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

Optimal. Leaf size=1172 \[ -\frac {B g^3 i^3 n \left (6 A+11 B n+6 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)^7}{420 b^4 d^4}-\frac {B^2 g^3 i^3 n^2 \log \left (\frac {a+b x}{c+d x}\right ) (b c-a d)^7}{210 b^4 d^4}-\frac {11 B^2 g^3 i^3 n^2 \log (c+d x) (b c-a d)^7}{420 b^4 d^4}-\frac {B^2 g^3 i^3 n^2 \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right ) (b c-a d)^7}{70 b^4 d^4}+\frac {5 B^2 g^3 i^3 n^2 x (b c-a d)^6}{84 b^3 d^3}-\frac {B g^3 i^3 n (a+b x) \left (6 A+5 B n+6 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) (b c-a d)^6}{420 b^4 d^3}-\frac {29 B^2 g^3 i^3 n^2 (c+d x)^2 (b c-a d)^5}{840 b^2 d^4}+\frac {B g^3 i^3 n (a+b x)^2 \left (3 A+B n+3 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) (b c-a d)^5}{420 b^4 d^2}+\frac {47 B^2 g^3 i^3 n^2 (c+d x)^3 (b c-a d)^4}{1260 b d^4}-\frac {B g^3 i^3 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)^4}{210 b^4 d}+\frac {2 B g^3 i^3 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)^4}{21 b d^4}+\frac {B^2 g^3 i^3 n^2 (a+b x)^4 (b c-a d)^3}{140 b^4}-\frac {13 B^2 g^3 i^3 n^2 (c+d x)^4 (b c-a d)^3}{420 d^4}+\frac {g^3 i^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 c-a d)^3}{140 b^4}-\frac {3 B g^3 i^3 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)^3}{140 b^4}-\frac {3 B g^3 i^3 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)^3}{14 d^4}+\frac {b B^2 g^3 i^3 n^2 (c+d x)^5 (b c-a d)^2}{105 d^4}+\frac {g^3 i^3 (a+b x)^4 (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)^2}{35 b^3}+\frac {6 b B g^3 i^3 n (c+d x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) (b c-a d)^2}{35 d^4}-\frac {B g^3 i^3 n (a+b x)^4 (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) (b c-a d)^2}{35 b^3}+\frac {g^3 i^3 (a+b x)^4 (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 (b c-a d)}{14 b^2}-\frac {b^2 B g^3 i^3 n (c+d x)^6 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) (b c-a d)}{21 d^4}+\frac {g^3 i^3 (a+b x)^4 (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{7 b} \]

[Out]

5/84*B^2*(-a*d+b*c)^6*g^3*i^3*n^2*x/b^3/d^3+1/140*B^2*(-a*d+b*c)^3*g^3*i^3*n^2*(b*x+a)^4/b^4-29/840*B^2*(-a*d+
b*c)^5*g^3*i^3*n^2*(d*x+c)^2/b^2/d^4+47/1260*B^2*(-a*d+b*c)^4*g^3*i^3*n^2*(d*x+c)^3/b/d^4-13/420*B^2*(-a*d+b*c
)^3*g^3*i^3*n^2*(d*x+c)^4/d^4+1/105*b*B^2*(-a*d+b*c)^2*g^3*i^3*n^2*(d*x+c)^5/d^4-1/210*B*(-a*d+b*c)^4*g^3*i^3*
n*(b*x+a)^3*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/b^4/d-3/140*B*(-a*d+b*c)^3*g^3*i^3*n*(b*x+a)^4*(A+B*ln(e*((b*x+a)/
(d*x+c))^n))/b^4-1/35*B*(-a*d+b*c)^2*g^3*i^3*n*(b*x+a)^4*(d*x+c)*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/b^3+2/21*B*(-
a*d+b*c)^4*g^3*i^3*n*(d*x+c)^3*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/b/d^4-3/14*B*(-a*d+b*c)^3*g^3*i^3*n*(d*x+c)^4*(
A+B*ln(e*((b*x+a)/(d*x+c))^n))/d^4+6/35*b*B*(-a*d+b*c)^2*g^3*i^3*n*(d*x+c)^5*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/d
^4-1/21*b^2*B*(-a*d+b*c)*g^3*i^3*n*(d*x+c)^6*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/d^4+1/140*(-a*d+b*c)^3*g^3*i^3*(b
*x+a)^4*(A+B*ln(e*((b*x+a)/(d*x+c))^n))^2/b^4+1/35*(-a*d+b*c)^2*g^3*i^3*(b*x+a)^4*(d*x+c)*(A+B*ln(e*((b*x+a)/(
d*x+c))^n))^2/b^3+1/14*(-a*d+b*c)*g^3*i^3*(b*x+a)^4*(d*x+c)^2*(A+B*ln(e*((b*x+a)/(d*x+c))^n))^2/b^2+1/7*g^3*i^
3*(b*x+a)^4*(d*x+c)^3*(A+B*ln(e*((b*x+a)/(d*x+c))^n))^2/b+1/420*B*(-a*d+b*c)^5*g^3*i^3*n*(b*x+a)^2*(3*A+B*n+3*
B*ln(e*((b*x+a)/(d*x+c))^n))/b^4/d^2-1/420*B*(-a*d+b*c)^6*g^3*i^3*n*(b*x+a)*(6*A+5*B*n+6*B*ln(e*((b*x+a)/(d*x+
c))^n))/b^4/d^3-1/420*B*(-a*d+b*c)^7*g^3*i^3*n*(6*A+11*B*n+6*B*ln(e*((b*x+a)/(d*x+c))^n))*ln((-a*d+b*c)/b/(d*x
+c))/b^4/d^4-1/210*B^2*(-a*d+b*c)^7*g^3*i^3*n^2*ln((b*x+a)/(d*x+c))/b^4/d^4-11/420*B^2*(-a*d+b*c)^7*g^3*i^3*n^
2*ln(d*x+c)/b^4/d^4-1/70*B^2*(-a*d+b*c)^7*g^3*i^3*n^2*polylog(2,d*(b*x+a)/b/(d*x+c))/b^4/d^4

________________________________________________________________________________________

Rubi [A]  time = 4.48, antiderivative size = 961, normalized size of antiderivative = 0.82, number of steps used = 118, 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^3 n^2 \log ^2(c+d x) (b c-a d)^7}{140 b^4 d^4}-\frac {B^2 g^3 i^3 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x) (b c-a d)^7}{70 b^4 d^4}+\frac {B g^3 i^3 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)^7}{70 b^4 d^4}-\frac {B^2 g^3 i^3 n^2 \text {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right ) (b c-a d)^7}{70 b^4 d^4}+\frac {B^2 g^3 i^3 n^2 x (b c-a d)^6}{70 b^3 d^3}-\frac {A B g^3 i^3 n x (b c-a d)^6}{70 b^3 d^3}-\frac {B^2 g^3 i^3 n (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) (b c-a d)^6}{70 b^4 d^3}-\frac {3 B^2 g^3 i^3 n^2 (a+b x)^2 (b c-a d)^5}{280 b^4 d^2}+\frac {B g^3 i^3 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)^5}{140 b^4 d^2}+\frac {11 B^2 g^3 i^3 n^2 (a+b x)^3 (b c-a d)^4}{1260 b^4 d}-\frac {B g^3 i^3 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)^4}{210 b^4 d}+\frac {B^2 g^3 i^3 n^2 (a+b x)^4 (b c-a d)^3}{42 b^4}+\frac {g^3 i^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 c-a d)^3}{4 b^4}-\frac {17 B g^3 i^3 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)^3}{140 b^4}+\frac {B^2 d g^3 i^3 n^2 (a+b x)^5 (b c-a d)^2}{105 b^4}+\frac {3 d g^3 i^3 (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)^2}{5 b^4}-\frac {B d g^3 i^3 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)^2}{7 b^4}+\frac {d^2 g^3 i^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 c-a d)}{2 b^4}-\frac {B d^2 g^3 i^3 n (a+b x)^6 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) (b c-a d)}{21 b^4}+\frac {d^3 g^3 i^3 (a+b x)^7 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{7 b^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(A*B*(b*c - a*d)^6*g^3*i^3*n*x)/(70*b^3*d^3) + (B^2*(b*c - a*d)^6*g^3*i^3*n^2*x)/(70*b^3*d^3) - (3*B^2*(b*c -
a*d)^5*g^3*i^3*n^2*(a + b*x)^2)/(280*b^4*d^2) + (11*B^2*(b*c - a*d)^4*g^3*i^3*n^2*(a + b*x)^3)/(1260*b^4*d) +
(B^2*(b*c - a*d)^3*g^3*i^3*n^2*(a + b*x)^4)/(42*b^4) + (B^2*d*(b*c - a*d)^2*g^3*i^3*n^2*(a + b*x)^5)/(105*b^4
) - (B^2*(b*c - a*d)^6*g^3*i^3*n*(a + b*x)*Log[e*((a + b*x)/(c + d*x))^n])/(70*b^4*d^3) + (B*(b*c - a*d)^5*g^3
*i^3*n*(a + b*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(140*b^4*d^2) - (B*(b*c - a*d)^4*g^3*i^3*n*(a + b*x
)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(210*b^4*d) - (17*B*(b*c - a*d)^3*g^3*i^3*n*(a + b*x)^4*(A + B*Log
[e*((a + b*x)/(c + d*x))^n]))/(140*b^4) - (B*d*(b*c - a*d)^2*g^3*i^3*n*(a + b*x)^5*(A + B*Log[e*((a + b*x)/(c
+ d*x))^n]))/(7*b^4) - (B*d^2*(b*c - a*d)*g^3*i^3*n*(a + b*x)^6*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(21*b^
4) + ((b*c - a*d)^3*g^3*i^3*(a + b*x)^4*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(4*b^4) + (3*d*(b*c - a*d)^2
*g^3*i^3*(a + b*x)^5*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(5*b^4) + (d^2*(b*c - a*d)*g^3*i^3*(a + b*x)^6*
(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(2*b^4) + (d^3*g^3*i^3*(a + b*x)^7*(A + B*Log[e*((a + b*x)/(c + d*x)
)^n])^2)/(7*b^4) - (B^2*(b*c - a*d)^7*g^3*i^3*n^2*Log[-((d*(a + b*x))/(b*c - a*d))]*Log[c + d*x])/(70*b^4*d^4)
+ (B*(b*c - a*d)^7*g^3*i^3*n*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*Log[c + d*x])/(70*b^4*d^4) + (B^2*(b*c -
a*d)^7*g^3*i^3*n^2*Log[c + d*x]^2)/(140*b^4*d^4) - (B^2*(b*c - a*d)^7*g^3*i^3*n^2*PolyLog[2, (b*(c + d*x))/(b*
c - a*d)])/(70*b^4*d^4)

Rule 12

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

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 2301

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

Rule 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 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 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 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 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 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 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 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 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]

Rubi steps

\begin {align*} \int (178 c+178 d x)^3 (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 {(-b c+a d)^3 g^3 (178 c+178 d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{d^3}+\frac {3 b (b c-a d)^2 g^3 (178 c+178 d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{178 d^3}-\frac {3 b^2 (b c-a d) g^3 (178 c+178 d x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{31684 d^3}+\frac {b^3 g^3 (178 c+178 d x)^6 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{5639752 d^3}\right ) \, dx\\ &=\frac {\left (b^3 g^3\right ) \int (178 c+178 d x)^6 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx}{5639752 d^3}-\frac {\left (3 b^2 (b c-a d) g^3\right ) \int (178 c+178 d x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx}{31684 d^3}+\frac {\left (3 b (b c-a d)^2 g^3\right ) \int (178 c+178 d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx}{178 d^3}-\frac {\left ((b c-a d)^3 g^3\right ) \int (178 c+178 d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx}{d^3}\\ &=-\frac {1409938 (b c-a d)^3 g^3 (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{d^4}+\frac {16919256 b (b c-a d)^2 g^3 (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^4}-\frac {2819876 b^2 (b c-a d) g^3 (c+d x)^6 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{d^4}+\frac {5639752 b^3 g^3 (c+d x)^7 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{7 d^4}-\frac {\left (b^3 B g^3 n\right ) \int \frac {5661610866627712 (b c-a d) (c+d x)^6 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{a+b x} \, dx}{3513565496 d^4}+\frac {\left (b^2 B (b c-a d) g^3 n\right ) \int \frac {31806802621504 (b c-a d) (c+d x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{a+b x} \, dx}{5639752 d^4}-\frac {\left (3 b B (b c-a d)^2 g^3 n\right ) \int \frac {178689902368 (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}{79210 d^4}+\frac {\left (B (b c-a d)^3 g^3 n\right ) \int \frac {1003875856 (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}{356 d^4}\\ &=-\frac {1409938 (b c-a d)^3 g^3 (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{d^4}+\frac {16919256 b (b c-a d)^2 g^3 (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^4}-\frac {2819876 b^2 (b c-a d) g^3 (c+d x)^6 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{d^4}+\frac {5639752 b^3 g^3 (c+d x)^7 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{7 d^4}-\frac {\left (11279504 b^3 B (b c-a d) g^3 n\right ) \int \frac {(c+d x)^6 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{a+b x} \, dx}{7 d^4}+\frac {\left (5639752 b^2 B (b c-a d)^2 g^3 n\right ) \int \frac {(c+d x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{a+b x} \, dx}{d^4}-\frac {\left (33838512 b B (b c-a d)^3 g^3 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^4}+\frac {\left (2819876 B (b c-a d)^4 g^3 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^4}\\ &=-\frac {1409938 (b c-a d)^3 g^3 (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{d^4}+\frac {16919256 b (b c-a d)^2 g^3 (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^4}-\frac {2819876 b^2 (b c-a d) g^3 (c+d x)^6 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{d^4}+\frac {5639752 b^3 g^3 (c+d x)^7 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{7 d^4}-\frac {\left (11279504 b^3 B (b c-a d) g^3 n\right ) \int \left (\frac {d (b c-a d)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^6}+\frac {(b c-a d)^6 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^6 (a+b x)}+\frac {d (b c-a d)^4 (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^5}+\frac {d (b c-a d)^3 (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^4}+\frac {d (b c-a d)^2 (c+d x)^3 \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)^4 \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)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b}\right ) \, dx}{7 d^4}+\frac {\left (5639752 b^2 B (b c-a d)^2 g^3 n\right ) \int \left (\frac {d (b c-a d)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^5}+\frac {(b c-a d)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^5 (a+b x)}+\frac {d (b c-a d)^3 (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^4}+\frac {d (b c-a d)^2 (c+d x)^2 \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)^3 \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)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b}\right ) \, dx}{d^4}-\frac {\left (33838512 b B (b c-a d)^3 g^3 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^4}+\frac {\left (2819876 B (b c-a d)^4 g^3 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^4}\\ &=-\frac {1409938 (b c-a d)^3 g^3 (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{d^4}+\frac {16919256 b (b c-a d)^2 g^3 (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^4}-\frac {2819876 b^2 (b c-a d) g^3 (c+d x)^6 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{d^4}+\frac {5639752 b^3 g^3 (c+d x)^7 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{7 d^4}-\frac {\left (11279504 b^2 B (b c-a d) g^3 n\right ) \int (c+d x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx}{7 d^3}-\frac {\left (11279504 b B (b c-a d)^2 g^3 n\right ) \int (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx}{7 d^3}+\frac {\left (5639752 b B (b c-a d)^2 g^3 n\right ) \int (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx}{d^3}-\frac {\left (11279504 B (b c-a d)^3 g^3 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}{7 d^3}+\frac {\left (5639752 B (b c-a d)^3 g^3 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}{d^3}-\frac {\left (33838512 B (b c-a d)^3 g^3 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^3}-\frac {\left (11279504 B (b c-a d)^4 g^3 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}{7 b d^3}+\frac {\left (2819876 B (b c-a d)^4 g^3 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}{b d^3}+\frac {\left (5639752 B (b c-a d)^4 g^3 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}{b d^3}-\frac {\left (33838512 B (b c-a d)^4 g^3 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 b d^3}-\frac {\left (11279504 B (b c-a d)^5 g^3 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}{7 b^2 d^3}+\frac {\left (2819876 B (b c-a d)^5 g^3 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^2 d^3}+\frac {\left (5639752 B (b c-a d)^5 g^3 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^2 d^3}-\frac {\left (33838512 B (b c-a d)^5 g^3 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^2 d^3}-\frac {\left (11279504 B (b c-a d)^6 g^3 n\right ) \int \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx}{7 b^3 d^3}+\frac {\left (2819876 B (b c-a d)^6 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^3 d^3}+\frac {\left (5639752 B (b c-a d)^6 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^3 d^3}-\frac {\left (33838512 B (b c-a d)^6 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^3 d^3}-\frac {\left (11279504 B (b c-a d)^7 g^3 n\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{a+b x} \, dx}{7 b^3 d^4}+\frac {\left (2819876 B (b c-a d)^7 g^3 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^3 d^4}+\frac {\left (5639752 B (b c-a d)^7 g^3 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^3 d^4}-\frac {\left (33838512 B (b c-a d)^7 g^3 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^3 d^4}\\ &=\frac {2819876 A B (b c-a d)^6 g^3 n x}{35 b^3 d^3}+\frac {1409938 B (b c-a d)^5 g^3 n (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{35 b^2 d^4}+\frac {2819876 B (b c-a d)^4 g^3 n (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{105 b d^4}-\frac {23968946 B (b c-a d)^3 g^3 n (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{35 d^4}+\frac {5639752 b B (b c-a d)^2 g^3 n (c+d x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{7 d^4}-\frac {5639752 b^2 B (b c-a d) g^3 n (c+d x)^6 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{21 d^4}+\frac {2819876 B (b c-a d)^7 g^3 n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{35 b^4 d^4}-\frac {1409938 (b c-a d)^3 g^3 (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{d^4}+\frac {16919256 b (b c-a d)^2 g^3 (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^4}-\frac {2819876 b^2 (b c-a d) g^3 (c+d x)^6 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{d^4}+\frac {5639752 b^3 g^3 (c+d x)^7 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{7 d^4}-\frac {\left (11279504 B^2 (b c-a d)^6 g^3 n\right ) \int \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \, dx}{7 b^3 d^3}+\frac {\left (2819876 B^2 (b c-a d)^6 g^3 n\right ) \int \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \, dx}{b^3 d^3}+\frac {\left (5639752 B^2 (b c-a d)^6 g^3 n\right ) \int \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \, dx}{b^3 d^3}-\frac {\left (33838512 B^2 (b c-a d)^6 g^3 n\right ) \int \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \, dx}{5 b^3 d^3}+\frac {\left (5639752 b^2 B^2 (b c-a d) g^3 n^2\right ) \int \frac {(b c-a d) (c+d x)^5}{a+b x} \, dx}{21 d^4}+\frac {\left (11279504 b B^2 (b c-a d)^2 g^3 n^2\right ) \int \frac {(b c-a d) (c+d x)^4}{a+b x} \, dx}{35 d^4}-\frac {\left (5639752 b B^2 (b c-a d)^2 g^3 n^2\right ) \int \frac {(b c-a d) (c+d x)^4}{a+b x} \, dx}{5 d^4}+\frac {\left (2819876 B^2 (b c-a d)^3 g^3 n^2\right ) \int \frac {(b c-a d) (c+d x)^3}{a+b x} \, dx}{7 d^4}-\frac {\left (1409938 B^2 (b c-a d)^3 g^3 n^2\right ) \int \frac {(b c-a d) (c+d x)^3}{a+b x} \, dx}{d^4}+\frac {\left (8459628 B^2 (b c-a d)^3 g^3 n^2\right ) \int \frac {(b c-a d) (c+d x)^3}{a+b x} \, dx}{5 d^4}+\frac {\left (11279504 B^2 (b c-a d)^4 g^3 n^2\right ) \int \frac {(b c-a d) (c+d x)^2}{a+b x} \, dx}{21 b d^4}-\frac {\left (2819876 B^2 (b c-a d)^4 g^3 n^2\right ) \int \frac {(b c-a d) (c+d x)^2}{a+b x} \, dx}{3 b d^4}-\frac {\left (5639752 B^2 (b c-a d)^4 g^3 n^2\right ) \int \frac {(b c-a d) (c+d x)^2}{a+b x} \, dx}{3 b d^4}+\frac {\left (11279504 B^2 (b c-a d)^4 g^3 n^2\right ) \int \frac {(b c-a d) (c+d x)^2}{a+b x} \, dx}{5 b d^4}+\frac {\left (5639752 B^2 (b c-a d)^5 g^3 n^2\right ) \int \frac {(b c-a d) (c+d x)}{a+b x} \, dx}{7 b^2 d^4}-\frac {\left (1409938 B^2 (b c-a d)^5 g^3 n^2\right ) \int \frac {(b c-a d) (c+d x)}{a+b x} \, dx}{b^2 d^4}-\frac {\left (2819876 B^2 (b c-a d)^5 g^3 n^2\right ) \int \frac {(b c-a d) (c+d x)}{a+b x} \, dx}{b^2 d^4}+\frac {\left (16919256 B^2 (b c-a d)^5 g^3 n^2\right ) \int \frac {(b c-a d) (c+d x)}{a+b x} \, dx}{5 b^2 d^4}+\frac {\left (11279504 B^2 (b c-a d)^7 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 (a+b x)}{a+b x} \, dx}{7 b^4 d^4}-\frac {\left (2819876 B^2 (b c-a d)^7 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 (a+b x)}{a+b x} \, dx}{b^4 d^4}-\frac {\left (5639752 B^2 (b c-a d)^7 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 (a+b x)}{a+b x} \, dx}{b^4 d^4}+\frac {\left (33838512 B^2 (b c-a d)^7 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 (a+b x)}{a+b x} \, dx}{5 b^4 d^4}\\ &=\frac {2819876 A B (b c-a d)^6 g^3 n x}{35 b^3 d^3}+\frac {2819876 B^2 (b c-a d)^6 g^3 n (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{35 b^4 d^3}+\frac {1409938 B (b c-a d)^5 g^3 n (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{35 b^2 d^4}+\frac {2819876 B (b c-a d)^4 g^3 n (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{105 b d^4}-\frac {23968946 B (b c-a d)^3 g^3 n (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{35 d^4}+\frac {5639752 b B (b c-a d)^2 g^3 n (c+d x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{7 d^4}-\frac {5639752 b^2 B (b c-a d) g^3 n (c+d x)^6 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{21 d^4}+\frac {2819876 B (b c-a d)^7 g^3 n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{35 b^4 d^4}-\frac {1409938 (b c-a d)^3 g^3 (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{d^4}+\frac {16919256 b (b c-a d)^2 g^3 (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^4}-\frac {2819876 b^2 (b c-a d) g^3 (c+d x)^6 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{d^4}+\frac {5639752 b^3 g^3 (c+d x)^7 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{7 d^4}+\frac {\left (5639752 b^2 B^2 (b c-a d)^2 g^3 n^2\right ) \int \frac {(c+d x)^5}{a+b x} \, dx}{21 d^4}+\frac {\left (11279504 b B^2 (b c-a d)^3 g^3 n^2\right ) \int \frac {(c+d x)^4}{a+b x} \, dx}{35 d^4}-\frac {\left (5639752 b B^2 (b c-a d)^3 g^3 n^2\right ) \int \frac {(c+d x)^4}{a+b x} \, dx}{5 d^4}+\frac {\left (2819876 B^2 (b c-a d)^4 g^3 n^2\right ) \int \frac {(c+d x)^3}{a+b x} \, dx}{7 d^4}-\frac {\left (1409938 B^2 (b c-a d)^4 g^3 n^2\right ) \int \frac {(c+d x)^3}{a+b x} \, dx}{d^4}+\frac {\left (8459628 B^2 (b c-a d)^4 g^3 n^2\right ) \int \frac {(c+d x)^3}{a+b x} \, dx}{5 d^4}+\frac {\left (11279504 B^2 (b c-a d)^5 g^3 n^2\right ) \int \frac {(c+d x)^2}{a+b x} \, dx}{21 b d^4}-\frac {\left (2819876 B^2 (b c-a d)^5 g^3 n^2\right ) \int \frac {(c+d x)^2}{a+b x} \, dx}{3 b d^4}-\frac {\left (5639752 B^2 (b c-a d)^5 g^3 n^2\right ) \int \frac {(c+d x)^2}{a+b x} \, dx}{3 b d^4}+\frac {\left (11279504 B^2 (b c-a d)^5 g^3 n^2\right ) \int \frac {(c+d x)^2}{a+b x} \, dx}{5 b d^4}+\frac {\left (5639752 B^2 (b c-a d)^6 g^3 n^2\right ) \int \frac {c+d x}{a+b x} \, dx}{7 b^2 d^4}-\frac {\left (1409938 B^2 (b c-a d)^6 g^3 n^2\right ) \int \frac {c+d x}{a+b x} \, dx}{b^2 d^4}-\frac {\left (2819876 B^2 (b c-a d)^6 g^3 n^2\right ) \int \frac {c+d x}{a+b x} \, dx}{b^2 d^4}+\frac {\left (16919256 B^2 (b c-a d)^6 g^3 n^2\right ) \int \frac {c+d x}{a+b x} \, dx}{5 b^2 d^4}+\frac {\left (11279504 B^2 (b c-a d)^7 g^3 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}{7 b^4 d^4}-\frac {\left (2819876 B^2 (b c-a d)^7 g^3 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^4 d^4}-\frac {\left (5639752 B^2 (b c-a d)^7 g^3 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^4 d^4}+\frac {\left (33838512 B^2 (b c-a d)^7 g^3 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^4 d^4}+\frac {\left (11279504 B^2 (b c-a d)^7 g^3 n^2\right ) \int \frac {1}{c+d x} \, dx}{7 b^4 d^3}-\frac {\left (2819876 B^2 (b c-a d)^7 g^3 n^2\right ) \int \frac {1}{c+d x} \, dx}{b^4 d^3}-\frac {\left (5639752 B^2 (b c-a d)^7 g^3 n^2\right ) \int \frac {1}{c+d x} \, dx}{b^4 d^3}+\frac {\left (33838512 B^2 (b c-a d)^7 g^3 n^2\right ) \int \frac {1}{c+d x} \, dx}{5 b^4 d^3}\\ &=\frac {2819876 A B (b c-a d)^6 g^3 n x}{35 b^3 d^3}+\frac {2819876 B^2 (b c-a d)^6 g^3 n (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{35 b^4 d^3}+\frac {1409938 B (b c-a d)^5 g^3 n (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{35 b^2 d^4}+\frac {2819876 B (b c-a d)^4 g^3 n (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{105 b d^4}-\frac {23968946 B (b c-a d)^3 g^3 n (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{35 d^4}+\frac {5639752 b B (b c-a d)^2 g^3 n (c+d x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{7 d^4}-\frac {5639752 b^2 B (b c-a d) g^3 n (c+d x)^6 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{21 d^4}+\frac {2819876 B (b c-a d)^7 g^3 n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{35 b^4 d^4}-\frac {1409938 (b c-a d)^3 g^3 (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{d^4}+\frac {16919256 b (b c-a d)^2 g^3 (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^4}-\frac {2819876 b^2 (b c-a d) g^3 (c+d x)^6 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{d^4}+\frac {5639752 b^3 g^3 (c+d x)^7 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{7 d^4}-\frac {2819876 B^2 (b c-a d)^7 g^3 n^2 \log (c+d x)}{35 b^4 d^4}+\frac {\left (5639752 b^2 B^2 (b c-a d)^2 g^3 n^2\right ) \int \left (\frac {d (b c-a d)^4}{b^5}+\frac {(b c-a d)^5}{b^5 (a+b x)}+\frac {d (b c-a d)^3 (c+d x)}{b^4}+\frac {d (b c-a d)^2 (c+d x)^2}{b^3}+\frac {d (b c-a d) (c+d x)^3}{b^2}+\frac {d (c+d x)^4}{b}\right ) \, dx}{21 d^4}+\frac {\left (11279504 b B^2 (b c-a d)^3 g^3 n^2\right ) \int \left (\frac {d (b c-a d)^3}{b^4}+\frac {(b c-a d)^4}{b^4 (a+b x)}+\frac {d (b c-a d)^2 (c+d x)}{b^3}+\frac {d (b c-a d) (c+d x)^2}{b^2}+\frac {d (c+d x)^3}{b}\right ) \, dx}{35 d^4}-\frac {\left (5639752 b B^2 (b c-a d)^3 g^3 n^2\right ) \int \left (\frac {d (b c-a d)^3}{b^4}+\frac {(b c-a d)^4}{b^4 (a+b x)}+\frac {d (b c-a d)^2 (c+d x)}{b^3}+\frac {d (b c-a d) (c+d x)^2}{b^2}+\frac {d (c+d x)^3}{b}\right ) \, dx}{5 d^4}+\frac {\left (2819876 B^2 (b c-a d)^4 g^3 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}{7 d^4}-\frac {\left (1409938 B^2 (b c-a d)^4 g^3 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}{d^4}+\frac {\left (8459628 B^2 (b c-a d)^4 g^3 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}{5 d^4}+\frac {\left (11279504 B^2 (b c-a d)^5 g^3 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}{21 b d^4}-\frac {\left (2819876 B^2 (b c-a d)^5 g^3 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 b d^4}-\frac {\left (5639752 B^2 (b c-a d)^5 g^3 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 b d^4}+\frac {\left (11279504 B^2 (b c-a d)^5 g^3 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}{5 b d^4}+\frac {\left (5639752 B^2 (b c-a d)^6 g^3 n^2\right ) \int \left (\frac {d}{b}+\frac {b c-a d}{b (a+b x)}\right ) \, dx}{7 b^2 d^4}-\frac {\left (1409938 B^2 (b c-a d)^6 g^3 n^2\right ) \int \left (\frac {d}{b}+\frac {b c-a d}{b (a+b x)}\right ) \, dx}{b^2 d^4}-\frac {\left (2819876 B^2 (b c-a d)^6 g^3 n^2\right ) \int \left (\frac {d}{b}+\frac {b c-a d}{b (a+b x)}\right ) \, dx}{b^2 d^4}+\frac {\left (16919256 B^2 (b c-a d)^6 g^3 n^2\right ) \int \left (\frac {d}{b}+\frac {b c-a d}{b (a+b x)}\right ) \, dx}{5 b^2 d^4}+\frac {\left (11279504 B^2 (b c-a d)^7 g^3 n^2\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{7 b^3 d^4}-\frac {\left (2819876 B^2 (b c-a d)^7 g^3 n^2\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{b^3 d^4}-\frac {\left (5639752 B^2 (b c-a d)^7 g^3 n^2\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{b^3 d^4}+\frac {\left (33838512 B^2 (b c-a d)^7 g^3 n^2\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{5 b^3 d^4}-\frac {\left (11279504 B^2 (b c-a d)^7 g^3 n^2\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{7 b^4 d^3}+\frac {\left (2819876 B^2 (b c-a d)^7 g^3 n^2\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{b^4 d^3}+\frac {\left (5639752 B^2 (b c-a d)^7 g^3 n^2\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{b^4 d^3}-\frac {\left (33838512 B^2 (b c-a d)^7 g^3 n^2\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{5 b^4 d^3}\\ &=\frac {2819876 A B (b c-a d)^6 g^3 n x}{35 b^3 d^3}+\frac {2819876 B^2 (b c-a d)^6 g^3 n^2 x}{35 b^3 d^3}+\frac {2114907 B^2 (b c-a d)^5 g^3 n^2 (c+d x)^2}{35 b^2 d^4}+\frac {15509318 B^2 (b c-a d)^4 g^3 n^2 (c+d x)^3}{315 b d^4}-\frac {2819876 B^2 (b c-a d)^3 g^3 n^2 (c+d x)^4}{21 d^4}+\frac {5639752 b B^2 (b c-a d)^2 g^3 n^2 (c+d x)^5}{105 d^4}+\frac {2819876 B^2 (b c-a d)^7 g^3 n^2 \log (a+b x)}{35 b^4 d^4}+\frac {2819876 B^2 (b c-a d)^6 g^3 n (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{35 b^4 d^3}+\frac {1409938 B (b c-a d)^5 g^3 n (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{35 b^2 d^4}+\frac {2819876 B (b c-a d)^4 g^3 n (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{105 b d^4}-\frac {23968946 B (b c-a d)^3 g^3 n (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{35 d^4}+\frac {5639752 b B (b c-a d)^2 g^3 n (c+d x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{7 d^4}-\frac {5639752 b^2 B (b c-a d) g^3 n (c+d x)^6 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{21 d^4}+\frac {2819876 B (b c-a d)^7 g^3 n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{35 b^4 d^4}-\frac {1409938 (b c-a d)^3 g^3 (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{d^4}+\frac {16919256 b (b c-a d)^2 g^3 (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^4}-\frac {2819876 b^2 (b c-a d) g^3 (c+d x)^6 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{d^4}+\frac {5639752 b^3 g^3 (c+d x)^7 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{7 d^4}-\frac {2819876 B^2 (b c-a d)^7 g^3 n^2 \log (c+d x)}{35 b^4 d^4}+\frac {2819876 B^2 (b c-a d)^7 g^3 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{35 b^4 d^4}+\frac {\left (11279504 B^2 (b c-a d)^7 g^3 n^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{7 b^4 d^4}-\frac {\left (2819876 B^2 (b c-a d)^7 g^3 n^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{b^4 d^4}-\frac {\left (5639752 B^2 (b c-a d)^7 g^3 n^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{b^4 d^4}+\frac {\left (33838512 B^2 (b c-a d)^7 g^3 n^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{5 b^4 d^4}+\frac {\left (11279504 B^2 (b c-a d)^7 g^3 n^2\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{7 b^3 d^4}-\frac {\left (2819876 B^2 (b c-a d)^7 g^3 n^2\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{b^3 d^4}-\frac {\left (5639752 B^2 (b c-a d)^7 g^3 n^2\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{b^3 d^4}+\frac {\left (33838512 B^2 (b c-a d)^7 g^3 n^2\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{5 b^3 d^4}\\ &=\frac {2819876 A B (b c-a d)^6 g^3 n x}{35 b^3 d^3}+\frac {2819876 B^2 (b c-a d)^6 g^3 n^2 x}{35 b^3 d^3}+\frac {2114907 B^2 (b c-a d)^5 g^3 n^2 (c+d x)^2}{35 b^2 d^4}+\frac {15509318 B^2 (b c-a d)^4 g^3 n^2 (c+d x)^3}{315 b d^4}-\frac {2819876 B^2 (b c-a d)^3 g^3 n^2 (c+d x)^4}{21 d^4}+\frac {5639752 b B^2 (b c-a d)^2 g^3 n^2 (c+d x)^5}{105 d^4}+\frac {2819876 B^2 (b c-a d)^7 g^3 n^2 \log (a+b x)}{35 b^4 d^4}-\frac {1409938 B^2 (b c-a d)^7 g^3 n^2 \log ^2(a+b x)}{35 b^4 d^4}+\frac {2819876 B^2 (b c-a d)^6 g^3 n (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{35 b^4 d^3}+\frac {1409938 B (b c-a d)^5 g^3 n (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{35 b^2 d^4}+\frac {2819876 B (b c-a d)^4 g^3 n (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{105 b d^4}-\frac {23968946 B (b c-a d)^3 g^3 n (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{35 d^4}+\frac {5639752 b B (b c-a d)^2 g^3 n (c+d x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{7 d^4}-\frac {5639752 b^2 B (b c-a d) g^3 n (c+d x)^6 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{21 d^4}+\frac {2819876 B (b c-a d)^7 g^3 n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{35 b^4 d^4}-\frac {1409938 (b c-a d)^3 g^3 (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{d^4}+\frac {16919256 b (b c-a d)^2 g^3 (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^4}-\frac {2819876 b^2 (b c-a d) g^3 (c+d x)^6 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{d^4}+\frac {5639752 b^3 g^3 (c+d x)^7 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{7 d^4}-\frac {2819876 B^2 (b c-a d)^7 g^3 n^2 \log (c+d x)}{35 b^4 d^4}+\frac {2819876 B^2 (b c-a d)^7 g^3 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{35 b^4 d^4}+\frac {\left (11279504 B^2 (b c-a d)^7 g^3 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 )}{7 b^4 d^4}-\frac {\left (2819876 B^2 (b c-a d)^7 g^3 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^4 d^4}-\frac {\left (5639752 B^2 (b c-a d)^7 g^3 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^4 d^4}+\frac {\left (33838512 B^2 (b c-a d)^7 g^3 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^4 d^4}\\ &=\frac {2819876 A B (b c-a d)^6 g^3 n x}{35 b^3 d^3}+\frac {2819876 B^2 (b c-a d)^6 g^3 n^2 x}{35 b^3 d^3}+\frac {2114907 B^2 (b c-a d)^5 g^3 n^2 (c+d x)^2}{35 b^2 d^4}+\frac {15509318 B^2 (b c-a d)^4 g^3 n^2 (c+d x)^3}{315 b d^4}-\frac {2819876 B^2 (b c-a d)^3 g^3 n^2 (c+d x)^4}{21 d^4}+\frac {5639752 b B^2 (b c-a d)^2 g^3 n^2 (c+d x)^5}{105 d^4}+\frac {2819876 B^2 (b c-a d)^7 g^3 n^2 \log (a+b x)}{35 b^4 d^4}-\frac {1409938 B^2 (b c-a d)^7 g^3 n^2 \log ^2(a+b x)}{35 b^4 d^4}+\frac {2819876 B^2 (b c-a d)^6 g^3 n (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{35 b^4 d^3}+\frac {1409938 B (b c-a d)^5 g^3 n (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{35 b^2 d^4}+\frac {2819876 B (b c-a d)^4 g^3 n (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{105 b d^4}-\frac {23968946 B (b c-a d)^3 g^3 n (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{35 d^4}+\frac {5639752 b B (b c-a d)^2 g^3 n (c+d x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{7 d^4}-\frac {5639752 b^2 B (b c-a d) g^3 n (c+d x)^6 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{21 d^4}+\frac {2819876 B (b c-a d)^7 g^3 n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{35 b^4 d^4}-\frac {1409938 (b c-a d)^3 g^3 (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{d^4}+\frac {16919256 b (b c-a d)^2 g^3 (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^4}-\frac {2819876 b^2 (b c-a d) g^3 (c+d x)^6 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{d^4}+\frac {5639752 b^3 g^3 (c+d x)^7 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{7 d^4}-\frac {2819876 B^2 (b c-a d)^7 g^3 n^2 \log (c+d x)}{35 b^4 d^4}+\frac {2819876 B^2 (b c-a d)^7 g^3 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{35 b^4 d^4}+\frac {2819876 B^2 (b c-a d)^7 g^3 n^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{35 b^4 d^4}\\ \end {align*}

________________________________________________________________________________________

Mathematica [B]  time = 3.57, size = 2448, normalized size = 2.09 \[ \text {Result too large to show} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(g^3*i^3*(35*(b*c - a*d)^3*(a + b*x)^4*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2 + 84*d*(b*c - a*d)^2*(a + b*x)
^5*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2 + 70*d^2*(b*c - a*d)*(a + b*x)^6*(A + B*Log[e*((a + b*x)/(c + d*x)
)^n])^2 + 20*d^3*(a + b*x)^7*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2 - (35*B*(b*c - a*d)^4*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)])))/(3*d^4) + (7*B*(b*c - a*d)^3*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 - (7*B*(b*c - a*d)^2*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*PolyLog[2, (b*(c + d*x))/(b*c
- a*d)])))/(6*d^4) + (B*(b*c - a*d)*n*(360*A*b*d*(b*c - a*d)^5*x + 60*b^2*B*c*d*(b*c - a*d)^4*n*x - 60*a*b*B*d
^2*(b*c - a*d)^4*n*x + 462*b*B*d*(b*c - a*d)^5*n*x - 30*b*B*c*d^2*(b*c - a*d)^3*n*(a + b*x)^2 + 30*a*B*d^3*(b*
c - a*d)^3*n*(a + b*x)^2 - 141*B*d^2*(b*c - a*d)^4*n*(a + b*x)^2 + 20*b*B*c*d^3*(b*c - a*d)^2*n*(a + b*x)^3 -
20*a*B*d^4*(b*c - a*d)^2*n*(a + b*x)^3 + 54*B*d^3*(b*c - a*d)^3*n*(a + b*x)^3 - 15*b*B*c*d^4*(b*c - a*d)*n*(a
+ b*x)^4 + 15*a*B*d^5*(b*c - a*d)*n*(a + b*x)^4 - 18*B*d^4*(b*c - a*d)^2*n*(a + b*x)^4 + 12*b*B*c*d^5*n*(a + b
*x)^5 - 12*a*B*d^6*n*(a + b*x)^5 + 360*B*d*(b*c - a*d)^5*(a + b*x)*Log[e*((a + b*x)/(c + d*x))^n] - 180*d^2*(b
*c - a*d)^4*(a + b*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) + 120*d^3*(b*c - a*d)^3*(a + b*x)^3*(A + B*Log[
e*((a + b*x)/(c + d*x))^n]) - 90*d^4*(b*c - a*d)^2*(a + b*x)^4*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) + 72*d^5
*(b*c - a*d)*(a + b*x)^5*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) - 60*d^6*(a + b*x)^6*(A + B*Log[e*((a + b*x)/(
c + d*x))^n]) - 60*b*B*c*(b*c - a*d)^5*n*Log[c + d*x] + 60*a*B*d*(b*c - a*d)^5*n*Log[c + d*x] - 822*B*(b*c - a
*d)^6*n*Log[c + d*x] - 360*(b*c - a*d)^6*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*Log[c + d*x] + 180*B*(b*c - a*
d)^6*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)])))/(9*d^4)))/(140*b^4)

________________________________________________________________________________________

fricas [F]  time = 1.03, size = 0, normalized size = 0.00 \[ {\rm integral}\left (A^{2} b^{3} d^{3} g^{3} i^{3} x^{6} + A^{2} a^{3} c^{3} g^{3} i^{3} + 3 \, {\left (A^{2} b^{3} c d^{2} + A^{2} a b^{2} d^{3}\right )} g^{3} i^{3} x^{5} + 3 \, {\left (A^{2} b^{3} c^{2} d + 3 \, A^{2} a b^{2} c d^{2} + A^{2} a^{2} b d^{3}\right )} g^{3} i^{3} x^{4} + {\left (A^{2} b^{3} c^{3} + 9 \, A^{2} a b^{2} c^{2} d + 9 \, A^{2} a^{2} b c d^{2} + A^{2} a^{3} d^{3}\right )} g^{3} i^{3} x^{3} + 3 \, {\left (A^{2} a b^{2} c^{3} + 3 \, A^{2} a^{2} b c^{2} d + A^{2} a^{3} c d^{2}\right )} g^{3} i^{3} x^{2} + 3 \, {\left (A^{2} a^{2} b c^{3} + A^{2} a^{3} c^{2} d\right )} g^{3} i^{3} x + {\left (B^{2} b^{3} d^{3} g^{3} i^{3} x^{6} + B^{2} a^{3} c^{3} g^{3} i^{3} + 3 \, {\left (B^{2} b^{3} c d^{2} + B^{2} a b^{2} d^{3}\right )} g^{3} i^{3} x^{5} + 3 \, {\left (B^{2} b^{3} c^{2} d + 3 \, B^{2} a b^{2} c d^{2} + B^{2} a^{2} b d^{3}\right )} g^{3} i^{3} x^{4} + {\left (B^{2} b^{3} c^{3} + 9 \, B^{2} a b^{2} c^{2} d + 9 \, B^{2} a^{2} b c d^{2} + B^{2} a^{3} d^{3}\right )} g^{3} i^{3} x^{3} + 3 \, {\left (B^{2} a b^{2} c^{3} + 3 \, B^{2} a^{2} b c^{2} d + B^{2} a^{3} c d^{2}\right )} g^{3} i^{3} x^{2} + 3 \, {\left (B^{2} a^{2} b c^{3} + B^{2} a^{3} c^{2} d\right )} g^{3} i^{3} x\right )} \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right )^{2} + 2 \, {\left (A B b^{3} d^{3} g^{3} i^{3} x^{6} + A B a^{3} c^{3} g^{3} i^{3} + 3 \, {\left (A B b^{3} c d^{2} + A B a b^{2} d^{3}\right )} g^{3} i^{3} x^{5} + 3 \, {\left (A B b^{3} c^{2} d + 3 \, A B a b^{2} c d^{2} + A B a^{2} b d^{3}\right )} g^{3} i^{3} x^{4} + {\left (A B b^{3} c^{3} + 9 \, A B a b^{2} c^{2} d + 9 \, A B a^{2} b c d^{2} + A B a^{3} d^{3}\right )} g^{3} i^{3} x^{3} + 3 \, {\left (A B a b^{2} c^{3} + 3 \, A B a^{2} b c^{2} d + A B a^{3} c d^{2}\right )} g^{3} i^{3} x^{2} + 3 \, {\left (A B a^{2} b c^{3} + A B a^{3} c^{2} d\right )} g^{3} i^{3} x\right )} \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right ), x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*g*x+a*g)^3*(d*i*x+c*i)^3*(A+B*log(e*((b*x+a)/(d*x+c))^n))^2,x, algorithm="fricas")

[Out]

integral(A^2*b^3*d^3*g^3*i^3*x^6 + A^2*a^3*c^3*g^3*i^3 + 3*(A^2*b^3*c*d^2 + A^2*a*b^2*d^3)*g^3*i^3*x^5 + 3*(A^
2*b^3*c^2*d + 3*A^2*a*b^2*c*d^2 + A^2*a^2*b*d^3)*g^3*i^3*x^4 + (A^2*b^3*c^3 + 9*A^2*a*b^2*c^2*d + 9*A^2*a^2*b*
c*d^2 + A^2*a^3*d^3)*g^3*i^3*x^3 + 3*(A^2*a*b^2*c^3 + 3*A^2*a^2*b*c^2*d + A^2*a^3*c*d^2)*g^3*i^3*x^2 + 3*(A^2*
a^2*b*c^3 + A^2*a^3*c^2*d)*g^3*i^3*x + (B^2*b^3*d^3*g^3*i^3*x^6 + B^2*a^3*c^3*g^3*i^3 + 3*(B^2*b^3*c*d^2 + B^2
*a*b^2*d^3)*g^3*i^3*x^5 + 3*(B^2*b^3*c^2*d + 3*B^2*a*b^2*c*d^2 + B^2*a^2*b*d^3)*g^3*i^3*x^4 + (B^2*b^3*c^3 + 9
*B^2*a*b^2*c^2*d + 9*B^2*a^2*b*c*d^2 + B^2*a^3*d^3)*g^3*i^3*x^3 + 3*(B^2*a*b^2*c^3 + 3*B^2*a^2*b*c^2*d + B^2*a
^3*c*d^2)*g^3*i^3*x^2 + 3*(B^2*a^2*b*c^3 + B^2*a^3*c^2*d)*g^3*i^3*x)*log(e*((b*x + a)/(d*x + c))^n)^2 + 2*(A*B
*b^3*d^3*g^3*i^3*x^6 + A*B*a^3*c^3*g^3*i^3 + 3*(A*B*b^3*c*d^2 + A*B*a*b^2*d^3)*g^3*i^3*x^5 + 3*(A*B*b^3*c^2*d
+ 3*A*B*a*b^2*c*d^2 + A*B*a^2*b*d^3)*g^3*i^3*x^4 + (A*B*b^3*c^3 + 9*A*B*a*b^2*c^2*d + 9*A*B*a^2*b*c*d^2 + A*B*
a^3*d^3)*g^3*i^3*x^3 + 3*(A*B*a*b^2*c^3 + 3*A*B*a^2*b*c^2*d + A*B*a^3*c*d^2)*g^3*i^3*x^2 + 3*(A*B*a^2*b*c^3 +
A*B*a^3*c^2*d)*g^3*i^3*x)*log(e*((b*x + a)/(d*x + c))^n), x)

________________________________________________________________________________________

giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*g*x+a*g)^3*(d*i*x+c*i)^3*(A+B*log(e*((b*x+a)/(d*x+c))^n))^2,x, algorithm="giac")

[Out]

Timed out

________________________________________________________________________________________

maple [F]  time = 0.48, size = 0, normalized size = 0.00 \[ \int \left (b g x +a g \right )^{3} \left (d i x +c i \right )^{3} \left (B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )+A \right )^{2}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [B]  time = 6.36, size = 7845, normalized size = 6.69 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*g*x+a*g)^3*(d*i*x+c*i)^3*(A+B*log(e*((b*x+a)/(d*x+c))^n))^2,x, algorithm="maxima")

[Out]

2/7*A*B*b^3*d^3*g^3*i^3*x^7*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 1/7*A^2*b^3*d^3*g^3*i^3*x^7 + A*B*b^3*c*d
^2*g^3*i^3*x^6*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + A*B*a*b^2*d^3*g^3*i^3*x^6*log(e*(b*x/(d*x + c) + a/(d*
x + c))^n) + 1/2*A^2*b^3*c*d^2*g^3*i^3*x^6 + 1/2*A^2*a*b^2*d^3*g^3*i^3*x^6 + 6/5*A*B*b^3*c^2*d*g^3*i^3*x^5*log
(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 18/5*A*B*a*b^2*c*d^2*g^3*i^3*x^5*log(e*(b*x/(d*x + c) + a/(d*x + c))^n)
+ 6/5*A*B*a^2*b*d^3*g^3*i^3*x^5*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 3/5*A^2*b^3*c^2*d*g^3*i^3*x^5 + 9/5*A
^2*a*b^2*c*d^2*g^3*i^3*x^5 + 3/5*A^2*a^2*b*d^3*g^3*i^3*x^5 + 1/2*A*B*b^3*c^3*g^3*i^3*x^4*log(e*(b*x/(d*x + c)
+ a/(d*x + c))^n) + 9/2*A*B*a*b^2*c^2*d*g^3*i^3*x^4*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 9/2*A*B*a^2*b*c*d
^2*g^3*i^3*x^4*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 1/2*A*B*a^3*d^3*g^3*i^3*x^4*log(e*(b*x/(d*x + c) + a/(
d*x + c))^n) + 1/4*A^2*b^3*c^3*g^3*i^3*x^4 + 9/4*A^2*a*b^2*c^2*d*g^3*i^3*x^4 + 9/4*A^2*a^2*b*c*d^2*g^3*i^3*x^4
+ 1/4*A^2*a^3*d^3*g^3*i^3*x^4 + 2*A*B*a*b^2*c^3*g^3*i^3*x^3*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 6*A*B*a^
2*b*c^2*d*g^3*i^3*x^3*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 2*A*B*a^3*c*d^2*g^3*i^3*x^3*log(e*(b*x/(d*x + c
) + a/(d*x + c))^n) + A^2*a*b^2*c^3*g^3*i^3*x^3 + 3*A^2*a^2*b*c^2*d*g^3*i^3*x^3 + A^2*a^3*c*d^2*g^3*i^3*x^3 +
3*A*B*a^2*b*c^3*g^3*i^3*x^2*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 3*A*B*a^3*c^2*d*g^3*i^3*x^2*log(e*(b*x/(d
*x + c) + a/(d*x + c))^n) + 3/2*A^2*a^2*b*c^3*g^3*i^3*x^2 + 3/2*A^2*a^3*c^2*d*g^3*i^3*x^2 + 1/210*A*B*b^3*d^3*
g^3*i^3*n*(60*a^7*log(b*x + a)/b^7 - 60*c^7*log(d*x + c)/d^7 - (10*(b^6*c*d^5 - a*b^5*d^6)*x^6 - 12*(b^6*c^2*d
^4 - a^2*b^4*d^6)*x^5 + 15*(b^6*c^3*d^3 - a^3*b^3*d^6)*x^4 - 20*(b^6*c^4*d^2 - a^4*b^2*d^6)*x^3 + 30*(b^6*c^5*
d - a^5*b*d^6)*x^2 - 60*(b^6*c^6 - a^6*d^6)*x)/(b^6*d^6)) - 1/60*A*B*b^3*c*d^2*g^3*i^3*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)) - 1/60*A*B
*a*b^2*d^3*g^3*i^3*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)) + 1/10*A*B*b^3*c^2*d*g^3*i^3*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)) + 3/10*A*B*a*b^2*c*d^2*g^3*i^3*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^2*b*d^3*g^3*i^3*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^3*g^3*i^3*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))
- 3/4*A*B*a*b^2*c^2*d*g^3*i^3*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)) - 3/4*A*B*a^2*b*c*d^2*g^3*i^3*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)) - 1/12*A*B*a^3*d^3*g^3*i^3*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^3*g^3*i^3*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*d*g^3*i^3*n*(2*a^3*log(b*x + a)/b^3 - 2*c^3*log(d*x + c)/d^3 -
((b^2*c*d - a*b*d^2)*x^2 - 2*(b^2*c^2 - a^2*d^2)*x)/(b^2*d^2)) + A*B*a^3*c*d^2*g^3*i^3*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^
3*g^3*i^3*n*(a^2*log(b*x + a)/b^2 - c^2*log(d*x + c)/d^2 + (b*c - a*d)*x/(b*d)) - 3*A*B*a^3*c^2*d*g^3*i^3*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^3*g^3*i^3*n*(a*log(b*x + a)/b
- c*log(d*x + c)/d) + 2*A*B*a^3*c^3*g^3*i^3*x*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + A^2*a^3*c^3*g^3*i^3*x -
1/420*(107*a^4*b^2*c^3*d^4*g^3*i^3*n^2 - 39*a^5*b*c^2*d^5*g^3*i^3*n^2 + 6*a^6*c*d^6*g^3*i^3*n^2 - 6*b^6*c^7*g
^3*i^3*n*log(e) - 6*(g^3*i^3*n^2 - 7*g^3*i^3*n*log(e))*a*b^5*c^6*d + 3*(13*g^3*i^3*n^2 - 42*g^3*i^3*n*log(e))*
a^2*b^4*c^5*d^2 - (107*g^3*i^3*n^2 - 210*g^3*i^3*n*log(e))*a^3*b^3*c^4*d^3)*B^2*log(d*x + c)/(b^3*d^4) + 1/70*
(b^7*c^7*g^3*i^3*n^2 - 7*a*b^6*c^6*d*g^3*i^3*n^2 + 21*a^2*b^5*c^5*d^2*g^3*i^3*n^2 - 35*a^3*b^4*c^4*d^3*g^3*i^3
*n^2 + 35*a^4*b^3*c^3*d^4*g^3*i^3*n^2 - 21*a^5*b^2*c^2*d^5*g^3*i^3*n^2 + 7*a^6*b*c*d^6*g^3*i^3*n^2 - a^7*d^7*g
^3*i^3*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^4*d^4
) + 1/2520*(360*B^2*b^7*d^7*g^3*i^3*x^7*log(e)^2 - 60*((2*g^3*i^3*n*log(e) - 21*g^3*i^3*log(e)^2)*b^7*c*d^6 -
(2*g^3*i^3*n*log(e) + 21*g^3*i^3*log(e)^2)*a*b^6*d^7)*B^2*x^6 + 24*((g^3*i^3*n^2 - 15*g^3*i^3*n*log(e) + 63*g^
3*i^3*log(e)^2)*b^7*c^2*d^5 - (2*g^3*i^3*n^2 - 189*g^3*i^3*log(e)^2)*a*b^6*c*d^6 + (g^3*i^3*n^2 + 15*g^3*i^3*n
*log(e) + 63*g^3*i^3*log(e)^2)*a^2*b^5*d^7)*B^2*x^5 + 6*((10*g^3*i^3*n^2 - 51*g^3*i^3*n*log(e) + 105*g^3*i^3*l
og(e)^2)*b^7*c^3*d^4 - (10*g^3*i^3*n^2 + 147*g^3*i^3*n*log(e) - 945*g^3*i^3*log(e)^2)*a*b^6*c^2*d^5 - (10*g^3*
i^3*n^2 - 147*g^3*i^3*n*log(e) - 945*g^3*i^3*log(e)^2)*a^2*b^5*c*d^6 + (10*g^3*i^3*n^2 + 51*g^3*i^3*n*log(e) +
105*g^3*i^3*log(e)^2)*a^3*b^4*d^7)*B^2*x^4 + 2*((11*g^3*i^3*n^2 - 6*g^3*i^3*n*log(e))*b^7*c^4*d^3 + 4*(19*g^3
*i^3*n^2 - 147*g^3*i^3*n*log(e) + 315*g^3*i^3*log(e)^2)*a*b^6*c^3*d^4 - 6*(29*g^3*i^3*n^2 - 630*g^3*i^3*log(e)
^2)*a^2*b^5*c^2*d^5 + 4*(19*g^3*i^3*n^2 + 147*g^3*i^3*n*log(e) + 315*g^3*i^3*log(e)^2)*a^3*b^4*c*d^6 + (11*g^3
*i^3*n^2 + 6*g^3*i^3*n*log(e))*a^4*b^3*d^7)*B^2*x^3 - 3*(3*(3*g^3*i^3*n^2 - 2*g^3*i^3*n*log(e))*b^7*c^5*d^2 -
(67*g^3*i^3*n^2 - 42*g^3*i^3*n*log(e))*a*b^6*c^4*d^3 + 2*(29*g^3*i^3*n^2 + 252*g^3*i^3*n*log(e) - 630*g^3*i^3*
log(e)^2)*a^2*b^5*c^3*d^4 + 2*(29*g^3*i^3*n^2 - 252*g^3*i^3*n*log(e) - 630*g^3*i^3*log(e)^2)*a^3*b^4*c^2*d^5 -
(67*g^3*i^3*n^2 + 42*g^3*i^3*n*log(e))*a^4*b^3*c*d^6 + 3*(3*g^3*i^3*n^2 + 2*g^3*i^3*n*log(e))*a^5*b^2*d^7)*B^
2*x^2 - 18*(35*a^4*b^3*c^3*d^4*g^3*i^3*n^2 - 21*a^5*b^2*c^2*d^5*g^3*i^3*n^2 + 7*a^6*b*c*d^6*g^3*i^3*n^2 - a^7*
d^7*g^3*i^3*n^2)*B^2*log(b*x + a)^2 - 36*(b^7*c^7*g^3*i^3*n^2 - 7*a*b^6*c^6*d*g^3*i^3*n^2 + 21*a^2*b^5*c^5*d^2
*g^3*i^3*n^2 - 35*a^3*b^4*c^4*d^3*g^3*i^3*n^2)*B^2*log(b*x + a)*log(d*x + c) + 18*(b^7*c^7*g^3*i^3*n^2 - 7*a*b
^6*c^6*d*g^3*i^3*n^2 + 21*a^2*b^5*c^5*d^2*g^3*i^3*n^2 - 35*a^3*b^4*c^4*d^3*g^3*i^3*n^2)*B^2*log(d*x + c)^2 + 6
*(6*(g^3*i^3*n^2 - g^3*i^3*n*log(e))*b^7*c^6*d - 3*(15*g^3*i^3*n^2 - 14*g^3*i^3*n*log(e))*a*b^6*c^5*d^2 + 2*(7
3*g^3*i^3*n^2 - 63*g^3*i^3*n*log(e))*a^2*b^5*c^4*d^3 - 2*(107*g^3*i^3*n^2 - 210*g^3*i^3*log(e)^2)*a^3*b^4*c^3*
d^4 + 2*(73*g^3*i^3*n^2 + 63*g^3*i^3*n*log(e))*a^4*b^3*c^2*d^5 - 3*(15*g^3*i^3*n^2 + 14*g^3*i^3*n*log(e))*a^5*
b^2*c*d^6 + 6*(g^3*i^3*n^2 + g^3*i^3*n*log(e))*a^6*b*d^7)*B^2*x - 6*(6*a*b^6*c^6*d*g^3*i^3*n^2 - 39*a^2*b^5*c^
5*d^2*g^3*i^3*n^2 + 107*a^3*b^4*c^4*d^3*g^3*i^3*n^2 + 6*a^7*d^7*g^3*i^3*n*log(e) - (107*g^3*i^3*n^2 + 210*g^3*
i^3*n*log(e))*a^4*b^3*c^3*d^4 + 3*(13*g^3*i^3*n^2 + 42*g^3*i^3*n*log(e))*a^5*b^2*c^2*d^5 - 6*(g^3*i^3*n^2 + 7*
g^3*i^3*n*log(e))*a^6*b*c*d^6)*B^2*log(b*x + a) + 18*(20*B^2*b^7*d^7*g^3*i^3*x^7 + 140*B^2*a^3*b^4*c^3*d^4*g^3
*i^3*x + 70*(b^7*c*d^6*g^3*i^3 + a*b^6*d^7*g^3*i^3)*B^2*x^6 + 84*(b^7*c^2*d^5*g^3*i^3 + 3*a*b^6*c*d^6*g^3*i^3
+ a^2*b^5*d^7*g^3*i^3)*B^2*x^5 + 35*(b^7*c^3*d^4*g^3*i^3 + 9*a*b^6*c^2*d^5*g^3*i^3 + 9*a^2*b^5*c*d^6*g^3*i^3 +
a^3*b^4*d^7*g^3*i^3)*B^2*x^4 + 140*(a*b^6*c^3*d^4*g^3*i^3 + 3*a^2*b^5*c^2*d^5*g^3*i^3 + a^3*b^4*c*d^6*g^3*i^3
)*B^2*x^3 + 210*(a^2*b^5*c^3*d^4*g^3*i^3 + a^3*b^4*c^2*d^5*g^3*i^3)*B^2*x^2)*log((b*x + a)^n)^2 + 18*(20*B^2*b
^7*d^7*g^3*i^3*x^7 + 140*B^2*a^3*b^4*c^3*d^4*g^3*i^3*x + 70*(b^7*c*d^6*g^3*i^3 + a*b^6*d^7*g^3*i^3)*B^2*x^6 +
84*(b^7*c^2*d^5*g^3*i^3 + 3*a*b^6*c*d^6*g^3*i^3 + a^2*b^5*d^7*g^3*i^3)*B^2*x^5 + 35*(b^7*c^3*d^4*g^3*i^3 + 9*a
*b^6*c^2*d^5*g^3*i^3 + 9*a^2*b^5*c*d^6*g^3*i^3 + a^3*b^4*d^7*g^3*i^3)*B^2*x^4 + 140*(a*b^6*c^3*d^4*g^3*i^3 + 3
*a^2*b^5*c^2*d^5*g^3*i^3 + a^3*b^4*c*d^6*g^3*i^3)*B^2*x^3 + 210*(a^2*b^5*c^3*d^4*g^3*i^3 + a^3*b^4*c^2*d^5*g^3
*i^3)*B^2*x^2)*log((d*x + c)^n)^2 + 6*(120*B^2*b^7*d^7*g^3*i^3*x^7*log(e) - 20*((g^3*i^3*n - 21*g^3*i^3*log(e)
)*b^7*c*d^6 - (g^3*i^3*n + 21*g^3*i^3*log(e))*a*b^6*d^7)*B^2*x^6 + 12*(126*a*b^6*c*d^6*g^3*i^3*log(e) - (5*g^3
*i^3*n - 42*g^3*i^3*log(e))*b^7*c^2*d^5 + (5*g^3*i^3*n + 42*g^3*i^3*log(e))*a^2*b^5*d^7)*B^2*x^5 - 3*((17*g^3*
i^3*n - 70*g^3*i^3*log(e))*b^7*c^3*d^4 + 7*(7*g^3*i^3*n - 90*g^3*i^3*log(e))*a*b^6*c^2*d^5 - 7*(7*g^3*i^3*n +
90*g^3*i^3*log(e))*a^2*b^5*c*d^6 - (17*g^3*i^3*n + 70*g^3*i^3*log(e))*a^3*b^4*d^7)*B^2*x^4 - 2*(b^7*c^4*d^3*g^
3*i^3*n - a^4*b^3*d^7*g^3*i^3*n - 1260*a^2*b^5*c^2*d^5*g^3*i^3*log(e) + 14*(7*g^3*i^3*n - 30*g^3*i^3*log(e))*a
*b^6*c^3*d^4 - 14*(7*g^3*i^3*n + 30*g^3*i^3*log(e))*a^3*b^4*c*d^6)*B^2*x^3 + 3*(b^7*c^5*d^2*g^3*i^3*n - 7*a*b^
6*c^4*d^3*g^3*i^3*n + 7*a^4*b^3*c*d^6*g^3*i^3*n - a^5*b^2*d^7*g^3*i^3*n - 84*(g^3*i^3*n - 5*g^3*i^3*log(e))*a^
2*b^5*c^3*d^4 + 84*(g^3*i^3*n + 5*g^3*i^3*log(e))*a^3*b^4*c^2*d^5)*B^2*x^2 - 6*(b^7*c^6*d*g^3*i^3*n - 7*a*b^6*
c^5*d^2*g^3*i^3*n + 21*a^2*b^5*c^4*d^3*g^3*i^3*n - 21*a^4*b^3*c^2*d^5*g^3*i^3*n + 7*a^5*b^2*c*d^6*g^3*i^3*n -
a^6*b*d^7*g^3*i^3*n - 140*a^3*b^4*c^3*d^4*g^3*i^3*log(e))*B^2*x + 6*(35*a^4*b^3*c^3*d^4*g^3*i^3*n - 21*a^5*b^2
*c^2*d^5*g^3*i^3*n + 7*a^6*b*c*d^6*g^3*i^3*n - a^7*d^7*g^3*i^3*n)*B^2*log(b*x + a) + 6*(b^7*c^7*g^3*i^3*n - 7*
a*b^6*c^6*d*g^3*i^3*n + 21*a^2*b^5*c^5*d^2*g^3*i^3*n - 35*a^3*b^4*c^4*d^3*g^3*i^3*n)*B^2*log(d*x + c))*log((b*
x + a)^n) - 6*(120*B^2*b^7*d^7*g^3*i^3*x^7*log(e) - 20*((g^3*i^3*n - 21*g^3*i^3*log(e))*b^7*c*d^6 - (g^3*i^3*n
+ 21*g^3*i^3*log(e))*a*b^6*d^7)*B^2*x^6 + 12*(126*a*b^6*c*d^6*g^3*i^3*log(e) - (5*g^3*i^3*n - 42*g^3*i^3*log(
e))*b^7*c^2*d^5 + (5*g^3*i^3*n + 42*g^3*i^3*log(e))*a^2*b^5*d^7)*B^2*x^5 - 3*((17*g^3*i^3*n - 70*g^3*i^3*log(e
))*b^7*c^3*d^4 + 7*(7*g^3*i^3*n - 90*g^3*i^3*log(e))*a*b^6*c^2*d^5 - 7*(7*g^3*i^3*n + 90*g^3*i^3*log(e))*a^2*b
^5*c*d^6 - (17*g^3*i^3*n + 70*g^3*i^3*log(e))*a^3*b^4*d^7)*B^2*x^4 - 2*(b^7*c^4*d^3*g^3*i^3*n - a^4*b^3*d^7*g^
3*i^3*n - 1260*a^2*b^5*c^2*d^5*g^3*i^3*log(e) + 14*(7*g^3*i^3*n - 30*g^3*i^3*log(e))*a*b^6*c^3*d^4 - 14*(7*g^3
*i^3*n + 30*g^3*i^3*log(e))*a^3*b^4*c*d^6)*B^2*x^3 + 3*(b^7*c^5*d^2*g^3*i^3*n - 7*a*b^6*c^4*d^3*g^3*i^3*n + 7*
a^4*b^3*c*d^6*g^3*i^3*n - a^5*b^2*d^7*g^3*i^3*n - 84*(g^3*i^3*n - 5*g^3*i^3*log(e))*a^2*b^5*c^3*d^4 + 84*(g^3*
i^3*n + 5*g^3*i^3*log(e))*a^3*b^4*c^2*d^5)*B^2*x^2 - 6*(b^7*c^6*d*g^3*i^3*n - 7*a*b^6*c^5*d^2*g^3*i^3*n + 21*a
^2*b^5*c^4*d^3*g^3*i^3*n - 21*a^4*b^3*c^2*d^5*g^3*i^3*n + 7*a^5*b^2*c*d^6*g^3*i^3*n - a^6*b*d^7*g^3*i^3*n - 14
0*a^3*b^4*c^3*d^4*g^3*i^3*log(e))*B^2*x + 6*(35*a^4*b^3*c^3*d^4*g^3*i^3*n - 21*a^5*b^2*c^2*d^5*g^3*i^3*n + 7*a
^6*b*c*d^6*g^3*i^3*n - a^7*d^7*g^3*i^3*n)*B^2*log(b*x + a) + 6*(b^7*c^7*g^3*i^3*n - 7*a*b^6*c^6*d*g^3*i^3*n +
21*a^2*b^5*c^5*d^2*g^3*i^3*n - 35*a^3*b^4*c^4*d^3*g^3*i^3*n)*B^2*log(d*x + c) + 6*(20*B^2*b^7*d^7*g^3*i^3*x^7
+ 140*B^2*a^3*b^4*c^3*d^4*g^3*i^3*x + 70*(b^7*c*d^6*g^3*i^3 + a*b^6*d^7*g^3*i^3)*B^2*x^6 + 84*(b^7*c^2*d^5*g^3
*i^3 + 3*a*b^6*c*d^6*g^3*i^3 + a^2*b^5*d^7*g^3*i^3)*B^2*x^5 + 35*(b^7*c^3*d^4*g^3*i^3 + 9*a*b^6*c^2*d^5*g^3*i^
3 + 9*a^2*b^5*c*d^6*g^3*i^3 + a^3*b^4*d^7*g^3*i^3)*B^2*x^4 + 140*(a*b^6*c^3*d^4*g^3*i^3 + 3*a^2*b^5*c^2*d^5*g^
3*i^3 + a^3*b^4*c*d^6*g^3*i^3)*B^2*x^3 + 210*(a^2*b^5*c^3*d^4*g^3*i^3 + a^3*b^4*c^2*d^5*g^3*i^3)*B^2*x^2)*log(
(b*x + a)^n))*log((d*x + c)^n))/(b^4*d^4)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int {\left (a\,g+b\,g\,x\right )}^3\,{\left (c\,i+d\,i\,x\right )}^3\,{\left (A+B\,\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\right )}^2 \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*g*x+a*g)**3*(d*i*x+c*i)**3*(A+B*ln(e*((b*x+a)/(d*x+c))**n))**2,x)

[Out]

Timed out

________________________________________________________________________________________