3.2.69 \(\int (a g+b g x)^2 (c i+d i x)^2 (A+B \log (e (\frac {a+b x}{c+d x})^n))^2 \, dx\) [169]

Optimal. Leaf size=819 \[ -\frac {B^2 (b c-a d)^4 g^2 i^2 n^2 x}{10 b^2 d^2}-\frac {B^2 (b c-a d)^3 g^2 i^2 n^2 (c+d x)^2}{20 b d^3}+\frac {B^2 (b c-a d)^2 g^2 i^2 n^2 (c+d x)^3}{30 d^3}-\frac {B (b c-a d)^3 g^2 i^2 n (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{30 b^3 d}-\frac {B (b c-a d)^2 g^2 i^2 n (a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{15 b^3}-\frac {B (b c-a d)^3 g^2 i^2 n (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{5 b d^3}+\frac {4 B (b c-a d)^2 g^2 i^2 n (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{15 d^3}-\frac {b B (b c-a d) g^2 i^2 n (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{10 d^3}+\frac {(b c-a d)^2 g^2 i^2 (a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{30 b^3}+\frac {(b c-a d) g^2 i^2 (a+b x)^3 (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{10 b^2}+\frac {g^2 i^2 (a+b x)^3 (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{5 b}+\frac {B (b c-a d)^4 g^2 i^2 n (a+b x) \left (2 A+B n+2 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{30 b^3 d^2}+\frac {B (b c-a d)^5 g^2 i^2 n \left (2 A+3 B n+2 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log \left (\frac {b c-a d}{b (c+d x)}\right )}{30 b^3 d^3}+\frac {B^2 (b c-a d)^5 g^2 i^2 n^2 \log \left (\frac {a+b x}{c+d x}\right )}{30 b^3 d^3}+\frac {B^2 (b c-a d)^5 g^2 i^2 n^2 \log (c+d x)}{10 b^3 d^3}+\frac {B^2 (b c-a d)^5 g^2 i^2 n^2 \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right )}{15 b^3 d^3} \]

[Out]

-1/10*B^2*(-a*d+b*c)^4*g^2*i^2*n^2*x/b^2/d^2-1/20*B^2*(-a*d+b*c)^3*g^2*i^2*n^2*(d*x+c)^2/b/d^3+1/30*B^2*(-a*d+
b*c)^2*g^2*i^2*n^2*(d*x+c)^3/d^3-1/30*B*(-a*d+b*c)^3*g^2*i^2*n*(b*x+a)^2*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/b^3/d
-1/15*B*(-a*d+b*c)^2*g^2*i^2*n*(b*x+a)^3*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/b^3-1/5*B*(-a*d+b*c)^3*g^2*i^2*n*(d*x
+c)^2*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/b/d^3+4/15*B*(-a*d+b*c)^2*g^2*i^2*n*(d*x+c)^3*(A+B*ln(e*((b*x+a)/(d*x+c)
)^n))/d^3-1/10*b*B*(-a*d+b*c)*g^2*i^2*n*(d*x+c)^4*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/d^3+1/30*(-a*d+b*c)^2*g^2*i^
2*(b*x+a)^3*(A+B*ln(e*((b*x+a)/(d*x+c))^n))^2/b^3+1/10*(-a*d+b*c)*g^2*i^2*(b*x+a)^3*(d*x+c)*(A+B*ln(e*((b*x+a)
/(d*x+c))^n))^2/b^2+1/5*g^2*i^2*(b*x+a)^3*(d*x+c)^2*(A+B*ln(e*((b*x+a)/(d*x+c))^n))^2/b+1/30*B*(-a*d+b*c)^4*g^
2*i^2*n*(b*x+a)*(2*A+B*n+2*B*ln(e*((b*x+a)/(d*x+c))^n))/b^3/d^2+1/30*B*(-a*d+b*c)^5*g^2*i^2*n*(2*A+3*B*n+2*B*l
n(e*((b*x+a)/(d*x+c))^n))*ln((-a*d+b*c)/b/(d*x+c))/b^3/d^3+1/30*B^2*(-a*d+b*c)^5*g^2*i^2*n^2*ln((b*x+a)/(d*x+c
))/b^3/d^3+1/10*B^2*(-a*d+b*c)^5*g^2*i^2*n^2*ln(d*x+c)/b^3/d^3+1/15*B^2*(-a*d+b*c)^5*g^2*i^2*n^2*polylog(2,d*(
b*x+a)/b/(d*x+c))/b^3/d^3

________________________________________________________________________________________

Rubi [A]
time = 0.67, antiderivative size = 819, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 11, integrand size = 45, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.244, Rules used = {2561, 2383, 2381, 2384, 2354, 2438, 2373, 45, 2382, 12, 907} \begin {gather*} \frac {B g^2 i^2 n \left (2 A+3 B n+2 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log \left (\frac {b c-a d}{b (c+d x)}\right ) (b c-a d)^5}{30 b^3 d^3}+\frac {B^2 g^2 i^2 n^2 \log \left (\frac {a+b x}{c+d x}\right ) (b c-a d)^5}{30 b^3 d^3}+\frac {B^2 g^2 i^2 n^2 \log (c+d x) (b c-a d)^5}{10 b^3 d^3}+\frac {B^2 g^2 i^2 n^2 \text {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right ) (b c-a d)^5}{15 b^3 d^3}-\frac {B^2 g^2 i^2 n^2 x (b c-a d)^4}{10 b^2 d^2}+\frac {B g^2 i^2 n (a+b x) \left (2 A+B n+2 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) (b c-a d)^4}{30 b^3 d^2}-\frac {B^2 g^2 i^2 n^2 (c+d x)^2 (b c-a d)^3}{20 b d^3}-\frac {B g^2 i^2 n (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) (b c-a d)^3}{30 b^3 d}-\frac {B g^2 i^2 n (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) (b c-a d)^3}{5 b d^3}+\frac {B^2 g^2 i^2 n^2 (c+d x)^3 (b c-a d)^2}{30 d^3}+\frac {g^2 i^2 (a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 (b c-a d)^2}{30 b^3}-\frac {B g^2 i^2 n (a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) (b c-a d)^2}{15 b^3}+\frac {4 B g^2 i^2 n (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) (b c-a d)^2}{15 d^3}+\frac {g^2 i^2 (a+b x)^3 (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 (b c-a d)}{10 b^2}-\frac {b B g^2 i^2 n (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) (b c-a d)}{10 d^3}+\frac {g^2 i^2 (a+b x)^3 (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{5 b} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/10*(B^2*(b*c - a*d)^4*g^2*i^2*n^2*x)/(b^2*d^2) - (B^2*(b*c - a*d)^3*g^2*i^2*n^2*(c + d*x)^2)/(20*b*d^3) + (
B^2*(b*c - a*d)^2*g^2*i^2*n^2*(c + d*x)^3)/(30*d^3) - (B*(b*c - a*d)^3*g^2*i^2*n*(a + b*x)^2*(A + B*Log[e*((a
+ b*x)/(c + d*x))^n]))/(30*b^3*d) - (B*(b*c - a*d)^2*g^2*i^2*n*(a + b*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^
n]))/(15*b^3) - (B*(b*c - a*d)^3*g^2*i^2*n*(c + d*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(5*b*d^3) + (4*
B*(b*c - a*d)^2*g^2*i^2*n*(c + d*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(15*d^3) - (b*B*(b*c - a*d)*g^2*
i^2*n*(c + d*x)^4*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(10*d^3) + ((b*c - a*d)^2*g^2*i^2*(a + b*x)^3*(A + B
*Log[e*((a + b*x)/(c + d*x))^n])^2)/(30*b^3) + ((b*c - a*d)*g^2*i^2*(a + b*x)^3*(c + d*x)*(A + B*Log[e*((a + b
*x)/(c + d*x))^n])^2)/(10*b^2) + (g^2*i^2*(a + b*x)^3*(c + d*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(5
*b) + (B*(b*c - a*d)^4*g^2*i^2*n*(a + b*x)*(2*A + B*n + 2*B*Log[e*((a + b*x)/(c + d*x))^n]))/(30*b^3*d^2) + (B
*(b*c - a*d)^5*g^2*i^2*n*(2*A + 3*B*n + 2*B*Log[e*((a + b*x)/(c + d*x))^n])*Log[(b*c - a*d)/(b*(c + d*x))])/(3
0*b^3*d^3) + (B^2*(b*c - a*d)^5*g^2*i^2*n^2*Log[(a + b*x)/(c + d*x)])/(30*b^3*d^3) + (B^2*(b*c - a*d)^5*g^2*i^
2*n^2*Log[c + d*x])/(10*b^3*d^3) + (B^2*(b*c - a*d)^5*g^2*i^2*n^2*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(15
*b^3*d^3)

Rule 12

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

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 907

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :
> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] &
& NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[p] && ((EqQ[p, 1] && I
ntegersQ[m, n]) || (ILtQ[m, 0] && ILtQ[n, 0]))

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

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Mathematica [A]
time = 0.66, size = 1254, normalized size = 1.53 \begin {gather*} \frac {g^2 i^2 \left (20 d^3 (b c-a d)^2 (a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2+30 d^4 (b c-a d) (a+b x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2+12 d^5 (a+b x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2+20 B (b c-a d)^3 n \left (2 A b d (b c-a d) x+2 B d (b c-a d) (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-d^2 (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )-2 B (b c-a d)^2 n \log (c+d x)-2 (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)+B (b c-a d) n (b d x+(-b c+a d) \log (c+d x))+B (b c-a d)^2 n \left (\left (2 \log \left (\frac {d (a+b x)}{-b c+a d}\right )-\log (c+d x)\right ) \log (c+d x)+2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )\right )\right )-10 B (b c-a d)^2 n \left (6 A b d (b c-a d)^2 x+6 B d (b c-a d)^2 (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+3 d^2 (-b c+a d) (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )+2 d^3 (a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )-6 B (b c-a d)^3 n \log (c+d x)-6 (b c-a d)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)+B (b c-a d) n \left (2 b d (b c-a d) x-d^2 (a+b x)^2-2 (b c-a d)^2 \log (c+d x)\right )+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 \left (\left (2 \log \left (\frac {d (a+b x)}{-b c+a d}\right )-\log (c+d x)\right ) \log (c+d x)+2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )\right )\right )+B (b c-a d) n \left (24 A b d (b c-a d)^3 x+24 B d (b c-a d)^3 (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-12 d^2 (b c-a d)^2 (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )+8 d^3 (b c-a d) (a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )-6 d^4 (a+b x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )-24 B (b c-a d)^4 n \log (c+d x)-24 (b c-a d)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)+4 B (b c-a d)^2 n \left (2 b d (b c-a d) x-d^2 (a+b x)^2-2 (b c-a d)^2 \log (c+d x)\right )+B (b c-a d) n \left (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)\right )+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 \left (\left (2 \log \left (\frac {d (a+b x)}{-b c+a d}\right )-\log (c+d x)\right ) \log (c+d x)+2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )\right )\right )\right )}{60 b^3 d^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]
time = 0.26, size = 0, normalized size = 0.00 \[\int \left (b g x +a g \right )^{2} \left (d i x +c i \right )^{2} \left (A +B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )\right )^{2}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 3499 vs. \(2 (748) = 1496\).
time = 0.81, size = 3499, normalized size = 4.27 \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)^2*(d*i*x+c*i)^2*(A+B*log(e*((b*x+a)/(d*x+c))^n))^2,x, algorithm="maxima")

[Out]

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

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

[Out]

-1/30*(6*B^2*b^2*d^2*g^2*n^2*x^5 + 30*B^2*a^2*c^2*g^2*n^2*x + 15*(B^2*b^2*c*d + B^2*a*b*d^2)*g^2*n^2*x^4 + 10*
(B^2*b^2*c^2 + 4*B^2*a*b*c*d + B^2*a^2*d^2)*g^2*n^2*x^3 + 30*(B^2*a*b*c^2 + B^2*a^2*c*d)*g^2*n^2*x^2)*log((b*x
+ a)/(d*x + c))^2 + integral(-1/15*(15*(A^2 + 2*A*B + B^2)*b^3*d^3*g^2*x^6 + 15*(A^2 + 2*A*B + B^2)*a^3*c^3*g
^2 + 45*((A^2 + 2*A*B + B^2)*b^3*c*d^2 + (A^2 + 2*A*B + B^2)*a*b^2*d^3)*g^2*x^5 + 45*((A^2 + 2*A*B + B^2)*b^3*
c^2*d + 3*(A^2 + 2*A*B + B^2)*a*b^2*c*d^2 + (A^2 + 2*A*B + B^2)*a^2*b*d^3)*g^2*x^4 + 15*((A^2 + 2*A*B + B^2)*b
^3*c^3 + 9*(A^2 + 2*A*B + B^2)*a*b^2*c^2*d + 9*(A^2 + 2*A*B + B^2)*a^2*b*c*d^2 + (A^2 + 2*A*B + B^2)*a^3*d^3)*
g^2*x^3 + 45*((A^2 + 2*A*B + B^2)*a*b^2*c^3 + 3*(A^2 + 2*A*B + B^2)*a^2*b*c^2*d + (A^2 + 2*A*B + B^2)*a^3*c*d^
2)*g^2*x^2 + 45*((A^2 + 2*A*B + B^2)*a^2*b*c^3 + (A^2 + 2*A*B + B^2)*a^3*c^2*d)*g^2*x + (30*(A*B + B^2)*b^3*d^
3*g^2*n*x^6 + 30*(A*B + B^2)*a^3*c^3*g^2*n - 6*((B^2*b^3*c*d^2 - B^2*a*b^2*d^3)*g^2*n^2 - 15*((A*B + B^2)*b^3*
c*d^2 + (A*B + B^2)*a*b^2*d^3)*g^2*n)*x^5 - 15*((B^2*b^3*c^2*d - B^2*a^2*b*d^3)*g^2*n^2 - 6*((A*B + B^2)*b^3*c
^2*d + 3*(A*B + B^2)*a*b^2*c*d^2 + (A*B + B^2)*a^2*b*d^3)*g^2*n)*x^4 - 10*((B^2*b^3*c^3 + 3*B^2*a*b^2*c^2*d -
3*B^2*a^2*b*c*d^2 - B^2*a^3*d^3)*g^2*n^2 - 3*((A*B + B^2)*b^3*c^3 + 9*(A*B + B^2)*a*b^2*c^2*d + 9*(A*B + B^2)*
a^2*b*c*d^2 + (A*B + B^2)*a^3*d^3)*g^2*n)*x^3 - 30*((B^2*a*b^2*c^3 - B^2*a^3*c*d^2)*g^2*n^2 - 3*((A*B + B^2)*a
*b^2*c^3 + 3*(A*B + B^2)*a^2*b*c^2*d + (A*B + B^2)*a^3*c*d^2)*g^2*n)*x^2 - 30*((B^2*a^2*b*c^3 - B^2*a^3*c^2*d)
*g^2*n^2 - 3*((A*B + B^2)*a^2*b*c^3 + (A*B + B^2)*a^3*c^2*d)*g^2*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)**2*(d*i*x+c*i)**2*(A+B*ln(e*((b*x+a)/(d*x+c))**n))**2,x)

[Out]

Timed out

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Giac [F(-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)^2*(d*i*x+c*i)^2*(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 )}^2\,{\left (c\,i+d\,i\,x\right )}^2\,{\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)^2*(c*i + d*i*x)^2*(A + B*log(e*((a + b*x)/(c + d*x))^n))^2,x)

[Out]

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

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