3.2.68 \(\int (a g+b g x)^3 (c i+d i x)^2 (A+B \log (e (\frac {a+b x}{c+d x})^n))^2 \, dx\) [168]

Optimal. Leaf size=766 \[ \frac {3 B^2 (b c-a d)^5 g^3 i^2 n^2 x}{20 b^2 d^3}+\frac {B^2 (b c-a d)^2 g^3 i^2 n^2 (a+b x)^4}{60 b^3}-\frac {3 B^2 (b c-a d)^4 g^3 i^2 n^2 (c+d x)^2}{40 b d^4}+\frac {B^2 (b c-a d)^3 g^3 i^2 n^2 (c+d x)^3}{60 d^4}-\frac {B (b c-a d)^3 g^3 i^2 n (a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{90 b^3 d}-\frac {B (b c-a d)^2 g^3 i^2 n (a+b x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{20 b^3}-\frac {B (b c-a d) g^3 i^2 n (a+b x)^4 (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{15 b^2}+\frac {(b c-a d)^2 g^3 i^2 (a+b x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{60 b^3}+\frac {(b c-a d) g^3 i^2 (a+b x)^4 (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{15 b^2}+\frac {g^3 i^2 (a+b x)^4 (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{6 b}+\frac {B (b c-a d)^4 g^3 i^2 n (a+b x)^2 \left (3 A+B n+3 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{180 b^3 d^2}-\frac {B (b c-a d)^5 g^3 i^2 n (a+b x) \left (6 A+5 B n+6 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{180 b^3 d^3}-\frac {B (b c-a d)^6 g^3 i^2 n \left (6 A+11 B n+6 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log \left (\frac {b c-a d}{b (c+d x)}\right )}{180 b^3 d^4}-\frac {B^2 (b c-a d)^6 g^3 i^2 n^2 \log (c+d x)}{20 b^3 d^4}-\frac {B^2 (b c-a d)^6 g^3 i^2 n^2 \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right )}{30 b^3 d^4} \]

[Out]

3/20*B^2*(-a*d+b*c)^5*g^3*i^2*n^2*x/b^2/d^3+1/60*B^2*(-a*d+b*c)^2*g^3*i^2*n^2*(b*x+a)^4/b^3-3/40*B^2*(-a*d+b*c
)^4*g^3*i^2*n^2*(d*x+c)^2/b/d^4+1/60*B^2*(-a*d+b*c)^3*g^3*i^2*n^2*(d*x+c)^3/d^4-1/90*B*(-a*d+b*c)^3*g^3*i^2*n*
(b*x+a)^3*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/b^3/d-1/20*B*(-a*d+b*c)^2*g^3*i^2*n*(b*x+a)^4*(A+B*ln(e*((b*x+a)/(d*
x+c))^n))/b^3-1/15*B*(-a*d+b*c)*g^3*i^2*n*(b*x+a)^4*(d*x+c)*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/b^2+1/60*(-a*d+b*c
)^2*g^3*i^2*(b*x+a)^4*(A+B*ln(e*((b*x+a)/(d*x+c))^n))^2/b^3+1/15*(-a*d+b*c)*g^3*i^2*(b*x+a)^4*(d*x+c)*(A+B*ln(
e*((b*x+a)/(d*x+c))^n))^2/b^2+1/6*g^3*i^2*(b*x+a)^4*(d*x+c)^2*(A+B*ln(e*((b*x+a)/(d*x+c))^n))^2/b+1/180*B*(-a*
d+b*c)^4*g^3*i^2*n*(b*x+a)^2*(3*A+B*n+3*B*ln(e*((b*x+a)/(d*x+c))^n))/b^3/d^2-1/180*B*(-a*d+b*c)^5*g^3*i^2*n*(b
*x+a)*(6*A+5*B*n+6*B*ln(e*((b*x+a)/(d*x+c))^n))/b^3/d^3-1/180*B*(-a*d+b*c)^6*g^3*i^2*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^3/d^4-1/20*B^2*(-a*d+b*c)^6*g^3*i^2*n^2*ln(d*x+c)/b^3/d^4-1/30*
B^2*(-a*d+b*c)^6*g^3*i^2*n^2*polylog(2,d*(b*x+a)/b/(d*x+c))/b^3/d^4

________________________________________________________________________________________

Rubi [A]
time = 0.71, antiderivative size = 766, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 13, integrand size = 45, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.289, Rules used = {2561, 2383, 2381, 2384, 2354, 2438, 2373, 45, 47, 37, 2382, 12, 79} \begin {gather*} -\frac {B^2 g^3 i^2 n^2 (b c-a d)^6 \text {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{30 b^3 d^4}-\frac {B g^3 i^2 n (b c-a d)^6 \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (6 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+6 A+11 B n\right )}{180 b^3 d^4}-\frac {B g^3 i^2 n (a+b x) (b c-a d)^5 \left (6 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+6 A+5 B n\right )}{180 b^3 d^3}+\frac {B g^3 i^2 n (a+b x)^2 (b c-a d)^4 \left (3 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+3 A+B n\right )}{180 b^3 d^2}-\frac {B g^3 i^2 n (a+b x)^3 (b c-a d)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{90 b^3 d}+\frac {g^3 i^2 (a+b x)^4 (b c-a d)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{60 b^3}-\frac {B g^3 i^2 n (a+b x)^4 (b c-a d)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{20 b^3}+\frac {g^3 i^2 (a+b x)^4 (c+d x) (b c-a d) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{15 b^2}-\frac {B g^3 i^2 n (a+b x)^4 (c+d x) (b c-a d) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{15 b^2}+\frac {g^3 i^2 (a+b x)^4 (c+d x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{6 b}-\frac {B^2 g^3 i^2 n^2 (b c-a d)^6 \log (c+d x)}{20 b^3 d^4}+\frac {B^2 g^3 i^2 n^2 (a+b x)^4 (b c-a d)^2}{60 b^3}+\frac {3 B^2 g^3 i^2 n^2 x (b c-a d)^5}{20 b^2 d^3}-\frac {3 B^2 g^3 i^2 n^2 (c+d x)^2 (b c-a d)^4}{40 b d^4}+\frac {B^2 g^3 i^2 n^2 (c+d x)^3 (b c-a d)^3}{60 d^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(1634\) vs. \(2(766)=1532\).
time = 0.96, size = 1634, normalized size = 2.13 \begin {gather*} \frac {g^3 i^2 \left (15 (b c-a d)^2 (a+b x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2+24 d (b c-a d) (a+b x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2+10 d^2 (a+b x)^6 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2-\frac {5 B (b c-a d)^3 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 )}{d^4}+\frac {2 B (b c-a d)^2 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 )}{d^4}-\frac {B (b c-a d) n \left (120 A b d (b c-a d)^4 x+120 B d (b c-a d)^4 (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+60 d^2 (-b c+a d)^3 (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )+40 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 )+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 )+24 d^5 (a+b x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )-120 B (b c-a d)^5 n \log (c+d x)-120 (b c-a d)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)+20 B (b c-a d)^3 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 )+5 B (b c-a d)^2 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 )+2 B (b c-a d) n \left (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)\right )+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 \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 )}{6 d^4}\right )}{60 b^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(g^3*i^2*(15*(b*c - a*d)^2*(a + b*x)^4*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2 + 24*d*(b*c - a*d)*(a + b*x)^5
*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2 + 10*d^2*(a + b*x)^6*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2 - (5*B
*(b*c - a*d)^3*n*(6*A*b*d*(b*c - a*d)^2*x + 6*B*d*(b*c - a*d)^2*(a + b*x)*Log[e*((a + b*x)/(c + d*x))^n] + 3*d
^2*(-(b*c) + a*d)*(a + b*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) + 2*d^3*(a + b*x)^3*(A + B*Log[e*((a + b*
x)/(c + d*x))^n]) - 6*B*(b*c - a*d)^3*n*Log[c + d*x] - 6*(b*c - a*d)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*
Log[c + d*x] + B*(b*c - a*d)*n*(2*b*d*(b*c - a*d)*x - d^2*(a + b*x)^2 - 2*(b*c - a*d)^2*Log[c + d*x]) + 3*B*(b
*c - a*d)^2*n*(b*d*x + (-(b*c) + a*d)*Log[c + d*x]) + 3*B*(b*c - a*d)^3*n*((2*Log[(d*(a + b*x))/(-(b*c) + a*d)
] - Log[c + d*x])*Log[c + d*x] + 2*PolyLog[2, (b*(c + d*x))/(b*c - a*d)])))/d^4 + (2*B*(b*c - a*d)^2*n*(24*A*b
*d*(b*c - a*d)^3*x + 24*B*d*(b*c - a*d)^3*(a + b*x)*Log[e*((a + b*x)/(c + d*x))^n] - 12*d^2*(b*c - a*d)^2*(a +
b*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) + 8*d^3*(b*c - a*d)*(a + b*x)^3*(A + B*Log[e*((a + b*x)/(c + d*
x))^n]) - 6*d^4*(a + b*x)^4*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) - 24*B*(b*c - a*d)^4*n*Log[c + d*x] - 24*(b
*c - a*d)^4*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*Log[c + d*x] + 4*B*(b*c - a*d)^2*n*(2*b*d*(b*c - a*d)*x - d
^2*(a + b*x)^2 - 2*(b*c - a*d)^2*Log[c + d*x]) + B*(b*c - a*d)*n*(6*b*d*(b*c - a*d)^2*x + 3*d^2*(-(b*c) + a*d)
*(a + b*x)^2 + 2*d^3*(a + b*x)^3 - 6*(b*c - a*d)^3*Log[c + d*x]) + 12*B*(b*c - a*d)^3*n*(b*d*x + (-(b*c) + a*d
)*Log[c + d*x]) + 12*B*(b*c - a*d)^4*n*((2*Log[(d*(a + b*x))/(-(b*c) + a*d)] - Log[c + d*x])*Log[c + d*x] + 2*
PolyLog[2, (b*(c + d*x))/(b*c - a*d)])))/d^4 - (B*(b*c - a*d)*n*(120*A*b*d*(b*c - a*d)^4*x + 120*B*d*(b*c - a*
d)^4*(a + b*x)*Log[e*((a + b*x)/(c + d*x))^n] + 60*d^2*(-(b*c) + a*d)^3*(a + b*x)^2*(A + B*Log[e*((a + b*x)/(c
+ d*x))^n]) + 40*d^3*(b*c - a*d)^2*(a + b*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) + 30*d^4*(-(b*c) + a*d)
*(a + b*x)^4*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) + 24*d^5*(a + b*x)^5*(A + B*Log[e*((a + b*x)/(c + d*x))^n]
) - 120*B*(b*c - a*d)^5*n*Log[c + d*x] - 120*(b*c - a*d)^5*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*Log[c + d*x]
+ 20*B*(b*c - a*d)^3*n*(2*b*d*(b*c - a*d)*x - d^2*(a + b*x)^2 - 2*(b*c - a*d)^2*Log[c + d*x]) + 5*B*(b*c - a*
d)^2*n*(6*b*d*(b*c - a*d)^2*x + 3*d^2*(-(b*c) + a*d)*(a + b*x)^2 + 2*d^3*(a + b*x)^3 - 6*(b*c - a*d)^3*Log[c +
d*x]) + 2*B*(b*c - a*d)*n*(12*b*d*(b*c - a*d)^3*x - 6*d^2*(b*c - a*d)^2*(a + b*x)^2 + 4*d^3*(b*c - a*d)*(a +
b*x)^3 - 3*d^4*(a + b*x)^4 - 12*(b*c - a*d)^4*Log[c + d*x]) + 60*B*(b*c - a*d)^4*n*(b*d*x + (-(b*c) + a*d)*Log
[c + d*x]) + 60*B*(b*c - a*d)^5*n*((2*Log[(d*(a + b*x))/(-(b*c) + a*d)] - Log[c + d*x])*Log[c + d*x] + 2*PolyL
og[2, (b*(c + d*x))/(b*c - a*d)])))/(6*d^4)))/(60*b^3)

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Maple [F]
time = 0.32, size = 0, normalized size = 0.00 \[\int \left (b g x +a g \right )^{3} \left (d i x +c i \right )^{2} \left (A +B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )\right )^{2}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 4882 vs. \(2 (699) = 1398\).
time = 0.85, size = 4882, normalized size = 6.37 \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)^2*(A+B*log(e*((b*x+a)/(d*x+c))^n))^2,x, algorithm="maxima")

[Out]

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

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

[Out]

-1/60*(10*B^2*b^3*d^2*g^3*n^2*x^6 + 60*B^2*a^3*c^2*g^3*n^2*x + 12*(2*B^2*b^3*c*d + 3*B^2*a*b^2*d^2)*g^3*n^2*x^
5 + 15*(B^2*b^3*c^2 + 6*B^2*a*b^2*c*d + 3*B^2*a^2*b*d^2)*g^3*n^2*x^4 + 20*(3*B^2*a*b^2*c^2 + 6*B^2*a^2*b*c*d +
B^2*a^3*d^2)*g^3*n^2*x^3 + 30*(3*B^2*a^2*b*c^2 + 2*B^2*a^3*c*d)*g^3*n^2*x^2)*log((b*x + a)/(d*x + c))^2 + int
egral(-1/30*(30*(A^2 + 2*A*B + B^2)*b^4*d^3*g^3*x^7 + 30*(A^2 + 2*A*B + B^2)*a^4*c^3*g^3 + 30*(3*(A^2 + 2*A*B
+ B^2)*b^4*c*d^2 + 4*(A^2 + 2*A*B + B^2)*a*b^3*d^3)*g^3*x^6 + 90*((A^2 + 2*A*B + B^2)*b^4*c^2*d + 4*(A^2 + 2*A
*B + B^2)*a*b^3*c*d^2 + 2*(A^2 + 2*A*B + B^2)*a^2*b^2*d^3)*g^3*x^5 + 30*((A^2 + 2*A*B + B^2)*b^4*c^3 + 12*(A^2
+ 2*A*B + B^2)*a*b^3*c^2*d + 18*(A^2 + 2*A*B + B^2)*a^2*b^2*c*d^2 + 4*(A^2 + 2*A*B + B^2)*a^3*b*d^3)*g^3*x^4
+ 30*(4*(A^2 + 2*A*B + B^2)*a*b^3*c^3 + 18*(A^2 + 2*A*B + B^2)*a^2*b^2*c^2*d + 12*(A^2 + 2*A*B + B^2)*a^3*b*c*
d^2 + (A^2 + 2*A*B + B^2)*a^4*d^3)*g^3*x^3 + 90*(2*(A^2 + 2*A*B + B^2)*a^2*b^2*c^3 + 4*(A^2 + 2*A*B + B^2)*a^3
*b*c^2*d + (A^2 + 2*A*B + B^2)*a^4*c*d^2)*g^3*x^2 + 30*(4*(A^2 + 2*A*B + B^2)*a^3*b*c^3 + 3*(A^2 + 2*A*B + B^2
)*a^4*c^2*d)*g^3*x + (60*(A*B + B^2)*b^4*d^3*g^3*n*x^7 + 60*(A*B + B^2)*a^4*c^3*g^3*n - 10*((B^2*b^4*c*d^2 - B
^2*a*b^3*d^3)*g^3*n^2 - 6*(3*(A*B + B^2)*b^4*c*d^2 + 4*(A*B + B^2)*a*b^3*d^3)*g^3*n)*x^6 - 12*((2*B^2*b^4*c^2*
d + B^2*a*b^3*c*d^2 - 3*B^2*a^2*b^2*d^3)*g^3*n^2 - 15*((A*B + B^2)*b^4*c^2*d + 4*(A*B + B^2)*a*b^3*c*d^2 + 2*(
A*B + B^2)*a^2*b^2*d^3)*g^3*n)*x^5 - 15*((B^2*b^4*c^3 + 5*B^2*a*b^3*c^2*d - 3*B^2*a^2*b^2*c*d^2 - 3*B^2*a^3*b*
d^3)*g^3*n^2 - 4*((A*B + B^2)*b^4*c^3 + 12*(A*B + B^2)*a*b^3*c^2*d + 18*(A*B + B^2)*a^2*b^2*c*d^2 + 4*(A*B + B
^2)*a^3*b*d^3)*g^3*n)*x^4 - 20*((3*B^2*a*b^3*c^3 + 3*B^2*a^2*b^2*c^2*d - 5*B^2*a^3*b*c*d^2 - B^2*a^4*d^3)*g^3*
n^2 - 3*(4*(A*B + B^2)*a*b^3*c^3 + 18*(A*B + B^2)*a^2*b^2*c^2*d + 12*(A*B + B^2)*a^3*b*c*d^2 + (A*B + B^2)*a^4
*d^3)*g^3*n)*x^3 - 30*((3*B^2*a^2*b^2*c^3 - B^2*a^3*b*c^2*d - 2*B^2*a^4*c*d^2)*g^3*n^2 - 6*(2*(A*B + B^2)*a^2*
b^2*c^3 + 4*(A*B + B^2)*a^3*b*c^2*d + (A*B + B^2)*a^4*c*d^2)*g^3*n)*x^2 - 60*((B^2*a^3*b*c^3 - B^2*a^4*c^2*d)*
g^3*n^2 - (4*(A*B + B^2)*a^3*b*c^3 + 3*(A*B + B^2)*a^4*c^2*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)**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)^3*(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 )}^3\,{\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)^3*(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)^3*(c*i + d*i*x)^2*(A + B*log(e*((a + b*x)/(c + d*x))^n))^2, x)

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