3.2.78 \(\int (a g+b g x)^3 (c i+d i x)^3 (A+B \log (e (\frac {a+b x}{c+d x})^n))^2 \, dx\) [178]

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

[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 = 1.05, antiderivative size = 1172, normalized size of antiderivative = 1.00, number of steps used = 22, number of rules used = 14, integrand size = 45, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.311, Rules used = {2561, 2383, 2381, 2384, 2354, 2438, 2373, 45, 47, 37, 2382, 12, 79, 1634} \begin {gather*} -\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 {PolyLog}\left (2,\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} \end {gather*}

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]

(5*B^2*(b*c - a*d)^6*g^3*i^3*n^2*x)/(84*b^3*d^3) + (B^2*(b*c - a*d)^3*g^3*i^3*n^2*(a + b*x)^4)/(140*b^4) - (29
*B^2*(b*c - a*d)^5*g^3*i^3*n^2*(c + d*x)^2)/(840*b^2*d^4) + (47*B^2*(b*c - a*d)^4*g^3*i^3*n^2*(c + d*x)^3)/(12
60*b*d^4) - (13*B^2*(b*c - a*d)^3*g^3*i^3*n^2*(c + d*x)^4)/(420*d^4) + (b*B^2*(b*c - a*d)^2*g^3*i^3*n^2*(c + d
*x)^5)/(105*d^4) - (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)
- (3*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*(b*c - a*d)^
2*g^3*i^3*n*(a + b*x)^4*(c + d*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(35*b^3) + (2*B*(b*c - a*d)^4*g^3*i^
3*n*(c + d*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(21*b*d^4) - (3*B*(b*c - a*d)^3*g^3*i^3*n*(c + d*x)^4*
(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(14*d^4) + (6*b*B*(b*c - a*d)^2*g^3*i^3*n*(c + d*x)^5*(A + B*Log[e*((a
+ b*x)/(c + d*x))^n]))/(35*d^4) - (b^2*B*(b*c - a*d)*g^3*i^3*n*(c + d*x)^6*(A + B*Log[e*((a + b*x)/(c + d*x))
^n]))/(21*d^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)/(140*b^4) + ((b*
c - a*d)^2*g^3*i^3*(a + b*x)^4*(c + d*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(35*b^3) + ((b*c - a*d)*g^3
*i^3*(a + b*x)^4*(c + d*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(14*b^2) + (g^3*i^3*(a + b*x)^4*(c + d*
x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(7*b) + (B*(b*c - a*d)^5*g^3*i^3*n*(a + b*x)^2*(3*A + B*n + 3*B
*Log[e*((a + b*x)/(c + d*x))^n]))/(420*b^4*d^2) - (B*(b*c - a*d)^6*g^3*i^3*n*(a + b*x)*(6*A + 5*B*n + 6*B*Log[
e*((a + b*x)/(c + d*x))^n]))/(420*b^4*d^3) - (B*(b*c - a*d)^7*g^3*i^3*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))])/(420*b^4*d^4) - (B^2*(b*c - a*d)^7*g^3*i^3*n^2*Log[(a + b*x)/(c
+ d*x)])/(210*b^4*d^4) - (11*B^2*(b*c - a*d)^7*g^3*i^3*n^2*Log[c + d*x])/(420*b^4*d^4) - (B^2*(b*c - a*d)^7*g^
3*i^3*n^2*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(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 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n +
1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rule 45

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 47

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n + 1
)/((b*c - a*d)*(m + 1))), x] - Dist[d*(Simplify[m + n + 2]/((b*c - a*d)*(m + 1))), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

Rule 79

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(-(b*e - a*f
))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1
) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e,
f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || L
tQ[p, n]))))

Rule 1634

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[Px*(a + b*x)
^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && PolyQ[Px, x] && (IntegersQ[m, n] || IGtQ[m, -2]) &&
GtQ[Expon[Px, x], 2]

Rule 2354

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

Rule 2373

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

Rule 2381

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_))^(q_), x_Symbol] :> Simp
[(-(f*x)^(m + 1))*(d + e*x)^(q + 1)*((a + b*Log[c*x^n])^p/(d*f*(q + 1))), x] + Dist[b*n*(p/(d*(q + 1))), Int[(
f*x)^m*(d + e*x)^(q + 1)*(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, f, m, n, q}, x] && EqQ[m
+ q + 2, 0] && IGtQ[p, 0] && LtQ[q, -1]

Rule 2382

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(x_)^(m_.)*((d_) + (e_.)*(x_))^(q_), x_Symbol] :> With[{u = IntHide[
x^m*(d + e*x)^q, x]}, Dist[a + b*Log[c*x^n], u, x] - Dist[b*n, Int[SimplifyIntegrand[u/x, x], x], x]] /; FreeQ
[{a, b, c, d, e, n}, x] && ILtQ[m + q + 2, 0] && IGtQ[m, 0]

Rule 2383

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_))^(q_), x_Symbol] :> Simp
[(-(f*x)^(m + 1))*(d + e*x)^(q + 1)*((a + b*Log[c*x^n])^p/(d*f*(q + 1))), x] + (Dist[(m + q + 2)/(d*(q + 1)),
Int[(f*x)^m*(d + e*x)^(q + 1)*(a + b*Log[c*x^n])^p, x], x] + Dist[b*n*(p/(d*(q + 1))), Int[(f*x)^m*(d + e*x)^(
q + 1)*(a + b*Log[c*x^n])^(p - 1), x], x]) /; FreeQ[{a, b, c, d, e, f, n}, x] && ILtQ[m + q + 2, 0] && IGtQ[p,
0] && LtQ[q, -1] && GtQ[m, 0]

Rule 2384

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_))^(q_.), x_Symbol] :> Simp[(f*x
)^m*(d + e*x)^(q + 1)*((a + b*Log[c*x^n])/(e*(q + 1))), x] - Dist[f/(e*(q + 1)), Int[(f*x)^(m - 1)*(d + e*x)^(
q + 1)*(a*m + b*n + b*m*Log[c*x^n]), x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && ILtQ[q, -1] && GtQ[m, 0]

Rule 2438

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 2561

Int[((A_.) + Log[(e_.)*(((a_.) + (b_.)*(x_))/((c_.) + (d_.)*(x_)))^(n_.)]*(B_.))^(p_.)*((f_.) + (g_.)*(x_))^(m
_.)*((h_.) + (i_.)*(x_))^(q_.), x_Symbol] :> Dist[(b*c - a*d)^(m + q + 1)*(g/b)^m*(i/d)^q, Subst[Int[x^m*((A +
B*Log[e*x^n])^p/(b - d*x)^(m + q + 2)), x], x, (a + b*x)/(c + d*x)], x] /; FreeQ[{a, b, c, d, e, f, g, h, i,
A, B, n, p}, x] && NeQ[b*c - a*d, 0] && EqQ[b*f - a*g, 0] && EqQ[d*h - c*i, 0] && IntegersQ[m, q]

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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(2448\) vs. \(2(1172)=2344\).
time = 2.20, size = 2448, normalized size = 2.09 \begin {gather*} \text {Result too large to show} \end {gather*}

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)

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Maple [F]
time = 0.37, size = 0, normalized size = 0.00 \[\int \left (b g x +a g \right )^{3} \left (d i x +c i \right )^{3} \left (A +B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )\right )^{2}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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Maxima [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 6587 vs. \(2 (1070) = 2140\).
time = 1.02, size = 6587, normalized size = 5.62 \begin {gather*} \text {Too large to display} \end {gather*}

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*I*A*B*b^3*d^3*g^3*x^7*log((b*x/(d*x + c) + a/(d*x + c))^n*e) - 1/7*I*A^2*b^3*d^3*g^3*x^7 - I*A*B*b^3*c*d^
2*g^3*x^6*log((b*x/(d*x + c) + a/(d*x + c))^n*e) - I*A*B*a*b^2*d^3*g^3*x^6*log((b*x/(d*x + c) + a/(d*x + c))^n
*e) - 1/2*I*A^2*b^3*c*d^2*g^3*x^6 - 1/2*I*A^2*a*b^2*d^3*g^3*x^6 - 6/5*I*A*B*b^3*c^2*d*g^3*x^5*log((b*x/(d*x +
c) + a/(d*x + c))^n*e) - 18/5*I*A*B*a*b^2*c*d^2*g^3*x^5*log((b*x/(d*x + c) + a/(d*x + c))^n*e) - 6/5*I*A*B*a^2
*b*d^3*g^3*x^5*log((b*x/(d*x + c) + a/(d*x + c))^n*e) - 3/5*I*A^2*b^3*c^2*d*g^3*x^5 - 9/5*I*A^2*a*b^2*c*d^2*g^
3*x^5 - 3/5*I*A^2*a^2*b*d^3*g^3*x^5 - 1/2*I*A*B*b^3*c^3*g^3*x^4*log((b*x/(d*x + c) + a/(d*x + c))^n*e) - 9/2*I
*A*B*a*b^2*c^2*d*g^3*x^4*log((b*x/(d*x + c) + a/(d*x + c))^n*e) - 9/2*I*A*B*a^2*b*c*d^2*g^3*x^4*log((b*x/(d*x
+ c) + a/(d*x + c))^n*e) - 1/2*I*A*B*a^3*d^3*g^3*x^4*log((b*x/(d*x + c) + a/(d*x + c))^n*e) - 1/4*I*A^2*b^3*c^
3*g^3*x^4 - 9/4*I*A^2*a*b^2*c^2*d*g^3*x^4 - 9/4*I*A^2*a^2*b*c*d^2*g^3*x^4 - 1/4*I*A^2*a^3*d^3*g^3*x^4 - 2*I*A*
B*a*b^2*c^3*g^3*x^3*log((b*x/(d*x + c) + a/(d*x + c))^n*e) - 6*I*A*B*a^2*b*c^2*d*g^3*x^3*log((b*x/(d*x + c) +
a/(d*x + c))^n*e) - 2*I*A*B*a^3*c*d^2*g^3*x^3*log((b*x/(d*x + c) + a/(d*x + c))^n*e) - I*A^2*a*b^2*c^3*g^3*x^3
- 3*I*A^2*a^2*b*c^2*d*g^3*x^3 - I*A^2*a^3*c*d^2*g^3*x^3 - 3*I*A*B*a^2*b*c^3*g^3*x^2*log((b*x/(d*x + c) + a/(d
*x + c))^n*e) - 3*I*A*B*a^3*c^2*d*g^3*x^2*log((b*x/(d*x + c) + a/(d*x + c))^n*e) - 3/2*I*A^2*a^2*b*c^3*g^3*x^2
- 3/2*I*A^2*a^3*c^2*d*g^3*x^2 - 1/210*I*A*B*b^3*d^3*g^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*I*A*B*b^3*c*d^2*g^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*I*A*B*a*b^2*d^3*g^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*I*A*B*b^3*c^2*d*g^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*I*A*B*a*b^2*c*d^2*g^3*n*(1
2*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*I*A*B*a^2*b*d^3*g^3*n*(1
2*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*I*A*B*b^3*c^3*g^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*I*A*B*a*b^2*c^2*d*g^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*I*A*B*a^2*b*c*d^2*g^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*I*A*B*a^3*d^3*g^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)) - I*A*B*a*b^2*c^3*g^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*I*A*B*a^2*b*c^2*d*g^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)) - I*A*B*a^3*c*d^2*g
^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*I*A*B*a^2*b*c^3*g^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*I*A
*B*a^3*c^2*d*g^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*I*A*B*a^3*c^3*g^3*n
*(a*log(b*x + a)/b - c*log(d*x + c)/d) - 2*I*A*B*a^3*c^3*g^3*x*log((b*x/(d*x + c) + a/(d*x + c))^n*e) - I*A^2*
a^3*c^3*g^3*x + 1/420*(107*I*a^4*b^2*c^3*d^4*g^3*n^2 - 39*I*a^5*b*c^2*d^5*g^3*n^2 + 6*I*a^6*c*d^6*g^3*n^2 - 6*
(I*n^2 - 7*I*n)*a*b^5*c^6*d*g^3 - 3*(-13*I*n^2 + 42*I*n)*a^2*b^4*c^5*d^2*g^3 + (-107*I*n^2 + 210*I*n)*a^3*b^3*
c^4*d^3*g^3 - 6*I*b^6*c^7*g^3*n)*B^2*log(d*x + c)/(b^3*d^4) + 1/70*(-I*b^7*c^7*g^3*n^2 + 7*I*a*b^6*c^6*d*g^3*n
^2 - 21*I*a^2*b^5*c^5*d^2*g^3*n^2 + 35*I*a^3*b^4*c^4*d^3*g^3*n^2 - 35*I*a^4*b^3*c^3*d^4*g^3*n^2 + 21*I*a^5*b^2
*c^2*d^5*g^3*n^2 - 7*I*a^6*b*c*d^6*g^3*n^2 + I*a^7*d^7*g^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^...

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/140*(-20*I*B^2*b^3*d^3*g^3*n^2*x^7 - 140*I*B^2*a^3*c^3*g^3*n^2*x - 70*(I*B^2*b^3*c*d^2 + I*B^2*a*b^2*d^3)*g^
3*n^2*x^6 - 84*(I*B^2*b^3*c^2*d + 3*I*B^2*a*b^2*c*d^2 + I*B^2*a^2*b*d^3)*g^3*n^2*x^5 - 35*(I*B^2*b^3*c^3 + 9*I
*B^2*a*b^2*c^2*d + 9*I*B^2*a^2*b*c*d^2 + I*B^2*a^3*d^3)*g^3*n^2*x^4 - 140*(I*B^2*a*b^2*c^3 + 3*I*B^2*a^2*b*c^2
*d + I*B^2*a^3*c*d^2)*g^3*n^2*x^3 - 210*(I*B^2*a^2*b*c^3 + I*B^2*a^3*c^2*d)*g^3*n^2*x^2)*log((b*x + a)/(d*x +
c))^2 + integral(-1/70*(70*(I*A^2 + 2*I*A*B + I*B^2)*b^4*d^4*g^3*x^8 + 70*(I*A^2 + 2*I*A*B + I*B^2)*a^4*c^4*g^
3 + 280*((I*A^2 + 2*I*A*B + I*B^2)*b^4*c*d^3 + (I*A^2 + 2*I*A*B + I*B^2)*a*b^3*d^4)*g^3*x^7 + 140*(3*(I*A^2 +
2*I*A*B + I*B^2)*b^4*c^2*d^2 + 8*(I*A^2 + 2*I*A*B + I*B^2)*a*b^3*c*d^3 + 3*(I*A^2 + 2*I*A*B + I*B^2)*a^2*b^2*d
^4)*g^3*x^6 + 280*((I*A^2 + 2*I*A*B + I*B^2)*b^4*c^3*d + 6*(I*A^2 + 2*I*A*B + I*B^2)*a*b^3*c^2*d^2 + 6*(I*A^2
+ 2*I*A*B + I*B^2)*a^2*b^2*c*d^3 + (I*A^2 + 2*I*A*B + I*B^2)*a^3*b*d^4)*g^3*x^5 + 70*((I*A^2 + 2*I*A*B + I*B^2
)*b^4*c^4 + 16*(I*A^2 + 2*I*A*B + I*B^2)*a*b^3*c^3*d + 36*(I*A^2 + 2*I*A*B + I*B^2)*a^2*b^2*c^2*d^2 + 16*(I*A^
2 + 2*I*A*B + I*B^2)*a^3*b*c*d^3 + (I*A^2 + 2*I*A*B + I*B^2)*a^4*d^4)*g^3*x^4 + 280*((I*A^2 + 2*I*A*B + I*B^2)
*a*b^3*c^4 + 6*(I*A^2 + 2*I*A*B + I*B^2)*a^2*b^2*c^3*d + 6*(I*A^2 + 2*I*A*B + I*B^2)*a^3*b*c^2*d^2 + (I*A^2 +
2*I*A*B + I*B^2)*a^4*c*d^3)*g^3*x^3 + 140*(3*(I*A^2 + 2*I*A*B + I*B^2)*a^2*b^2*c^4 + 8*(I*A^2 + 2*I*A*B + I*B^
2)*a^3*b*c^3*d + 3*(I*A^2 + 2*I*A*B + I*B^2)*a^4*c^2*d^2)*g^3*x^2 + 280*((I*A^2 + 2*I*A*B + I*B^2)*a^3*b*c^4 +
(I*A^2 + 2*I*A*B + I*B^2)*a^4*c^3*d)*g^3*x + (140*(I*A*B + I*B^2)*b^4*d^4*g^3*n*x^8 + 140*(I*A*B + I*B^2)*a^4
*c^4*g^3*n + 20*((-I*B^2*b^4*c*d^3 + I*B^2*a*b^3*d^4)*g^3*n^2 + 28*((I*A*B + I*B^2)*b^4*c*d^3 + (I*A*B + I*B^2
)*a*b^3*d^4)*g^3*n)*x^7 + 70*((-I*B^2*b^4*c^2*d^2 + I*B^2*a^2*b^2*d^4)*g^3*n^2 + 4*(3*(I*A*B + I*B^2)*b^4*c^2*
d^2 + 8*(I*A*B + I*B^2)*a*b^3*c*d^3 + 3*(I*A*B + I*B^2)*a^2*b^2*d^4)*g^3*n)*x^6 + 28*(3*(-I*B^2*b^4*c^3*d - 2*
I*B^2*a*b^3*c^2*d^2 + 2*I*B^2*a^2*b^2*c*d^3 + I*B^2*a^3*b*d^4)*g^3*n^2 + 20*((I*A*B + I*B^2)*b^4*c^3*d + 6*(I*
A*B + I*B^2)*a*b^3*c^2*d^2 + 6*(I*A*B + I*B^2)*a^2*b^2*c*d^3 + (I*A*B + I*B^2)*a^3*b*d^4)*g^3*n)*x^5 + 35*((-I
*B^2*b^4*c^4 - 8*I*B^2*a*b^3*c^3*d + 8*I*B^2*a^3*b*c*d^3 + I*B^2*a^4*d^4)*g^3*n^2 + 4*((I*A*B + I*B^2)*b^4*c^4
+ 16*(I*A*B + I*B^2)*a*b^3*c^3*d + 36*(I*A*B + I*B^2)*a^2*b^2*c^2*d^2 + 16*(I*A*B + I*B^2)*a^3*b*c*d^3 + (I*A
*B + I*B^2)*a^4*d^4)*g^3*n)*x^4 + 140*((-I*B^2*a*b^3*c^4 - 2*I*B^2*a^2*b^2*c^3*d + 2*I*B^2*a^3*b*c^2*d^2 + I*B
^2*a^4*c*d^3)*g^3*n^2 + 4*((I*A*B + I*B^2)*a*b^3*c^4 + 6*(I*A*B + I*B^2)*a^2*b^2*c^3*d + 6*(I*A*B + I*B^2)*a^3
*b*c^2*d^2 + (I*A*B + I*B^2)*a^4*c*d^3)*g^3*n)*x^3 + 70*(3*(-I*B^2*a^2*b^2*c^4 + I*B^2*a^4*c^2*d^2)*g^3*n^2 +
4*(3*(I*A*B + I*B^2)*a^2*b^2*c^4 + 8*(I*A*B + I*B^2)*a^3*b*c^3*d + 3*(I*A*B + I*B^2)*a^4*c^2*d^2)*g^3*n)*x^2 +
140*((-I*B^2*a^3*b*c^4 + I*B^2*a^4*c^3*d)*g^3*n^2 + 4*((I*A*B + I*B^2)*a^3*b*c^4 + (I*A*B + I*B^2)*a^4*c^3*d)
*g^3*n)*x)*log((b*x + a)/(d*x + c)))/(b*d*x^2 + a*c + (b*c + a*d)*x), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*g*x+a*g)^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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \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 \end {gather*}

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)

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