Integrand size = 43, antiderivative size = 475 \[ \int \frac {(c i+d i x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^5} \, dx=-\frac {B^2 d^2 i n^2 (c+d x)^2}{4 (b c-a d)^3 g^5 (a+b x)^2}+\frac {4 b B^2 d i n^2 (c+d x)^3}{27 (b c-a d)^3 g^5 (a+b x)^3}-\frac {b^2 B^2 i n^2 (c+d x)^4}{32 (b c-a d)^3 g^5 (a+b x)^4}-\frac {B d^2 i n (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 (b c-a d)^3 g^5 (a+b x)^2}+\frac {4 b B d i n (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{9 (b c-a d)^3 g^5 (a+b x)^3}-\frac {b^2 B i n (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8 (b c-a d)^3 g^5 (a+b x)^4}-\frac {d^2 i (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2 (b c-a d)^3 g^5 (a+b x)^2}+\frac {2 b d i (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{3 (b c-a d)^3 g^5 (a+b x)^3}-\frac {b^2 i (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{4 (b c-a d)^3 g^5 (a+b x)^4} \]
-1/4*B^2*d^2*i*n^2*(d*x+c)^2/(-a*d+b*c)^3/g^5/(b*x+a)^2+4/27*b*B^2*d*i*n^2 *(d*x+c)^3/(-a*d+b*c)^3/g^5/(b*x+a)^3-1/32*b^2*B^2*i*n^2*(d*x+c)^4/(-a*d+b *c)^3/g^5/(b*x+a)^4-1/2*B*d^2*i*n*(d*x+c)^2*(A+B*ln(e*((b*x+a)/(d*x+c))^n) )/(-a*d+b*c)^3/g^5/(b*x+a)^2+4/9*b*B*d*i*n*(d*x+c)^3*(A+B*ln(e*((b*x+a)/(d *x+c))^n))/(-a*d+b*c)^3/g^5/(b*x+a)^3-1/8*b^2*B*i*n*(d*x+c)^4*(A+B*ln(e*(( b*x+a)/(d*x+c))^n))/(-a*d+b*c)^3/g^5/(b*x+a)^4-1/2*d^2*i*(d*x+c)^2*(A+B*ln (e*((b*x+a)/(d*x+c))^n))^2/(-a*d+b*c)^3/g^5/(b*x+a)^2+2/3*b*d*i*(d*x+c)^3* (A+B*ln(e*((b*x+a)/(d*x+c))^n))^2/(-a*d+b*c)^3/g^5/(b*x+a)^3-1/4*b^2*i*(d* x+c)^4*(A+B*ln(e*((b*x+a)/(d*x+c))^n))^2/(-a*d+b*c)^3/g^5/(b*x+a)^4
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 0.71 (sec) , antiderivative size = 1319, normalized size of antiderivative = 2.78 \[ \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)^5} \, dx=-\frac {i \left (216 (b c-a d)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2-288 d (-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 )^2+16 B d n (a+b x) \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 )+3 B n \left (36 A (b c-a d)^4+9 B (b c-a d)^4 n+48 A d (-b c+a d)^3 (a+b x)+28 B d (-b c+a d)^3 n (a+b x)+72 A d^2 (b c-a d)^2 (a+b x)^2+78 B d^2 (b c-a d)^2 n (a+b x)^2+144 A d^3 (-b c+a d) (a+b x)^3+300 B d^3 (-b c+a d) n (a+b x)^3-144 A d^4 (a+b x)^4 \log (a+b x)-300 B d^4 n (a+b x)^4 \log (a+b x)+72 B d^4 n (a+b x)^4 \log ^2(a+b x)+36 B (b c-a d)^4 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+48 B d (-b c+a d)^3 (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+72 B d^2 (b c-a d)^2 (a+b x)^2 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+144 B d^3 (-b c+a d) (a+b x)^3 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-144 B d^4 (a+b x)^4 \log (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+144 A d^4 (a+b x)^4 \log (c+d x)+300 B d^4 n (a+b x)^4 \log (c+d x)-144 B d^4 n (a+b x)^4 \log \left (\frac {d (a+b x)}{-b c+a d}\right ) \log (c+d x)+144 B d^4 (a+b x)^4 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log (c+d x)+72 B d^4 n (a+b x)^4 \log ^2(c+d x)-144 B d^4 n (a+b x)^4 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )-144 B d^4 n (a+b x)^4 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{-b c+a d}\right )-144 B d^4 n (a+b x)^4 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )\right )\right )}{864 b^2 (b c-a d)^3 g^5 (a+b x)^4} \]
-1/864*(i*(216*(b*c - a*d)^4*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2 - 28 8*d*(-(b*c) + a*d)^3*(a + b*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2 + 16*B*d*n*(a + b*x)*(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[a + b*x]^2 + 12*B*(b*c - a*d)^3*Log[e*((a + b*x)/(c + d*x))^n] - 1 8*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*PolyLog[2, (b*(c + d*x))/(b*c - a*d)]) + 3*B*n*(36*A*(b*c - a*d)^4 + 9*B*(b*c - a*d)^4*n + 48*A*d*(-(b*c) + a*d)^3*(a + b*x) + 28*B*d *(-(b*c) + a*d)^3*n*(a + b*x) + 72*A*d^2*(b*c - a*d)^2*(a + b*x)^2 + 78*B* d^2*(b*c - a*d)^2*n*(a + b*x)^2 + 144*A*d^3*(-(b*c) + a*d)*(a + b*x)^3 + 3 00*B*d^3*(-(b*c) + a*d)*n*(a + b*x)^3 - 144*A*d^4*(a + b*x)^4*Log[a + b...
Time = 0.54 (sec) , antiderivative size = 364, normalized size of antiderivative = 0.77, 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.
\(\displaystyle \int \frac {(c i+d i x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{(a g+b g x)^5} \, dx\) |
\(\Big \downarrow \) 2961 |
\(\displaystyle \frac {i \int \frac {(c+d x)^5 \left (b-\frac {d (a+b x)}{c+d x}\right )^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a+b x)^5}d\frac {a+b x}{c+d x}}{g^5 (b c-a d)^3}\) |
\(\Big \downarrow \) 2795 |
\(\displaystyle \frac {i \int \left (\frac {b^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 (c+d x)^5}{(a+b x)^5}-\frac {2 b d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 (c+d x)^4}{(a+b x)^4}+\frac {d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 (c+d x)^3}{(a+b x)^3}\right )d\frac {a+b x}{c+d x}}{g^5 (b c-a d)^3}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {i \left (-\frac {b^2 (c+d x)^4 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{4 (a+b x)^4}-\frac {b^2 B n (c+d x)^4 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{8 (a+b x)^4}-\frac {d^2 (c+d x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{2 (a+b x)^2}-\frac {B d^2 n (c+d x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{2 (a+b x)^2}+\frac {2 b d (c+d x)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{3 (a+b x)^3}+\frac {4 b B d n (c+d x)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{9 (a+b x)^3}-\frac {b^2 B^2 n^2 (c+d x)^4}{32 (a+b x)^4}-\frac {B^2 d^2 n^2 (c+d x)^2}{4 (a+b x)^2}+\frac {4 b B^2 d n^2 (c+d x)^3}{27 (a+b x)^3}\right )}{g^5 (b c-a d)^3}\) |
(i*(-1/4*(B^2*d^2*n^2*(c + d*x)^2)/(a + b*x)^2 + (4*b*B^2*d*n^2*(c + d*x)^ 3)/(27*(a + b*x)^3) - (b^2*B^2*n^2*(c + d*x)^4)/(32*(a + b*x)^4) - (B*d^2* n*(c + d*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(2*(a + b*x)^2) + (4 *b*B*d*n*(c + d*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(9*(a + b*x)^ 3) - (b^2*B*n*(c + d*x)^4*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(8*(a + b*x)^4) - (d^2*(c + d*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(2*(a + b*x)^2) + (2*b*d*(c + d*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/ (3*(a + b*x)^3) - (b^2*(c + d*x)^4*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^ 2)/(4*(a + b*x)^4)))/((b*c - a*d)^3*g^5)
3.2.67.3.1 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. \(2196\) vs. \(2(457)=914\).
Time = 28.62 (sec) , antiderivative size = 2197, normalized size of antiderivative = 4.63
1/864*(480*A*B*x^3*a^7*b*c*d^4*i*n^2+144*A*B*x^3*a^6*b^2*c^2*d^3*i*n^2-172 8*A*B*x^3*a^5*b^3*c^3*d^2*i*n^2+1536*A*B*x^3*a^4*b^4*c^4*d*i*n^2+576*B^2*x ^2*ln(e*((b*x+a)/(d*x+c))^n)*a^7*b*c^2*d^3*i*n^2-72*B^2*x^2*ln(e*((b*x+a)/ (d*x+c))^n)*a^6*b^2*c^3*d^2*i*n^2+864*A*B*x^2*ln(e*((b*x+a)/(d*x+c))^n)*a^ 8*c*d^4*i*n+576*A*B*x^2*a^7*b*c^2*d^3*i*n^2-2664*A*B*x^2*a^6*b^2*c^3*d^2*i *n^2+2304*A*B*x^2*a^5*b^3*c^4*d*i*n^2-864*B^2*x*ln(e*((b*x+a)/(d*x+c))^n)^ 2*a^7*b*c^3*d^2*i*n+288*B^2*x*ln(e*((b*x+a)/(d*x+c))^n)^2*a^6*b^2*c^4*d*i* n-288*B^2*x*ln(e*((b*x+a)/(d*x+c))^n)*a^7*b*c^3*d^2*i*n^2+48*B^2*x*ln(e*(( b*x+a)/(d*x+c))^n)*a^6*b^2*c^4*d*i*n^2+1728*A*B*x*ln(e*((b*x+a)/(d*x+c))^n )*a^8*c^2*d^3*i*n-2016*A*B*x*a^7*b*c^3*d^2*i*n^2+1584*A*B*x*a^6*b^2*c^4*d* i*n^2-1152*A*B*ln(e*((b*x+a)/(d*x+c))^n)*a^7*b*c^4*d*i*n+72*B^2*x^4*ln(e*( (b*x+a)/(d*x+c))^n)^2*a^6*b^2*c*d^4*i*n+156*B^2*x^4*ln(e*((b*x+a)/(d*x+c)) ^n)*a^6*b^2*c*d^4*i*n^2+156*A*B*x^4*a^6*b^2*c*d^4*i*n^2-432*A*B*x^4*a^4*b^ 4*c^3*d^2*i*n^2+384*A*B*x^4*a^3*b^5*c^4*d*i*n^2+288*B^2*x^3*ln(e*((b*x+a)/ (d*x+c))^n)^2*a^7*b*c*d^4*i*n+480*B^2*x^3*ln(e*((b*x+a)/(d*x+c))^n)*a^7*b* c*d^4*i*n^2+144*B^2*x^3*ln(e*((b*x+a)/(d*x+c))^n)*a^6*b^2*c^2*d^3*i*n^2-17 28*A*B*x*ln(e*((b*x+a)/(d*x+c))^n)*a^7*b*c^3*d^2*i*n+576*A*B*x*ln(e*((b*x+ a)/(d*x+c))^n)*a^6*b^2*c^4*d*i*n+144*A*B*x^4*ln(e*((b*x+a)/(d*x+c))^n)*a^6 *b^2*c*d^4*i*n+576*A*B*x^3*ln(e*((b*x+a)/(d*x+c))^n)*a^7*b*c*d^4*i*n+288*A ^2*x^3*a^7*b*c*d^4*i*n-1728*A^2*x^3*a^5*b^3*c^3*d^2*i*n+2304*A^2*x^3*a^...
Leaf count of result is larger than twice the leaf count of optimal. 1868 vs. \(2 (457) = 914\).
Time = 0.42 (sec) , antiderivative size = 1868, normalized size of antiderivative = 3.93 \[ \int \frac {(c i+d i x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^5} \, dx=\text {Too large to display} \]
-1/864*((27*B^2*b^4*c^4 - 128*B^2*a*b^3*c^3*d + 216*B^2*a^2*b^2*c^2*d^2 - 115*B^2*a^4*d^4)*i*n^2 + 12*(13*(B^2*b^4*c*d^3 - B^2*a*b^3*d^4)*i*n^2 + 12 *(A*B*b^4*c*d^3 - A*B*a*b^3*d^4)*i*n)*x^3 + 12*(9*A*B*b^4*c^4 - 32*A*B*a*b ^3*c^3*d + 36*A*B*a^2*b^2*c^2*d^2 - 13*A*B*a^4*d^4)*i*n - 6*((B^2*b^4*c^2* d^2 - 80*B^2*a*b^3*c*d^3 + 79*B^2*a^2*b^2*d^4)*i*n^2 + 12*(A*B*b^4*c^2*d^2 - 8*A*B*a*b^3*c*d^3 + 7*A*B*a^2*b^2*d^4)*i*n)*x^2 + 72*(4*(B^2*b^4*c^3*d - 3*B^2*a*b^3*c^2*d^2 + 3*B^2*a^2*b^2*c*d^3 - B^2*a^3*b*d^4)*i*x + (3*B^2* b^4*c^4 - 8*B^2*a*b^3*c^3*d + 6*B^2*a^2*b^2*c^2*d^2 - B^2*a^4*d^4)*i)*log( e)^2 + 72*(B^2*b^4*d^4*i*n^2*x^4 + 4*B^2*a*b^3*d^4*i*n^2*x^3 + 6*B^2*a^2*b ^2*d^4*i*n^2*x^2 + 4*(B^2*b^4*c^3*d - 3*B^2*a*b^3*c^2*d^2 + 3*B^2*a^2*b^2* c*d^3)*i*n^2*x + (3*B^2*b^4*c^4 - 8*B^2*a*b^3*c^3*d + 6*B^2*a^2*b^2*c^2*d^ 2)*i*n^2)*log((b*x + a)/(d*x + c))^2 + 72*(3*A^2*b^4*c^4 - 8*A^2*a*b^3*c^3 *d + 6*A^2*a^2*b^2*c^2*d^2 - A^2*a^4*d^4)*i - 4*((5*B^2*b^4*c^3*d - 12*B^2 *a*b^3*c^2*d^2 - 108*B^2*a^2*b^2*c*d^3 + 115*B^2*a^3*b*d^4)*i*n^2 - 12*(A* B*b^4*c^3*d - 6*A*B*a*b^3*c^2*d^2 + 18*A*B*a^2*b^2*c*d^3 - 13*A*B*a^3*b*d^ 4)*i*n - 72*(A^2*b^4*c^3*d - 3*A^2*a*b^3*c^2*d^2 + 3*A^2*a^2*b^2*c*d^3 - A ^2*a^3*b*d^4)*i)*x + 12*(12*(B^2*b^4*c*d^3 - B^2*a*b^3*d^4)*i*n*x^3 - 6*(B ^2*b^4*c^2*d^2 - 8*B^2*a*b^3*c*d^3 + 7*B^2*a^2*b^2*d^4)*i*n*x^2 + (9*B^2*b ^4*c^4 - 32*B^2*a*b^3*c^3*d + 36*B^2*a^2*b^2*c^2*d^2 - 13*B^2*a^4*d^4)*i*n + 12*(3*A*B*b^4*c^4 - 8*A*B*a*b^3*c^3*d + 6*A*B*a^2*b^2*c^2*d^2 - A*B*...
Timed out. \[ \int \frac {(c i+d i x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^5} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 4838 vs. \(2 (457) = 914\).
Time = 0.54 (sec) , antiderivative size = 4838, normalized size of antiderivative = 10.19 \[ \int \frac {(c i+d i x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^5} \, dx=\text {Too large to display} \]
1/24*A*B*c*i*n*((12*b^3*d^3*x^3 - 3*b^3*c^3 + 13*a*b^2*c^2*d - 23*a^2*b*c* d^2 + 25*a^3*d^3 - 6*(b^3*c*d^2 - 7*a*b^2*d^3)*x^2 + 4*(b^3*c^2*d - 5*a*b^ 2*c*d^2 + 13*a^2*b*d^3)*x)/((b^8*c^3 - 3*a*b^7*c^2*d + 3*a^2*b^6*c*d^2 - a ^3*b^5*d^3)*g^5*x^4 + 4*(a*b^7*c^3 - 3*a^2*b^6*c^2*d + 3*a^3*b^5*c*d^2 - a ^4*b^4*d^3)*g^5*x^3 + 6*(a^2*b^6*c^3 - 3*a^3*b^5*c^2*d + 3*a^4*b^4*c*d^2 - a^5*b^3*d^3)*g^5*x^2 + 4*(a^3*b^5*c^3 - 3*a^4*b^4*c^2*d + 3*a^5*b^3*c*d^2 - a^6*b^2*d^3)*g^5*x + (a^4*b^4*c^3 - 3*a^5*b^3*c^2*d + 3*a^6*b^2*c*d^2 - a^7*b*d^3)*g^5) + 12*d^4*log(b*x + a)/((b^5*c^4 - 4*a*b^4*c^3*d + 6*a^2*b ^3*c^2*d^2 - 4*a^3*b^2*c*d^3 + a^4*b*d^4)*g^5) - 12*d^4*log(d*x + c)/((b^5 *c^4 - 4*a*b^4*c^3*d + 6*a^2*b^3*c^2*d^2 - 4*a^3*b^2*c*d^3 + a^4*b*d^4)*g^ 5)) - 1/72*A*B*d*i*n*((7*a*b^3*c^3 - 33*a^2*b^2*c^2*d + 75*a^3*b*c*d^2 - 1 3*a^4*d^3 + 12*(4*b^4*c*d^2 - a*b^3*d^3)*x^3 - 6*(4*b^4*c^2*d - 29*a*b^3*c *d^2 + 7*a^2*b^2*d^3)*x^2 + 4*(4*b^4*c^3 - 21*a*b^3*c^2*d + 57*a^2*b^2*c*d ^2 - 13*a^3*b*d^3)*x)/((b^9*c^3 - 3*a*b^8*c^2*d + 3*a^2*b^7*c*d^2 - a^3*b^ 6*d^3)*g^5*x^4 + 4*(a*b^8*c^3 - 3*a^2*b^7*c^2*d + 3*a^3*b^6*c*d^2 - a^4*b^ 5*d^3)*g^5*x^3 + 6*(a^2*b^7*c^3 - 3*a^3*b^6*c^2*d + 3*a^4*b^5*c*d^2 - a^5* b^4*d^3)*g^5*x^2 + 4*(a^3*b^6*c^3 - 3*a^4*b^5*c^2*d + 3*a^5*b^4*c*d^2 - a^ 6*b^3*d^3)*g^5*x + (a^4*b^5*c^3 - 3*a^5*b^4*c^2*d + 3*a^6*b^3*c*d^2 - a^7* b^2*d^3)*g^5) + 12*(4*b*c*d^3 - a*d^4)*log(b*x + a)/((b^6*c^4 - 4*a*b^5*c^ 3*d + 6*a^2*b^4*c^2*d^2 - 4*a^3*b^3*c*d^3 + a^4*b^2*d^4)*g^5) - 12*(4*b...
Time = 3.07 (sec) , antiderivative size = 871, normalized size of antiderivative = 1.83 \[ \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)^5} \, dx=-\frac {1}{864} \, {\left (\frac {72 \, {\left (3 \, B^{2} b^{2} i n^{2} - \frac {8 \, {\left (b x + a\right )} B^{2} b d i n^{2}}{d x + c} + \frac {6 \, {\left (b x + a\right )}^{2} B^{2} d^{2} i n^{2}}{{\left (d x + c\right )}^{2}}\right )} \log \left (\frac {b x + a}{d x + c}\right )^{2}}{\frac {{\left (b x + a\right )}^{4} b^{2} c^{2} g^{5}}{{\left (d x + c\right )}^{4}} - \frac {2 \, {\left (b x + a\right )}^{4} a b c d g^{5}}{{\left (d x + c\right )}^{4}} + \frac {{\left (b x + a\right )}^{4} a^{2} d^{2} g^{5}}{{\left (d x + c\right )}^{4}}} + \frac {12 \, {\left (9 \, B^{2} b^{2} i n^{2} - \frac {32 \, {\left (b x + a\right )} B^{2} b d i n^{2}}{d x + c} + \frac {36 \, {\left (b x + a\right )}^{2} B^{2} d^{2} i n^{2}}{{\left (d x + c\right )}^{2}} + 36 \, B^{2} b^{2} i n \log \left (e\right ) - \frac {96 \, {\left (b x + a\right )} B^{2} b d i n \log \left (e\right )}{d x + c} + \frac {72 \, {\left (b x + a\right )}^{2} B^{2} d^{2} i n \log \left (e\right )}{{\left (d x + c\right )}^{2}} + 36 \, A B b^{2} i n - \frac {96 \, {\left (b x + a\right )} A B b d i n}{d x + c} + \frac {72 \, {\left (b x + a\right )}^{2} A B d^{2} i n}{{\left (d x + c\right )}^{2}}\right )} \log \left (\frac {b x + a}{d x + c}\right )}{\frac {{\left (b x + a\right )}^{4} b^{2} c^{2} g^{5}}{{\left (d x + c\right )}^{4}} - \frac {2 \, {\left (b x + a\right )}^{4} a b c d g^{5}}{{\left (d x + c\right )}^{4}} + \frac {{\left (b x + a\right )}^{4} a^{2} d^{2} g^{5}}{{\left (d x + c\right )}^{4}}} + \frac {27 \, B^{2} b^{2} i n^{2} - \frac {128 \, {\left (b x + a\right )} B^{2} b d i n^{2}}{d x + c} + \frac {216 \, {\left (b x + a\right )}^{2} B^{2} d^{2} i n^{2}}{{\left (d x + c\right )}^{2}} + 108 \, B^{2} b^{2} i n \log \left (e\right ) - \frac {384 \, {\left (b x + a\right )} B^{2} b d i n \log \left (e\right )}{d x + c} + \frac {432 \, {\left (b x + a\right )}^{2} B^{2} d^{2} i n \log \left (e\right )}{{\left (d x + c\right )}^{2}} + 216 \, B^{2} b^{2} i \log \left (e\right )^{2} - \frac {576 \, {\left (b x + a\right )} B^{2} b d i \log \left (e\right )^{2}}{d x + c} + \frac {432 \, {\left (b x + a\right )}^{2} B^{2} d^{2} i \log \left (e\right )^{2}}{{\left (d x + c\right )}^{2}} + 108 \, A B b^{2} i n - \frac {384 \, {\left (b x + a\right )} A B b d i n}{d x + c} + \frac {432 \, {\left (b x + a\right )}^{2} A B d^{2} i n}{{\left (d x + c\right )}^{2}} + 432 \, A B b^{2} i \log \left (e\right ) - \frac {1152 \, {\left (b x + a\right )} A B b d i \log \left (e\right )}{d x + c} + \frac {864 \, {\left (b x + a\right )}^{2} A B d^{2} i \log \left (e\right )}{{\left (d x + c\right )}^{2}} + 216 \, A^{2} b^{2} i - \frac {576 \, {\left (b x + a\right )} A^{2} b d i}{d x + c} + \frac {432 \, {\left (b x + a\right )}^{2} A^{2} d^{2} i}{{\left (d x + c\right )}^{2}}}{\frac {{\left (b x + a\right )}^{4} b^{2} c^{2} g^{5}}{{\left (d x + c\right )}^{4}} - \frac {2 \, {\left (b x + a\right )}^{4} a b c d g^{5}}{{\left (d x + c\right )}^{4}} + \frac {{\left (b x + a\right )}^{4} a^{2} d^{2} g^{5}}{{\left (d x + c\right )}^{4}}}\right )} {\left (\frac {b c}{{\left (b c - a d\right )}^{2}} - \frac {a d}{{\left (b c - a d\right )}^{2}}\right )} \]
-1/864*(72*(3*B^2*b^2*i*n^2 - 8*(b*x + a)*B^2*b*d*i*n^2/(d*x + c) + 6*(b*x + a)^2*B^2*d^2*i*n^2/(d*x + c)^2)*log((b*x + a)/(d*x + c))^2/((b*x + a)^4 *b^2*c^2*g^5/(d*x + c)^4 - 2*(b*x + a)^4*a*b*c*d*g^5/(d*x + c)^4 + (b*x + a)^4*a^2*d^2*g^5/(d*x + c)^4) + 12*(9*B^2*b^2*i*n^2 - 32*(b*x + a)*B^2*b*d *i*n^2/(d*x + c) + 36*(b*x + a)^2*B^2*d^2*i*n^2/(d*x + c)^2 + 36*B^2*b^2*i *n*log(e) - 96*(b*x + a)*B^2*b*d*i*n*log(e)/(d*x + c) + 72*(b*x + a)^2*B^2 *d^2*i*n*log(e)/(d*x + c)^2 + 36*A*B*b^2*i*n - 96*(b*x + a)*A*B*b*d*i*n/(d *x + c) + 72*(b*x + a)^2*A*B*d^2*i*n/(d*x + c)^2)*log((b*x + a)/(d*x + c)) /((b*x + a)^4*b^2*c^2*g^5/(d*x + c)^4 - 2*(b*x + a)^4*a*b*c*d*g^5/(d*x + c )^4 + (b*x + a)^4*a^2*d^2*g^5/(d*x + c)^4) + (27*B^2*b^2*i*n^2 - 128*(b*x + a)*B^2*b*d*i*n^2/(d*x + c) + 216*(b*x + a)^2*B^2*d^2*i*n^2/(d*x + c)^2 + 108*B^2*b^2*i*n*log(e) - 384*(b*x + a)*B^2*b*d*i*n*log(e)/(d*x + c) + 432 *(b*x + a)^2*B^2*d^2*i*n*log(e)/(d*x + c)^2 + 216*B^2*b^2*i*log(e)^2 - 576 *(b*x + a)*B^2*b*d*i*log(e)^2/(d*x + c) + 432*(b*x + a)^2*B^2*d^2*i*log(e) ^2/(d*x + c)^2 + 108*A*B*b^2*i*n - 384*(b*x + a)*A*B*b*d*i*n/(d*x + c) + 4 32*(b*x + a)^2*A*B*d^2*i*n/(d*x + c)^2 + 432*A*B*b^2*i*log(e) - 1152*(b*x + a)*A*B*b*d*i*log(e)/(d*x + c) + 864*(b*x + a)^2*A*B*d^2*i*log(e)/(d*x + c)^2 + 216*A^2*b^2*i - 576*(b*x + a)*A^2*b*d*i/(d*x + c) + 432*(b*x + a)^2 *A^2*d^2*i/(d*x + c)^2)/((b*x + a)^4*b^2*c^2*g^5/(d*x + c)^4 - 2*(b*x + a) ^4*a*b*c*d*g^5/(d*x + c)^4 + (b*x + a)^4*a^2*d^2*g^5/(d*x + c)^4))*(b*c...
Time = 6.36 (sec) , antiderivative size = 1794, normalized size of antiderivative = 3.78 \[ \int \frac {(c i+d i x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^5} \, dx=\text {Too large to display} \]
((72*A^2*a^3*d^3*i + 216*A^2*b^3*c^3*i + 115*B^2*a^3*d^3*i*n^2 + 27*B^2*b^ 3*c^3*i*n^2 + 156*A*B*a^3*d^3*i*n + 108*A*B*b^3*c^3*i*n - 360*A^2*a*b^2*c^ 2*d*i + 72*A^2*a^2*b*c*d^2*i - 101*B^2*a*b^2*c^2*d*i*n^2 + 115*B^2*a^2*b*c *d^2*i*n^2 - 276*A*B*a*b^2*c^2*d*i*n + 156*A*B*a^2*b*c*d^2*i*n)/(12*(a*d - b*c)) + (x^2*(79*B^2*a*b^2*d^3*i*n^2 - B^2*b^3*c*d^2*i*n^2 + 84*A*B*a*b^2 *d^3*i*n - 12*A*B*b^3*c*d^2*i*n))/(2*(a*d - b*c)) + (x*(72*A^2*a^2*b*d^3*i + 72*A^2*b^3*c^2*d*i + 115*B^2*a^2*b*d^3*i*n^2 - 5*B^2*b^3*c^2*d*i*n^2 - 144*A^2*a*b^2*c*d^2*i + 7*B^2*a*b^2*c*d^2*i*n^2 + 156*A*B*a^2*b*d^3*i*n + 12*A*B*b^3*c^2*d*i*n - 60*A*B*a*b^2*c*d^2*i*n))/(3*(a*d - b*c)) + (d*x^3*( 13*B^2*b^3*d^2*i*n^2 + 12*A*B*b^3*d^2*i*n))/(a*d - b*c))/(x*(288*a^3*b^4*c *g^5 - 288*a^4*b^3*d*g^5) - x^3*(288*a^2*b^5*d*g^5 - 288*a*b^6*c*g^5) + x^ 4*(72*b^7*c*g^5 - 72*a*b^6*d*g^5) + x^2*(432*a^2*b^5*c*g^5 - 432*a^3*b^4*d *g^5) + 72*a^4*b^3*c*g^5 - 72*a^5*b^2*d*g^5) - log(e*((a + b*x)/(c + d*x)) ^n)^2*(((B^2*c*i)/(4*b) + (B^2*d*i*x)/(3*b) + (B^2*a*d*i)/(12*b^2))/(a^4*g ^5 + b^4*g^5*x^4 + 4*a*b^3*g^5*x^3 + 6*a^2*b^2*g^5*x^2 + 4*a^3*b*g^5*x) - (B^2*d^4*i)/(12*b^2*g^5*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2 ))) - log(e*((a + b*x)/(c + d*x))^n)*((A*B*a*d*i + 3*A*B*b*c*i - B^2*a*d*i *n + B^2*b*c*i*n + 4*A*B*b*d*i*x)/(6*a^4*b^2*g^5 + 6*b^6*g^5*x^4 + 24*a^3* b^3*g^5*x + 24*a*b^5*g^5*x^3 + 36*a^2*b^4*g^5*x^2) + (B^2*d^4*i*(x^2*(b*(b *((3*a*b^2*g^5*n*(a*d - b*c))/(2*d) + (b^2*g^5*n*(a*d - b*c)*(4*a*d - b...