*x + c) + 6*(10*B^2*b^6*d^6*g^3*i^2*x^6 + 60*B^2*a^3*b^3*c^2*d^4*g^3*i^
2*x + 12*(2*b^6*c*d^5*g^3*i^2 + 3*a*b^5*d^6*g^3*i^2)*B^2*x^5 + 15*(b^6*c^2*d^4*g^3*i^2 + 6*a*b^5*c*d^5*g^3*i^2
+ 3*a^2*b^4*d^6*g^3*i^2)*B^2*x^4 + 20*(3*a*b^5*c^2*d^4*g^3*i^2 + 6*a^2*b^4*c*d^5*g^3*i^2 + a^3*b^3*d^6*g^3*i^
2)*B^2*x^3 + 30*(3*a^2*b^4*c^2*d^4*g^3*i^2 + 2*a^3*b^3*c*d^5*g^3*i^2)*B^2*x^2)*log((b*x + a)^n))*log((d*x + c)
^n))/(b^3*d^4)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (A^{2} b^{3} d^{2} g^{3} i^{2} x^{5} + A^{2} a^{3} c^{2} g^{3} i^{2} +{\left (2 \, A^{2} b^{3} c d + 3 \, A^{2} a b^{2} d^{2}\right )} g^{3} i^{2} x^{4} +{\left (A^{2} b^{3} c^{2} + 6 \, A^{2} a b^{2} c d + 3 \, A^{2} a^{2} b d^{2}\right )} g^{3} i^{2} x^{3} +{\left (3 \, A^{2} a b^{2} c^{2} + 6 \, A^{2} a^{2} b c d + A^{2} a^{3} d^{2}\right )} g^{3} i^{2} x^{2} +{\left (3 \, A^{2} a^{2} b c^{2} + 2 \, A^{2} a^{3} c d\right )} g^{3} i^{2} x +{\left (B^{2} b^{3} d^{2} g^{3} i^{2} x^{5} + B^{2} a^{3} c^{2} g^{3} i^{2} +{\left (2 \, B^{2} b^{3} c d + 3 \, B^{2} a b^{2} d^{2}\right )} g^{3} i^{2} x^{4} +{\left (B^{2} b^{3} c^{2} + 6 \, B^{2} a b^{2} c d + 3 \, B^{2} a^{2} b d^{2}\right )} g^{3} i^{2} x^{3} +{\left (3 \, B^{2} a b^{2} c^{2} + 6 \, B^{2} a^{2} b c d + B^{2} a^{3} d^{2}\right )} g^{3} i^{2} x^{2} +{\left (3 \, B^{2} a^{2} b c^{2} + 2 \, B^{2} a^{3} c d\right )} g^{3} i^{2} x\right )} \log \left (e \left (\frac{b x + a}{d x + c}\right )^{n}\right )^{2} + 2 \,{\left (A B b^{3} d^{2} g^{3} i^{2} x^{5} + A B a^{3} c^{2} g^{3} i^{2} +{\left (2 \, A B b^{3} c d + 3 \, A B a b^{2} d^{2}\right )} g^{3} i^{2} x^{4} +{\left (A B b^{3} c^{2} + 6 \, A B a b^{2} c d + 3 \, A B a^{2} b d^{2}\right )} g^{3} i^{2} x^{3} +{\left (3 \, A B a b^{2} c^{2} + 6 \, A B a^{2} b c d + A B a^{3} d^{2}\right )} g^{3} i^{2} x^{2} +{\left (3 \, A B a^{2} b c^{2} + 2 \, A B a^{3} c d\right )} g^{3} 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)^3*(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^3*d^2*g^3*i^2*x^5 + A^2*a^3*c^2*g^3*i^2 + (2*A^2*b^3*c*d + 3*A^2*a*b^2*d^2)*g^3*i^2*x^4 + (A^2*
b^3*c^2 + 6*A^2*a*b^2*c*d + 3*A^2*a^2*b*d^2)*g^3*i^2*x^3 + (3*A^2*a*b^2*c^2 + 6*A^2*a^2*b*c*d + A^2*a^3*d^2)*g
^3*i^2*x^2 + (3*A^2*a^2*b*c^2 + 2*A^2*a^3*c*d)*g^3*i^2*x + (B^2*b^3*d^2*g^3*i^2*x^5 + B^2*a^3*c^2*g^3*i^2 + (2
*B^2*b^3*c*d + 3*B^2*a*b^2*d^2)*g^3*i^2*x^4 + (B^2*b^3*c^2 + 6*B^2*a*b^2*c*d + 3*B^2*a^2*b*d^2)*g^3*i^2*x^3 +
(3*B^2*a*b^2*c^2 + 6*B^2*a^2*b*c*d + B^2*a^3*d^2)*g^3*i^2*x^2 + (3*B^2*a^2*b*c^2 + 2*B^2*a^3*c*d)*g^3*i^2*x)*l
og(e*((b*x + a)/(d*x + c))^n)^2 + 2*(A*B*b^3*d^2*g^3*i^2*x^5 + A*B*a^3*c^2*g^3*i^2 + (2*A*B*b^3*c*d + 3*A*B*a*
b^2*d^2)*g^3*i^2*x^4 + (A*B*b^3*c^2 + 6*A*B*a*b^2*c*d + 3*A*B*a^2*b*d^2)*g^3*i^2*x^3 + (3*A*B*a*b^2*c^2 + 6*A*
B*a^2*b*c*d + A*B*a^3*d^2)*g^3*i^2*x^2 + (3*A*B*a^2*b*c^2 + 2*A*B*a^3*c*d)*g^3*i^2*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)**3*(d*i*x+c*i)**2*(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 )}^{3}{\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)^3*(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)^3*(d*i*x + c*i)^2*(B*log(e*((b*x + a)/(d*x + c))^n) + A)^2, x)