Integrand size = 43, antiderivative size = 307 \[ \int \frac {(c i+d i x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^4} \, dx=\frac {B^2 d i n^2 (c+d x)^2}{4 (b c-a d)^2 g^4 (a+b x)^2}-\frac {2 b B^2 i n^2 (c+d x)^3}{27 (b c-a d)^2 g^4 (a+b x)^3}+\frac {B d i n (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 (b c-a d)^2 g^4 (a+b x)^2}-\frac {2 b B i n (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{9 (b c-a d)^2 g^4 (a+b x)^3}+\frac {d i (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2 (b c-a d)^2 g^4 (a+b x)^2}-\frac {b i (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{3 (b c-a d)^2 g^4 (a+b x)^3} \]
1/4*B^2*d*i*n^2*(d*x+c)^2/(-a*d+b*c)^2/g^4/(b*x+a)^2-2/27*b*B^2*i*n^2*(d*x +c)^3/(-a*d+b*c)^2/g^4/(b*x+a)^3+1/2*B*d*i*n*(d*x+c)^2*(A+B*ln(e*((b*x+a)/ (d*x+c))^n))/(-a*d+b*c)^2/g^4/(b*x+a)^2-2/9*b*B*i*n*(d*x+c)^3*(A+B*ln(e*(( b*x+a)/(d*x+c))^n))/(-a*d+b*c)^2/g^4/(b*x+a)^3+1/2*d*i*(d*x+c)^2*(A+B*ln(e *((b*x+a)/(d*x+c))^n))^2/(-a*d+b*c)^2/g^4/(b*x+a)^2-1/3*b*i*(d*x+c)^3*(A+B *ln(e*((b*x+a)/(d*x+c))^n))^2/(-a*d+b*c)^2/g^4/(b*x+a)^3
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 0.63 (sec) , antiderivative size = 1082, normalized size of antiderivative = 3.52 \[ \int \frac {(c i+d i x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^4} \, dx=-\frac {i \left (36 (b c-a d)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2+54 d (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 )^2+2 B n \left (12 A (b c-a d)^3+4 B (b c-a d)^3 n-18 A d (b c-a d)^2 (a+b x)-15 B d (b c-a d)^2 n (a+b x)+36 A d^2 (b c-a d) (a+b x)^2+66 B d^2 (b c-a d) n (a+b x)^2+36 A d^3 (a+b x)^3 \log (a+b x)+66 B d^3 n (a+b x)^3 \log (a+b x)-18 B d^3 n (a+b x)^3 \log ^2(a+b x)+12 B (b c-a d)^3 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-18 B d (b c-a d)^2 (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+36 B d^2 (b c-a d) (a+b x)^2 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+36 B d^3 (a+b x)^3 \log (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-36 A d^3 (a+b x)^3 \log (c+d x)-66 B d^3 n (a+b x)^3 \log (c+d x)+36 B d^3 n (a+b x)^3 \log \left (\frac {d (a+b x)}{-b c+a d}\right ) \log (c+d x)-36 B d^3 (a+b x)^3 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log (c+d x)-18 B d^3 n (a+b x)^3 \log ^2(c+d x)+36 B d^3 n (a+b x)^3 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )+36 B d^3 n (a+b x)^3 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{-b c+a d}\right )+36 B d^3 n (a+b x)^3 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )\right )+27 B d n (a+b x) \left (2 (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )+4 d (-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 )-4 d^2 (a+b x)^2 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )+4 d^2 (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)-4 B d n (a+b x) (b c-a d+d (a+b x) \log (a+b x)-d (a+b x) \log (c+d x))+B n \left ((b c-a d)^2+2 d (-b c+a d) (a+b x)-2 d^2 (a+b x)^2 \log (a+b x)+2 d^2 (a+b x)^2 \log (c+d x)\right )+2 B d^2 n (a+b x)^2 \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{-b c+a d}\right )\right )-2 B d^2 n (a+b x)^2 \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 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )\right )\right )\right )}{108 b^2 (b c-a d)^2 g^4 (a+b x)^3} \]
-1/108*(i*(36*(b*c - a*d)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2 + 54* d*(b*c - a*d)^2*(a + b*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2 + 2*B*n *(12*A*(b*c - a*d)^3 + 4*B*(b*c - a*d)^3*n - 18*A*d*(b*c - a*d)^2*(a + b*x ) - 15*B*d*(b*c - a*d)^2*n*(a + b*x) + 36*A*d^2*(b*c - a*d)*(a + b*x)^2 + 66*B*d^2*(b*c - a*d)*n*(a + b*x)^2 + 36*A*d^3*(a + b*x)^3*Log[a + b*x] + 6 6*B*d^3*n*(a + b*x)^3*Log[a + b*x] - 18*B*d^3*n*(a + b*x)^3*Log[a + b*x]^2 + 12*B*(b*c - a*d)^3*Log[e*((a + b*x)/(c + d*x))^n] - 18*B*d*(b*c - a*d)^ 2*(a + b*x)*Log[e*((a + b*x)/(c + d*x))^n] + 36*B*d^2*(b*c - a*d)*(a + b*x )^2*Log[e*((a + b*x)/(c + d*x))^n] + 36*B*d^3*(a + b*x)^3*Log[a + b*x]*Log [e*((a + b*x)/(c + d*x))^n] - 36*A*d^3*(a + b*x)^3*Log[c + d*x] - 66*B*d^3 *n*(a + b*x)^3*Log[c + d*x] + 36*B*d^3*n*(a + b*x)^3*Log[(d*(a + b*x))/(-( b*c) + a*d)]*Log[c + d*x] - 36*B*d^3*(a + b*x)^3*Log[e*((a + b*x)/(c + d*x ))^n]*Log[c + d*x] - 18*B*d^3*n*(a + b*x)^3*Log[c + d*x]^2 + 36*B*d^3*n*(a + b*x)^3*Log[a + b*x]*Log[(b*(c + d*x))/(b*c - a*d)] + 36*B*d^3*n*(a + b* x)^3*PolyLog[2, (d*(a + b*x))/(-(b*c) + a*d)] + 36*B*d^3*n*(a + b*x)^3*Pol yLog[2, (b*(c + d*x))/(b*c - a*d)]) + 27*B*d*n*(a + b*x)*(2*(b*c - a*d)^2* (A + B*Log[e*((a + b*x)/(c + d*x))^n]) + 4*d*(-(b*c) + a*d)*(a + b*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) - 4*d^2*(a + b*x)^2*Log[a + b*x]*(A + B *Log[e*((a + b*x)/(c + d*x))^n]) + 4*d^2*(a + b*x)^2*(A + B*Log[e*((a + b* x)/(c + d*x))^n])*Log[c + d*x] - 4*B*d*n*(a + b*x)*(b*c - a*d + d*(a + ...
Time = 0.45 (sec) , antiderivative size = 238, normalized size of antiderivative = 0.78, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.070, Rules used = {2961, 2795, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {(c i+d i x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{(a g+b g x)^4} \, dx\) |
\(\Big \downarrow \) 2961 |
\(\displaystyle \frac {i \int \frac {(c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right ) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a+b x)^4}d\frac {a+b x}{c+d x}}{g^4 (b c-a d)^2}\) |
\(\Big \downarrow \) 2795 |
\(\displaystyle \frac {i \int \left (\frac {b (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a+b x)^4}-\frac {d (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a+b x)^3}\right )d\frac {a+b x}{c+d x}}{g^4 (b c-a d)^2}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {i \left (-\frac {b (c+d x)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{3 (a+b x)^3}-\frac {2 b B n (c+d x)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{9 (a+b x)^3}+\frac {d (c+d x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{2 (a+b x)^2}+\frac {B d n (c+d x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{2 (a+b x)^2}-\frac {2 b B^2 n^2 (c+d x)^3}{27 (a+b x)^3}+\frac {B^2 d n^2 (c+d x)^2}{4 (a+b x)^2}\right )}{g^4 (b c-a d)^2}\) |
(i*((B^2*d*n^2*(c + d*x)^2)/(4*(a + b*x)^2) - (2*b*B^2*n^2*(c + d*x)^3)/(2 7*(a + b*x)^3) + (B*d*n*(c + d*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) )/(2*(a + b*x)^2) - (2*b*B*n*(c + d*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x) )^n]))/(9*(a + b*x)^3) + (d*(c + d*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x)) ^n])^2)/(2*(a + b*x)^2) - (b*(c + d*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x) )^n])^2)/(3*(a + b*x)^3)))/((b*c - a*d)^2*g^4)
3.2.66.3.1 Defintions of rubi rules used
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*Log[ c*x^n])^p, (f*x)^m*(d + e*x^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b , c, d, e, f, m, n, p, q, r}, x] && IntegerQ[q] && (GtQ[q, 0] || (IGtQ[p, 0 ] && IntegerQ[m] && IntegerQ[r]))
Int[((A_.) + Log[(e_.)*(((a_.) + (b_.)*(x_))/((c_.) + (d_.)*(x_)))^(n_.)]*( B_.))^(p_.)*((f_.) + (g_.)*(x_))^(m_.)*((h_.) + (i_.)*(x_))^(q_.), x_Symbol ] :> Simp[(b*c - a*d)^(m + q + 1)*(g/b)^m*(i/d)^q Subst[Int[x^m*((A + B*L og[e*x^n])^p/(b - d*x)^(m + q + 2)), x], x, (a + b*x)/(c + d*x)], x] /; Fre eQ[{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]
Leaf count of result is larger than twice the leaf count of optimal. \(1098\) vs. \(2(295)=590\).
Time = 10.49 (sec) , antiderivative size = 1099, normalized size of antiderivative = 3.58
-1/108*(-108*A*B*x^2*ln(e*((b*x+a)/(d*x+c))^n)*a*b^5*d^4*i*n-108*B^2*x*ln( e*((b*x+a)/(d*x+c))^n)^2*a*b^5*c*d^3*i*n-108*B^2*x*ln(e*((b*x+a)/(d*x+c))^ n)*a*b^5*c*d^3*i*n^2+108*A*B*x*ln(e*((b*x+a)/(d*x+c))^n)*b^6*c^2*d^2*i*n-1 08*A*B*x*a*b^5*c*d^3*i*n^2-108*A*B*ln(e*((b*x+a)/(d*x+c))^n)*a*b^5*c^2*d^2 *i*n-216*A*B*x*ln(e*((b*x+a)/(d*x+c))^n)*a*b^5*c*d^3*i*n-54*B^2*x*a*b^5*c* d^3*i*n^3+90*A*B*x*a^2*b^4*d^4*i*n^2+18*A*B*x*b^6*c^2*d^2*i*n^2-54*B^2*ln( e*((b*x+a)/(d*x+c))^n)^2*a*b^5*c^2*d^2*i*n-54*B^2*ln(e*((b*x+a)/(d*x+c))^n )*a*b^5*c^2*d^2*i*n^2-108*A^2*x*a*b^5*c*d^3*i*n+72*A*B*ln(e*((b*x+a)/(d*x+ c))^n)*b^6*c^3*d*i*n+19*B^2*a^3*b^3*d^4*i*n^3+8*B^2*b^6*c^3*d*i*n^3+18*A^2 *a^3*b^3*d^4*i*n+36*A^2*b^6*c^3*d*i*n-36*A*B*x^3*ln(e*((b*x+a)/(d*x+c))^n) *b^6*d^4*i*n-54*B^2*x^2*ln(e*((b*x+a)/(d*x+c))^n)^2*a*b^5*d^4*i*n-54*B^2*x ^2*ln(e*((b*x+a)/(d*x+c))^n)*a*b^5*d^4*i*n^2-36*B^2*x^2*ln(e*((b*x+a)/(d*x +c))^n)*b^6*c*d^3*i*n^2+36*A*B*x^2*a*b^5*d^4*i*n^2-36*A*B*x^2*b^6*c*d^3*i* n^2+54*B^2*x*ln(e*((b*x+a)/(d*x+c))^n)^2*b^6*c^2*d^2*i*n+18*B^2*x*ln(e*((b *x+a)/(d*x+c))^n)*b^6*c^2*d^2*i*n^2-18*B^2*x^3*ln(e*((b*x+a)/(d*x+c))^n)^2 *b^6*d^4*i*n-30*B^2*x^3*ln(e*((b*x+a)/(d*x+c))^n)*b^6*d^4*i*n^2+30*B^2*x^2 *a*b^5*d^4*i*n^3-30*B^2*x^2*b^6*c*d^3*i*n^3+57*B^2*x*a^2*b^4*d^4*i*n^3-3*B ^2*x*b^6*c^2*d^2*i*n^3+36*B^2*ln(e*((b*x+a)/(d*x+c))^n)^2*b^6*c^3*d*i*n+24 *B^2*ln(e*((b*x+a)/(d*x+c))^n)*b^6*c^3*d*i*n^2+54*A^2*x*a^2*b^4*d^4*i*n+54 *A^2*x*b^6*c^2*d^2*i*n-54*A*B*a*b^5*c^2*d^2*i*n^2-27*B^2*a*b^5*c^2*d^2*...
Leaf count of result is larger than twice the leaf count of optimal. 1167 vs. \(2 (295) = 590\).
Time = 0.40 (sec) , antiderivative size = 1167, normalized size of antiderivative = 3.80 \[ \int \frac {(c i+d i x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^4} \, dx=\text {Too large to display} \]
-1/108*((8*B^2*b^3*c^3 - 27*B^2*a*b^2*c^2*d + 19*B^2*a^3*d^3)*i*n^2 + 6*(4 *A*B*b^3*c^3 - 9*A*B*a*b^2*c^2*d + 5*A*B*a^3*d^3)*i*n - 6*(5*(B^2*b^3*c*d^ 2 - B^2*a*b^2*d^3)*i*n^2 + 6*(A*B*b^3*c*d^2 - A*B*a*b^2*d^3)*i*n)*x^2 + 18 *(3*(B^2*b^3*c^2*d - 2*B^2*a*b^2*c*d^2 + B^2*a^2*b*d^3)*i*x + (2*B^2*b^3*c ^3 - 3*B^2*a*b^2*c^2*d + B^2*a^3*d^3)*i)*log(e)^2 - 18*(B^2*b^3*d^3*i*n^2* x^3 + 3*B^2*a*b^2*d^3*i*n^2*x^2 - 3*(B^2*b^3*c^2*d - 2*B^2*a*b^2*c*d^2)*i* n^2*x - (2*B^2*b^3*c^3 - 3*B^2*a*b^2*c^2*d)*i*n^2)*log((b*x + a)/(d*x + c) )^2 + 18*(2*A^2*b^3*c^3 - 3*A^2*a*b^2*c^2*d + A^2*a^3*d^3)*i - 3*((B^2*b^3 *c^2*d + 18*B^2*a*b^2*c*d^2 - 19*B^2*a^2*b*d^3)*i*n^2 - 6*(A*B*b^3*c^2*d - 6*A*B*a*b^2*c*d^2 + 5*A*B*a^2*b*d^3)*i*n - 18*(A^2*b^3*c^2*d - 2*A^2*a*b^ 2*c*d^2 + A^2*a^2*b*d^3)*i)*x - 6*(6*(B^2*b^3*c*d^2 - B^2*a*b^2*d^3)*i*n*x ^2 - (4*B^2*b^3*c^3 - 9*B^2*a*b^2*c^2*d + 5*B^2*a^3*d^3)*i*n - 6*(2*A*B*b^ 3*c^3 - 3*A*B*a*b^2*c^2*d + A*B*a^3*d^3)*i - 3*((B^2*b^3*c^2*d - 6*B^2*a*b ^2*c*d^2 + 5*B^2*a^2*b*d^3)*i*n + 6*(A*B*b^3*c^2*d - 2*A*B*a*b^2*c*d^2 + A *B*a^2*b*d^3)*i)*x + 6*(B^2*b^3*d^3*i*n*x^3 + 3*B^2*a*b^2*d^3*i*n*x^2 - 3* (B^2*b^3*c^2*d - 2*B^2*a*b^2*c*d^2)*i*n*x - (2*B^2*b^3*c^3 - 3*B^2*a*b^2*c ^2*d)*i*n)*log((b*x + a)/(d*x + c)))*log(e) + 6*((4*B^2*b^3*c^3 - 9*B^2*a* b^2*c^2*d)*i*n^2 - (5*B^2*b^3*d^3*i*n^2 + 6*A*B*b^3*d^3*i*n)*x^3 + 6*(2*A* B*b^3*c^3 - 3*A*B*a*b^2*c^2*d)*i*n - 3*(6*A*B*a*b^2*d^3*i*n + (2*B^2*b^3*c *d^2 + 3*B^2*a*b^2*d^3)*i*n^2)*x^2 + 3*((B^2*b^3*c^2*d - 6*B^2*a*b^2*c*...
Timed out. \[ \int \frac {(c i+d i x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^4} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 3312 vs. \(2 (295) = 590\).
Time = 0.41 (sec) , antiderivative size = 3312, normalized size of antiderivative = 10.79 \[ \int \frac {(c i+d i x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^4} \, dx=\text {Too large to display} \]
-1/9*A*B*c*i*n*((6*b^2*d^2*x^2 + 2*b^2*c^2 - 7*a*b*c*d + 11*a^2*d^2 - 3*(b ^2*c*d - 5*a*b*d^2)*x)/((b^6*c^2 - 2*a*b^5*c*d + a^2*b^4*d^2)*g^4*x^3 + 3* (a*b^5*c^2 - 2*a^2*b^4*c*d + a^3*b^3*d^2)*g^4*x^2 + 3*(a^2*b^4*c^2 - 2*a^3 *b^3*c*d + a^4*b^2*d^2)*g^4*x + (a^3*b^3*c^2 - 2*a^4*b^2*c*d + a^5*b*d^2)* g^4) + 6*d^3*log(b*x + a)/((b^4*c^3 - 3*a*b^3*c^2*d + 3*a^2*b^2*c*d^2 - a^ 3*b*d^3)*g^4) - 6*d^3*log(d*x + c)/((b^4*c^3 - 3*a*b^3*c^2*d + 3*a^2*b^2*c *d^2 - a^3*b*d^3)*g^4)) - 1/18*A*B*d*i*n*((5*a*b^2*c^2 - 22*a^2*b*c*d + 5* a^3*d^2 - 6*(3*b^3*c*d - a*b^2*d^2)*x^2 + 3*(3*b^3*c^2 - 16*a*b^2*c*d + 5* a^2*b*d^2)*x)/((b^7*c^2 - 2*a*b^6*c*d + a^2*b^5*d^2)*g^4*x^3 + 3*(a*b^6*c^ 2 - 2*a^2*b^5*c*d + a^3*b^4*d^2)*g^4*x^2 + 3*(a^2*b^5*c^2 - 2*a^3*b^4*c*d + a^4*b^3*d^2)*g^4*x + (a^3*b^4*c^2 - 2*a^4*b^3*c*d + a^5*b^2*d^2)*g^4) - 6*(3*b*c*d^2 - a*d^3)*log(b*x + a)/((b^5*c^3 - 3*a*b^4*c^2*d + 3*a^2*b^3*c *d^2 - a^3*b^2*d^3)*g^4) + 6*(3*b*c*d^2 - a*d^3)*log(d*x + c)/((b^5*c^3 - 3*a*b^4*c^2*d + 3*a^2*b^3*c*d^2 - a^3*b^2*d^3)*g^4)) - 1/6*(3*b*x + a)*B^2 *d*i*log(e*(b*x/(d*x + c) + a/(d*x + c))^n)^2/(b^5*g^4*x^3 + 3*a*b^4*g^4*x ^2 + 3*a^2*b^3*g^4*x + a^3*b^2*g^4) - 1/54*(6*n*((6*b^2*d^2*x^2 + 2*b^2*c^ 2 - 7*a*b*c*d + 11*a^2*d^2 - 3*(b^2*c*d - 5*a*b*d^2)*x)/((b^6*c^2 - 2*a*b^ 5*c*d + a^2*b^4*d^2)*g^4*x^3 + 3*(a*b^5*c^2 - 2*a^2*b^4*c*d + a^3*b^3*d^2) *g^4*x^2 + 3*(a^2*b^4*c^2 - 2*a^3*b^3*c*d + a^4*b^2*d^2)*g^4*x + (a^3*b^3* c^2 - 2*a^4*b^2*c*d + a^5*b*d^2)*g^4) + 6*d^3*log(b*x + a)/((b^4*c^3 - ...
Time = 2.31 (sec) , antiderivative size = 501, normalized size of antiderivative = 1.63 \[ \int \frac {(c i+d i x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^4} \, dx=-\frac {1}{108} \, {\left (\frac {18 \, {\left (2 \, B^{2} b i n^{2} - \frac {3 \, {\left (b x + a\right )} B^{2} d i n^{2}}{d x + c}\right )} \log \left (\frac {b x + a}{d x + c}\right )^{2}}{\frac {{\left (b x + a\right )}^{3} b c g^{4}}{{\left (d x + c\right )}^{3}} - \frac {{\left (b x + a\right )}^{3} a d g^{4}}{{\left (d x + c\right )}^{3}}} + \frac {6 \, {\left (4 \, B^{2} b i n^{2} - \frac {9 \, {\left (b x + a\right )} B^{2} d i n^{2}}{d x + c} + 12 \, B^{2} b i n \log \left (e\right ) - \frac {18 \, {\left (b x + a\right )} B^{2} d i n \log \left (e\right )}{d x + c} + 12 \, A B b i n - \frac {18 \, {\left (b x + a\right )} A B d i n}{d x + c}\right )} \log \left (\frac {b x + a}{d x + c}\right )}{\frac {{\left (b x + a\right )}^{3} b c g^{4}}{{\left (d x + c\right )}^{3}} - \frac {{\left (b x + a\right )}^{3} a d g^{4}}{{\left (d x + c\right )}^{3}}} + \frac {8 \, B^{2} b i n^{2} - \frac {27 \, {\left (b x + a\right )} B^{2} d i n^{2}}{d x + c} + 24 \, B^{2} b i n \log \left (e\right ) - \frac {54 \, {\left (b x + a\right )} B^{2} d i n \log \left (e\right )}{d x + c} + 36 \, B^{2} b i \log \left (e\right )^{2} - \frac {54 \, {\left (b x + a\right )} B^{2} d i \log \left (e\right )^{2}}{d x + c} + 24 \, A B b i n - \frac {54 \, {\left (b x + a\right )} A B d i n}{d x + c} + 72 \, A B b i \log \left (e\right ) - \frac {108 \, {\left (b x + a\right )} A B d i \log \left (e\right )}{d x + c} + 36 \, A^{2} b i - \frac {54 \, {\left (b x + a\right )} A^{2} d i}{d x + c}}{\frac {{\left (b x + a\right )}^{3} b c g^{4}}{{\left (d x + c\right )}^{3}} - \frac {{\left (b x + a\right )}^{3} a d g^{4}}{{\left (d x + c\right )}^{3}}}\right )} {\left (\frac {b c}{{\left (b c - a d\right )}^{2}} - \frac {a d}{{\left (b c - a d\right )}^{2}}\right )} \]
-1/108*(18*(2*B^2*b*i*n^2 - 3*(b*x + a)*B^2*d*i*n^2/(d*x + c))*log((b*x + a)/(d*x + c))^2/((b*x + a)^3*b*c*g^4/(d*x + c)^3 - (b*x + a)^3*a*d*g^4/(d* x + c)^3) + 6*(4*B^2*b*i*n^2 - 9*(b*x + a)*B^2*d*i*n^2/(d*x + c) + 12*B^2* b*i*n*log(e) - 18*(b*x + a)*B^2*d*i*n*log(e)/(d*x + c) + 12*A*B*b*i*n - 18 *(b*x + a)*A*B*d*i*n/(d*x + c))*log((b*x + a)/(d*x + c))/((b*x + a)^3*b*c* g^4/(d*x + c)^3 - (b*x + a)^3*a*d*g^4/(d*x + c)^3) + (8*B^2*b*i*n^2 - 27*( b*x + a)*B^2*d*i*n^2/(d*x + c) + 24*B^2*b*i*n*log(e) - 54*(b*x + a)*B^2*d* i*n*log(e)/(d*x + c) + 36*B^2*b*i*log(e)^2 - 54*(b*x + a)*B^2*d*i*log(e)^2 /(d*x + c) + 24*A*B*b*i*n - 54*(b*x + a)*A*B*d*i*n/(d*x + c) + 72*A*B*b*i* log(e) - 108*(b*x + a)*A*B*d*i*log(e)/(d*x + c) + 36*A^2*b*i - 54*(b*x + a )*A^2*d*i/(d*x + c))/((b*x + a)^3*b*c*g^4/(d*x + c)^3 - (b*x + a)^3*a*d*g^ 4/(d*x + c)^3))*(b*c/(b*c - a*d)^2 - a*d/(b*c - a*d)^2)
Time = 4.26 (sec) , antiderivative size = 993, normalized size of antiderivative = 3.23 \[ \int \frac {(c i+d i x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^4} \, dx=-\frac {\frac {18\,i\,A^2\,a^2\,d^2+18\,i\,A^2\,a\,b\,c\,d-36\,i\,A^2\,b^2\,c^2+30\,i\,A\,B\,a^2\,d^2\,n+30\,i\,A\,B\,a\,b\,c\,d\,n-24\,i\,A\,B\,b^2\,c^2\,n+19\,i\,B^2\,a^2\,d^2\,n^2+19\,i\,B^2\,a\,b\,c\,d\,n^2-8\,i\,B^2\,b^2\,c^2\,n^2}{6\,\left (a\,d-b\,c\right )}+\frac {x\,\left (-18\,c\,i\,A^2\,b^2\,d+18\,a\,i\,A^2\,b\,d^2-6\,c\,i\,A\,B\,b^2\,d\,n+30\,a\,i\,A\,B\,b\,d^2\,n+c\,i\,B^2\,b^2\,d\,n^2+19\,a\,i\,B^2\,b\,d^2\,n^2\right )}{2\,\left (a\,d-b\,c\right )}+\frac {x^2\,\left (5\,i\,B^2\,b^2\,d^2\,n^2+6\,A\,i\,B\,b^2\,d^2\,n\right )}{a\,d-b\,c}}{18\,a^3\,b^2\,g^4+54\,a^2\,b^3\,g^4\,x+54\,a\,b^4\,g^4\,x^2+18\,b^5\,g^4\,x^3}-{\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )}^2\,\left (\frac {\frac {B^2\,c\,i}{3\,b}+\frac {B^2\,d\,i\,x}{2\,b}+\frac {B^2\,a\,d\,i}{6\,b^2}}{a^3\,g^4+3\,a^2\,b\,g^4\,x+3\,a\,b^2\,g^4\,x^2+b^3\,g^4\,x^3}-\frac {B^2\,d^3\,i}{6\,b^2\,g^4\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}\right )-\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\,\left (\frac {A\,B\,a\,d\,i+2\,A\,B\,b\,c\,i-B^2\,a\,d\,i\,n+B^2\,b\,c\,i\,n+3\,A\,B\,b\,d\,i\,x}{3\,a^3\,b^2\,g^4+9\,a^2\,b^3\,g^4\,x+9\,a\,b^4\,g^4\,x^2+3\,b^5\,g^4\,x^3}+\frac {B^2\,d^3\,i\,\left (x\,\left (b\,\left (\frac {a\,b^2\,g^4\,n\,\left (a\,d-b\,c\right )}{d}+\frac {b^2\,g^4\,n\,\left (a\,d-b\,c\right )\,\left (3\,a\,d-b\,c\right )}{2\,d^2}\right )+\frac {2\,a\,b^3\,g^4\,n\,\left (a\,d-b\,c\right )}{d}+\frac {b^3\,g^4\,n\,\left (a\,d-b\,c\right )\,\left (3\,a\,d-b\,c\right )}{d^2}\right )+a\,\left (\frac {a\,b^2\,g^4\,n\,\left (a\,d-b\,c\right )}{d}+\frac {b^2\,g^4\,n\,\left (a\,d-b\,c\right )\,\left (3\,a\,d-b\,c\right )}{2\,d^2}\right )+\frac {3\,b^4\,g^4\,n\,x^2\,\left (a\,d-b\,c\right )}{d}+\frac {b^2\,g^4\,n\,\left (a\,d-b\,c\right )\,\left (3\,a^2\,d^2-3\,a\,b\,c\,d+b^2\,c^2\right )}{d^3}\right )}{3\,b^2\,g^4\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )\,\left (3\,a^3\,b^2\,g^4+9\,a^2\,b^3\,g^4\,x+9\,a\,b^4\,g^4\,x^2+3\,b^5\,g^4\,x^3\right )}\right )-\frac {B\,d^3\,i\,n\,\mathrm {atan}\left (\frac {B\,d^3\,i\,n\,\left (6\,A+5\,B\,n\right )\,\left (2\,b\,d\,x-\frac {b^4\,c^2\,g^4-a^2\,b^2\,d^2\,g^4}{b^2\,g^4\,\left (a\,d-b\,c\right )}\right )\,1{}\mathrm {i}}{\left (a\,d-b\,c\right )\,\left (5\,i\,B^2\,d^3\,n^2+6\,A\,i\,B\,d^3\,n\right )}\right )\,\left (6\,A+5\,B\,n\right )\,1{}\mathrm {i}}{9\,b^2\,g^4\,{\left (a\,d-b\,c\right )}^2} \]
- ((18*A^2*a^2*d^2*i - 36*A^2*b^2*c^2*i + 19*B^2*a^2*d^2*i*n^2 - 8*B^2*b^2 *c^2*i*n^2 + 30*A*B*a^2*d^2*i*n - 24*A*B*b^2*c^2*i*n + 18*A^2*a*b*c*d*i + 19*B^2*a*b*c*d*i*n^2 + 30*A*B*a*b*c*d*i*n)/(6*(a*d - b*c)) + (x*(18*A^2*a* b*d^2*i - 18*A^2*b^2*c*d*i + 19*B^2*a*b*d^2*i*n^2 + B^2*b^2*c*d*i*n^2 + 30 *A*B*a*b*d^2*i*n - 6*A*B*b^2*c*d*i*n))/(2*(a*d - b*c)) + (x^2*(5*B^2*b^2*d ^2*i*n^2 + 6*A*B*b^2*d^2*i*n))/(a*d - b*c))/(18*a^3*b^2*g^4 + 18*b^5*g^4*x ^3 + 54*a^2*b^3*g^4*x + 54*a*b^4*g^4*x^2) - log(e*((a + b*x)/(c + d*x))^n) ^2*(((B^2*c*i)/(3*b) + (B^2*d*i*x)/(2*b) + (B^2*a*d*i)/(6*b^2))/(a^3*g^4 + b^3*g^4*x^3 + 3*a*b^2*g^4*x^2 + 3*a^2*b*g^4*x) - (B^2*d^3*i)/(6*b^2*g^4*( a^2*d^2 + b^2*c^2 - 2*a*b*c*d))) - log(e*((a + b*x)/(c + d*x))^n)*((A*B*a* d*i + 2*A*B*b*c*i - B^2*a*d*i*n + B^2*b*c*i*n + 3*A*B*b*d*i*x)/(3*a^3*b^2* g^4 + 3*b^5*g^4*x^3 + 9*a^2*b^3*g^4*x + 9*a*b^4*g^4*x^2) + (B^2*d^3*i*(x*( b*((a*b^2*g^4*n*(a*d - b*c))/d + (b^2*g^4*n*(a*d - b*c)*(3*a*d - b*c))/(2* d^2)) + (2*a*b^3*g^4*n*(a*d - b*c))/d + (b^3*g^4*n*(a*d - b*c)*(3*a*d - b* c))/d^2) + a*((a*b^2*g^4*n*(a*d - b*c))/d + (b^2*g^4*n*(a*d - b*c)*(3*a*d - b*c))/(2*d^2)) + (3*b^4*g^4*n*x^2*(a*d - b*c))/d + (b^2*g^4*n*(a*d - b*c )*(3*a^2*d^2 + b^2*c^2 - 3*a*b*c*d))/d^3))/(3*b^2*g^4*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)*(3*a^3*b^2*g^4 + 3*b^5*g^4*x^3 + 9*a^2*b^3*g^4*x + 9*a*b^4*g^4 *x^2))) - (B*d^3*i*n*atan((B*d^3*i*n*(6*A + 5*B*n)*(2*b*d*x - (b^4*c^2*g^4 - a^2*b^2*d^2*g^4)/(b^2*g^4*(a*d - b*c)))*1i)/((a*d - b*c)*(5*B^2*d^3*...