Integrand size = 45, antiderivative size = 402 \[ \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x) (c i+d i x)^3} \, dx=\frac {B^2 d^2 n^2 (a+b x)^2}{4 (b c-a d)^3 g i^3 (c+d x)^2}+\frac {4 A b B d n (a+b x)}{(b c-a d)^3 g i^3 (c+d x)}-\frac {4 b B^2 d n^2 (a+b x)}{(b c-a d)^3 g i^3 (c+d x)}+\frac {4 b B^2 d n (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(b c-a d)^3 g i^3 (c+d x)}-\frac {B d^2 n (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 (b c-a d)^3 g i^3 (c+d x)^2}+\frac {d^2 (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2 (b c-a d)^3 g i^3 (c+d x)^2}-\frac {2 b d (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(b c-a d)^3 g i^3 (c+d x)}+\frac {b^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^3}{3 B (b c-a d)^3 g i^3 n} \]
1/4*B^2*d^2*n^2*(b*x+a)^2/(-a*d+b*c)^3/g/i^3/(d*x+c)^2+4*A*b*B*d*n*(b*x+a) /(-a*d+b*c)^3/g/i^3/(d*x+c)-4*b*B^2*d*n^2*(b*x+a)/(-a*d+b*c)^3/g/i^3/(d*x+ c)+4*b*B^2*d*n*(b*x+a)*ln(e*((b*x+a)/(d*x+c))^n)/(-a*d+b*c)^3/g/i^3/(d*x+c )-1/2*B*d^2*n*(b*x+a)^2*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/(-a*d+b*c)^3/g/i^3 /(d*x+c)^2+1/2*d^2*(b*x+a)^2*(A+B*ln(e*((b*x+a)/(d*x+c))^n))^2/(-a*d+b*c)^ 3/g/i^3/(d*x+c)^2-2*b*d*(b*x+a)*(A+B*ln(e*((b*x+a)/(d*x+c))^n))^2/(-a*d+b* c)^3/g/i^3/(d*x+c)+1/3*b^2*(A+B*ln(e*((b*x+a)/(d*x+c))^n))^3/B/(-a*d+b*c)^ 3/g/i^3/n
Leaf count is larger than twice the leaf count of optimal. \(971\) vs. \(2(402)=804\).
Time = 0.52 (sec) , antiderivative size = 971, normalized size of antiderivative = 2.42 \[ \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x) (c i+d i x)^3} \, dx=\frac {4 b^2 B^2 n^2 \log ^3\left (\frac {a+b x}{c+d x}\right )-\frac {6 B n \log ^2\left (\frac {a+b x}{c+d x}\right ) \left (-2 A b^2 c^2+4 a b B c d n-a^2 B d^2 n-4 A b^2 c d x+4 b^2 B c d n x+2 a b B d^2 n x-2 A b^2 d^2 x^2+3 b^2 B d^2 n x^2-2 b^2 B (c+d x)^2 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+2 b^2 B n (c+d x)^2 \log \left (\frac {a+b x}{c+d x}\right )\right )}{(c+d x)^2}-\frac {6 B (b c-a d) n \log \left (\frac {a+b x}{c+d x}\right ) \left (-6 A b c+2 a A d+7 b B c n-a B d n-4 A b d x+6 b B d n x+2 B (-3 b c+a d-2 b d x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+2 B n (3 b c-a d+2 b d x) \log \left (\frac {a+b x}{c+d x}\right )\right )}{(c+d x)^2}+\frac {3 (b c-a d)^2 \left (2 A^2-2 A B n+B^2 n^2+2 B^2 \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+2 B n (-2 A+B n) \log \left (\frac {a+b x}{c+d x}\right )+2 B^2 n^2 \log ^2\left (\frac {a+b x}{c+d x}\right )-2 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \left (-2 A+B n+2 B n \log \left (\frac {a+b x}{c+d x}\right )\right )\right )}{(c+d x)^2}+\frac {6 b (b c-a d) \left (2 A^2-6 A B n+7 B^2 n^2+2 B^2 \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+2 B n (-2 A+3 B n) \log \left (\frac {a+b x}{c+d x}\right )+2 B^2 n^2 \log ^2\left (\frac {a+b x}{c+d x}\right )-2 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \left (-2 A+3 B n+2 B n \log \left (\frac {a+b x}{c+d x}\right )\right )\right )}{c+d x}+6 b^2 \log (a+b x) \left (2 A^2-6 A B n+7 B^2 n^2+2 B^2 \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+2 B n (-2 A+3 B n) \log \left (\frac {a+b x}{c+d x}\right )+2 B^2 n^2 \log ^2\left (\frac {a+b x}{c+d x}\right )-2 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \left (-2 A+3 B n+2 B n \log \left (\frac {a+b x}{c+d x}\right )\right )\right )-6 b^2 \left (2 A^2-6 A B n+7 B^2 n^2+2 B^2 \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+2 B n (-2 A+3 B n) \log \left (\frac {a+b x}{c+d x}\right )+2 B^2 n^2 \log ^2\left (\frac {a+b x}{c+d x}\right )-2 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \left (-2 A+3 B n+2 B n \log \left (\frac {a+b x}{c+d x}\right )\right )\right ) \log (c+d x)}{12 (b c-a d)^3 g i^3} \]
(4*b^2*B^2*n^2*Log[(a + b*x)/(c + d*x)]^3 - (6*B*n*Log[(a + b*x)/(c + d*x) ]^2*(-2*A*b^2*c^2 + 4*a*b*B*c*d*n - a^2*B*d^2*n - 4*A*b^2*c*d*x + 4*b^2*B* c*d*n*x + 2*a*b*B*d^2*n*x - 2*A*b^2*d^2*x^2 + 3*b^2*B*d^2*n*x^2 - 2*b^2*B* (c + d*x)^2*Log[e*((a + b*x)/(c + d*x))^n] + 2*b^2*B*n*(c + d*x)^2*Log[(a + b*x)/(c + d*x)]))/(c + d*x)^2 - (6*B*(b*c - a*d)*n*Log[(a + b*x)/(c + d* x)]*(-6*A*b*c + 2*a*A*d + 7*b*B*c*n - a*B*d*n - 4*A*b*d*x + 6*b*B*d*n*x + 2*B*(-3*b*c + a*d - 2*b*d*x)*Log[e*((a + b*x)/(c + d*x))^n] + 2*B*n*(3*b*c - a*d + 2*b*d*x)*Log[(a + b*x)/(c + d*x)]))/(c + d*x)^2 + (3*(b*c - a*d)^ 2*(2*A^2 - 2*A*B*n + B^2*n^2 + 2*B^2*Log[e*((a + b*x)/(c + d*x))^n]^2 + 2* B*n*(-2*A + B*n)*Log[(a + b*x)/(c + d*x)] + 2*B^2*n^2*Log[(a + b*x)/(c + d *x)]^2 - 2*B*Log[e*((a + b*x)/(c + d*x))^n]*(-2*A + B*n + 2*B*n*Log[(a + b *x)/(c + d*x)])))/(c + d*x)^2 + (6*b*(b*c - a*d)*(2*A^2 - 6*A*B*n + 7*B^2* n^2 + 2*B^2*Log[e*((a + b*x)/(c + d*x))^n]^2 + 2*B*n*(-2*A + 3*B*n)*Log[(a + b*x)/(c + d*x)] + 2*B^2*n^2*Log[(a + b*x)/(c + d*x)]^2 - 2*B*Log[e*((a + b*x)/(c + d*x))^n]*(-2*A + 3*B*n + 2*B*n*Log[(a + b*x)/(c + d*x)])))/(c + d*x) + 6*b^2*Log[a + b*x]*(2*A^2 - 6*A*B*n + 7*B^2*n^2 + 2*B^2*Log[e*((a + b*x)/(c + d*x))^n]^2 + 2*B*n*(-2*A + 3*B*n)*Log[(a + b*x)/(c + d*x)] + 2*B^2*n^2*Log[(a + b*x)/(c + d*x)]^2 - 2*B*Log[e*((a + b*x)/(c + d*x))^n]* (-2*A + 3*B*n + 2*B*n*Log[(a + b*x)/(c + d*x)])) - 6*b^2*(2*A^2 - 6*A*B*n + 7*B^2*n^2 + 2*B^2*Log[e*((a + b*x)/(c + d*x))^n]^2 + 2*B*n*(-2*A + 3*...
Time = 0.89 (sec) , antiderivative size = 383, normalized size of antiderivative = 0.95, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2961, 2788, 2767, 2009, 2788, 2733, 2009, 2739, 15}
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.
| \(\displaystyle \int \frac {\left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{(a g+b g x) (c i+d i x)^3} \, dx\) |
| \(\Big \downarrow \) 2961 |
| \(\displaystyle \frac {\int \frac {(c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{a+b x}d\frac {a+b x}{c+d x}}{g i^3 (b c-a d)^3}\) |
| \(\Big \downarrow \) 2788 |
| \(\displaystyle \frac {b \int \frac {(c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right ) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{a+b x}d\frac {a+b x}{c+d x}-d \int \left (b-\frac {d (a+b x)}{c+d x}\right ) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2d\frac {a+b x}{c+d x}}{g i^3 (b c-a d)^3}\) |
| \(\Big \downarrow \) 2767 |
| \(\displaystyle \frac {b \int \frac {(c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right ) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{a+b x}d\frac {a+b x}{c+d x}-d \int \left (b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2-\frac {d (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{c+d x}\right )d\frac {a+b x}{c+d x}}{g i^3 (b c-a d)^3}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {b \int \frac {(c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right ) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{a+b x}d\frac {a+b x}{c+d x}-d \left (\frac {B d n (a+b x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{2 (c+d x)^2}+\frac {b (a+b x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{c+d x}-\frac {d (a+b x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{2 (c+d x)^2}-\frac {2 A b B n (a+b x)}{c+d x}-\frac {2 b B^2 n (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{c+d x}+\frac {2 b B^2 n^2 (a+b x)}{c+d x}-\frac {B^2 d n^2 (a+b x)^2}{4 (c+d x)^2}\right )}{g i^3 (b c-a d)^3}\) |
| \(\Big \downarrow \) 2788 |
| \(\displaystyle \frac {b \left (b \int \frac {(c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{a+b x}d\frac {a+b x}{c+d x}-d \int \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2d\frac {a+b x}{c+d x}\right )-d \left (\frac {B d n (a+b x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{2 (c+d x)^2}+\frac {b (a+b x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{c+d x}-\frac {d (a+b x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{2 (c+d x)^2}-\frac {2 A b B n (a+b x)}{c+d x}-\frac {2 b B^2 n (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{c+d x}+\frac {2 b B^2 n^2 (a+b x)}{c+d x}-\frac {B^2 d n^2 (a+b x)^2}{4 (c+d x)^2}\right )}{g i^3 (b c-a d)^3}\) |
| \(\Big \downarrow \) 2733 |
| \(\displaystyle \frac {b \left (b \int \frac {(c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{a+b x}d\frac {a+b x}{c+d x}-d \left (\frac {(a+b x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{c+d x}-2 B n \int \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )d\frac {a+b x}{c+d x}\right )\right )-d \left (\frac {B d n (a+b x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{2 (c+d x)^2}+\frac {b (a+b x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{c+d x}-\frac {d (a+b x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{2 (c+d x)^2}-\frac {2 A b B n (a+b x)}{c+d x}-\frac {2 b B^2 n (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{c+d x}+\frac {2 b B^2 n^2 (a+b x)}{c+d x}-\frac {B^2 d n^2 (a+b x)^2}{4 (c+d x)^2}\right )}{g i^3 (b c-a d)^3}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {b \left (b \int \frac {(c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{a+b x}d\frac {a+b x}{c+d x}-d \left (\frac {(a+b x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{c+d x}-2 B n \left (\frac {A (a+b x)}{c+d x}+\frac {B (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{c+d x}-\frac {B n (a+b x)}{c+d x}\right )\right )\right )-d \left (\frac {B d n (a+b x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{2 (c+d x)^2}+\frac {b (a+b x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{c+d x}-\frac {d (a+b x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{2 (c+d x)^2}-\frac {2 A b B n (a+b x)}{c+d x}-\frac {2 b B^2 n (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{c+d x}+\frac {2 b B^2 n^2 (a+b x)}{c+d x}-\frac {B^2 d n^2 (a+b x)^2}{4 (c+d x)^2}\right )}{g i^3 (b c-a d)^3}\) |
| \(\Big \downarrow \) 2739 |
| \(\displaystyle \frac {b \left (\frac {b \int \frac {(a+b x)^2}{(c+d x)^2}d\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{B n}-d \left (\frac {(a+b x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{c+d x}-2 B n \left (\frac {A (a+b x)}{c+d x}+\frac {B (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{c+d x}-\frac {B n (a+b x)}{c+d x}\right )\right )\right )-d \left (\frac {B d n (a+b x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{2 (c+d x)^2}+\frac {b (a+b x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{c+d x}-\frac {d (a+b x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{2 (c+d x)^2}-\frac {2 A b B n (a+b x)}{c+d x}-\frac {2 b B^2 n (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{c+d x}+\frac {2 b B^2 n^2 (a+b x)}{c+d x}-\frac {B^2 d n^2 (a+b x)^2}{4 (c+d x)^2}\right )}{g i^3 (b c-a d)^3}\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle \frac {b \left (\frac {b (a+b x)^3}{3 B n (c+d x)^3}-d \left (\frac {(a+b x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{c+d x}-2 B n \left (\frac {A (a+b x)}{c+d x}+\frac {B (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{c+d x}-\frac {B n (a+b x)}{c+d x}\right )\right )\right )-d \left (\frac {B d n (a+b x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{2 (c+d x)^2}+\frac {b (a+b x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{c+d x}-\frac {d (a+b x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{2 (c+d x)^2}-\frac {2 A b B n (a+b x)}{c+d x}-\frac {2 b B^2 n (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{c+d x}+\frac {2 b B^2 n^2 (a+b x)}{c+d x}-\frac {B^2 d n^2 (a+b x)^2}{4 (c+d x)^2}\right )}{g i^3 (b c-a d)^3}\) |
(-(d*(-1/4*(B^2*d*n^2*(a + b*x)^2)/(c + d*x)^2 - (2*A*b*B*n*(a + b*x))/(c + d*x) + (2*b*B^2*n^2*(a + b*x))/(c + d*x) - (2*b*B^2*n*(a + b*x)*Log[e*(( a + b*x)/(c + d*x))^n])/(c + d*x) + (B*d*n*(a + b*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(2*(c + d*x)^2) - (d*(a + b*x)^2*(A + B*Log[e*((a + b *x)/(c + d*x))^n])^2)/(2*(c + d*x)^2) + (b*(a + b*x)*(A + B*Log[e*((a + b* x)/(c + d*x))^n])^2)/(c + d*x))) + b*((b*(a + b*x)^3)/(3*B*n*(c + d*x)^3) - d*(((a + b*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(c + d*x) - 2*B* n*((A*(a + b*x))/(c + d*x) - (B*n*(a + b*x))/(c + d*x) + (B*(a + b*x)*Log[ e*((a + b*x)/(c + d*x))^n])/(c + d*x)))))/((b*c - a*d)^3*g*i^3)
3.3.6.3.1 Defintions of rubi rules used
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b *Log[c*x^n])^p, x] - Simp[b*n*p Int[(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{a, b, c, n}, x] && GtQ[p, 0] && IntegerQ[2*p]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/(x_), x_Symbol] :> Simp[1/( b*n) Subst[Int[x^p, x], x, a + b*Log[c*x^n]], x] /; FreeQ[{a, b, c, n, p} , x]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_)^(r_.))^( q_.), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*Log[c*x^n])^p, (d + e*x ^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, n, p, q, r}, x] && IntegerQ[q] && (GtQ[q, 0] || (IGtQ[p, 0] && IntegerQ[r]))
Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_.)) /(x_), x_Symbol] :> Simp[d Int[(d + e*x)^(q - 1)*((a + b*Log[c*x^n])^p/x) , x], x] + Simp[e Int[(d + e*x)^(q - 1)*(a + b*Log[c*x^n])^p, x], x] /; F reeQ[{a, b, c, d, e, n}, x] && IGtQ[p, 0] && GtQ[q, 0] && IntegerQ[2*q]
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. \(1235\) vs. \(2(394)=788\).
Time = 11.42 (sec) , antiderivative size = 1236, normalized size of antiderivative = 3.07
-1/12*(4*B^2*x^2*ln(e*((b*x+a)/(d*x+c))^n)^3*a^2*b^2*c^4*d^2+48*B^2*x^2*a^ 3*b*c^3*d^3*n^3-45*B^2*x^2*a^2*b^2*c^4*d^2*n^3+6*A*B*x^2*a^4*c^2*d^4*n^2+8 *B^2*x*ln(e*((b*x+a)/(d*x+c))^n)^3*a^2*b^2*c^5*d+54*B^2*x*a^3*b*c^4*d^2*n^ 3-48*B^2*x*a^2*b^2*c^5*d*n^3+12*A^2*x^2*ln(e*((b*x+a)/(d*x+c))^n)*a^2*b^2* c^4*d^2+24*A^2*x^2*a^3*b*c^3*d^3*n-36*A*B*x^2*ln(e*((b*x+a)/(d*x+c))^n)*a^ 2*b^2*c^4*d^2*n-24*A*B*x*ln(e*((b*x+a)/(d*x+c))^n)*a^3*b*c^4*d^2*n-48*A*B* x*ln(e*((b*x+a)/(d*x+c))^n)*a^2*b^2*c^5*d*n+4*B^2*ln(e*((b*x+a)/(d*x+c))^n )^3*a^2*b^2*c^6+12*A^2*ln(e*((b*x+a)/(d*x+c))^n)*a^2*b^2*c^6-24*B^2*x*ln(e *((b*x+a)/(d*x+c))^n)^2*a^2*b^2*c^5*d*n+36*B^2*x*ln(e*((b*x+a)/(d*x+c))^n) *a^3*b*c^4*d^2*n^2+48*B^2*x*ln(e*((b*x+a)/(d*x+c))^n)*a^2*b^2*c^5*d*n^2+24 *A*B*x*ln(e*((b*x+a)/(d*x+c))^n)^2*a^2*b^2*c^5*d-60*A*B*x*a^3*b*c^4*d^2*n^ 2+48*A*B*x*a^2*b^2*c^5*d*n^2-48*A*B*ln(e*((b*x+a)/(d*x+c))^n)*a^3*b*c^5*d* n-18*B^2*x^2*ln(e*((b*x+a)/(d*x+c))^n)^2*a^2*b^2*c^4*d^2*n+42*B^2*x^2*ln(e *((b*x+a)/(d*x+c))^n)*a^2*b^2*c^4*d^2*n^2+12*A*B*x^2*ln(e*((b*x+a)/(d*x+c) )^n)^2*a^2*b^2*c^4*d^2-48*A*B*x^2*a^3*b*c^3*d^3*n^2+42*A*B*x^2*a^2*b^2*c^4 *d^2*n^2-12*B^2*x*ln(e*((b*x+a)/(d*x+c))^n)^2*a^3*b*c^4*d^2*n-3*B^2*x^2*a^ 4*c^2*d^4*n^3-6*B^2*x*a^4*c^3*d^3*n^3-6*A^2*x^2*a^4*c^2*d^4*n+6*B^2*ln(e*( (b*x+a)/(d*x+c))^n)^2*a^4*c^4*d^2*n-6*B^2*ln(e*((b*x+a)/(d*x+c))^n)*a^4*c^ 4*d^2*n^2-12*A^2*x*a^4*c^3*d^3*n+12*A*B*ln(e*((b*x+a)/(d*x+c))^n)^2*a^2*b^ 2*c^6-18*A^2*x^2*a^2*b^2*c^4*d^2*n+12*A*B*x*a^4*c^3*d^3*n^2-24*B^2*ln(e...
Leaf count of result is larger than twice the leaf count of optimal. 1076 vs. \(2 (394) = 788\).
Time = 0.32 (sec) , antiderivative size = 1076, normalized size of antiderivative = 2.68 \[ \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x) (c i+d i x)^3} \, dx=\frac {18 \, A^{2} b^{2} c^{2} - 24 \, A^{2} a b c d + 6 \, A^{2} a^{2} d^{2} + 4 \, {\left (B^{2} b^{2} d^{2} n^{2} x^{2} + 2 \, B^{2} b^{2} c d n^{2} x + B^{2} b^{2} c^{2} n^{2}\right )} \log \left (\frac {b x + a}{d x + c}\right )^{3} + 3 \, {\left (15 \, B^{2} b^{2} c^{2} - 16 \, B^{2} a b c d + B^{2} a^{2} d^{2}\right )} n^{2} + 6 \, {\left (3 \, B^{2} b^{2} c^{2} - 4 \, B^{2} a b c d + B^{2} a^{2} d^{2} + 2 \, {\left (B^{2} b^{2} c d - B^{2} a b d^{2}\right )} x + 2 \, {\left (B^{2} b^{2} d^{2} x^{2} + 2 \, B^{2} b^{2} c d x + B^{2} b^{2} c^{2}\right )} \log \left (\frac {b x + a}{d x + c}\right )\right )} \log \left (e\right )^{2} + 6 \, {\left (2 \, A B b^{2} c^{2} n - {\left (4 \, B^{2} a b c d - B^{2} a^{2} d^{2}\right )} n^{2} - {\left (3 \, B^{2} b^{2} d^{2} n^{2} - 2 \, A B b^{2} d^{2} n\right )} x^{2} + 2 \, {\left (2 \, A B b^{2} c d n - {\left (2 \, B^{2} b^{2} c d + B^{2} a b d^{2}\right )} n^{2}\right )} x\right )} \log \left (\frac {b x + a}{d x + c}\right )^{2} - 6 \, {\left (7 \, A B b^{2} c^{2} - 8 \, A B a b c d + A B a^{2} d^{2}\right )} n + 6 \, {\left (2 \, A^{2} b^{2} c d - 2 \, A^{2} a b d^{2} + 7 \, {\left (B^{2} b^{2} c d - B^{2} a b d^{2}\right )} n^{2} - 6 \, {\left (A B b^{2} c d - A B a b d^{2}\right )} n\right )} x + 6 \, {\left (6 \, A B b^{2} c^{2} - 8 \, A B a b c d + 2 \, A B a^{2} d^{2} + 2 \, {\left (B^{2} b^{2} d^{2} n x^{2} + 2 \, B^{2} b^{2} c d n x + B^{2} b^{2} c^{2} n\right )} \log \left (\frac {b x + a}{d x + c}\right )^{2} - {\left (7 \, B^{2} b^{2} c^{2} - 8 \, B^{2} a b c d + B^{2} a^{2} d^{2}\right )} n + 2 \, {\left (2 \, A B b^{2} c d - 2 \, A B a b d^{2} - 3 \, {\left (B^{2} b^{2} c d - B^{2} a b d^{2}\right )} n\right )} x + 2 \, {\left (2 \, A B b^{2} c^{2} - {\left (3 \, B^{2} b^{2} d^{2} n - 2 \, A B b^{2} d^{2}\right )} x^{2} - {\left (4 \, B^{2} a b c d - B^{2} a^{2} d^{2}\right )} n + 2 \, {\left (2 \, A B b^{2} c d - {\left (2 \, B^{2} b^{2} c d + B^{2} a b d^{2}\right )} n\right )} x\right )} \log \left (\frac {b x + a}{d x + c}\right )\right )} \log \left (e\right ) + 6 \, {\left (2 \, A^{2} b^{2} c^{2} + {\left (8 \, B^{2} a b c d - B^{2} a^{2} d^{2}\right )} n^{2} + {\left (7 \, B^{2} b^{2} d^{2} n^{2} - 6 \, A B b^{2} d^{2} n + 2 \, A^{2} b^{2} d^{2}\right )} x^{2} - 2 \, {\left (4 \, A B a b c d - A B a^{2} d^{2}\right )} n + 2 \, {\left (2 \, A^{2} b^{2} c d + {\left (4 \, B^{2} b^{2} c d + 3 \, B^{2} a b d^{2}\right )} n^{2} - 2 \, {\left (2 \, A B b^{2} c d + A B a b d^{2}\right )} n\right )} x\right )} \log \left (\frac {b x + a}{d x + c}\right )}{12 \, {\left ({\left (b^{3} c^{3} d^{2} - 3 \, a b^{2} c^{2} d^{3} + 3 \, a^{2} b c d^{4} - a^{3} d^{5}\right )} g i^{3} x^{2} + 2 \, {\left (b^{3} c^{4} d - 3 \, a b^{2} c^{3} d^{2} + 3 \, a^{2} b c^{2} d^{3} - a^{3} c d^{4}\right )} g i^{3} x + {\left (b^{3} c^{5} - 3 \, a b^{2} c^{4} d + 3 \, a^{2} b c^{3} d^{2} - a^{3} c^{2} d^{3}\right )} g i^{3}\right )}} \]
1/12*(18*A^2*b^2*c^2 - 24*A^2*a*b*c*d + 6*A^2*a^2*d^2 + 4*(B^2*b^2*d^2*n^2 *x^2 + 2*B^2*b^2*c*d*n^2*x + B^2*b^2*c^2*n^2)*log((b*x + a)/(d*x + c))^3 + 3*(15*B^2*b^2*c^2 - 16*B^2*a*b*c*d + B^2*a^2*d^2)*n^2 + 6*(3*B^2*b^2*c^2 - 4*B^2*a*b*c*d + B^2*a^2*d^2 + 2*(B^2*b^2*c*d - B^2*a*b*d^2)*x + 2*(B^2*b ^2*d^2*x^2 + 2*B^2*b^2*c*d*x + B^2*b^2*c^2)*log((b*x + a)/(d*x + c)))*log( e)^2 + 6*(2*A*B*b^2*c^2*n - (4*B^2*a*b*c*d - B^2*a^2*d^2)*n^2 - (3*B^2*b^2 *d^2*n^2 - 2*A*B*b^2*d^2*n)*x^2 + 2*(2*A*B*b^2*c*d*n - (2*B^2*b^2*c*d + B^ 2*a*b*d^2)*n^2)*x)*log((b*x + a)/(d*x + c))^2 - 6*(7*A*B*b^2*c^2 - 8*A*B*a *b*c*d + A*B*a^2*d^2)*n + 6*(2*A^2*b^2*c*d - 2*A^2*a*b*d^2 + 7*(B^2*b^2*c* d - B^2*a*b*d^2)*n^2 - 6*(A*B*b^2*c*d - A*B*a*b*d^2)*n)*x + 6*(6*A*B*b^2*c ^2 - 8*A*B*a*b*c*d + 2*A*B*a^2*d^2 + 2*(B^2*b^2*d^2*n*x^2 + 2*B^2*b^2*c*d* n*x + B^2*b^2*c^2*n)*log((b*x + a)/(d*x + c))^2 - (7*B^2*b^2*c^2 - 8*B^2*a *b*c*d + B^2*a^2*d^2)*n + 2*(2*A*B*b^2*c*d - 2*A*B*a*b*d^2 - 3*(B^2*b^2*c* d - B^2*a*b*d^2)*n)*x + 2*(2*A*B*b^2*c^2 - (3*B^2*b^2*d^2*n - 2*A*B*b^2*d^ 2)*x^2 - (4*B^2*a*b*c*d - B^2*a^2*d^2)*n + 2*(2*A*B*b^2*c*d - (2*B^2*b^2*c *d + B^2*a*b*d^2)*n)*x)*log((b*x + a)/(d*x + c)))*log(e) + 6*(2*A^2*b^2*c^ 2 + (8*B^2*a*b*c*d - B^2*a^2*d^2)*n^2 + (7*B^2*b^2*d^2*n^2 - 6*A*B*b^2*d^2 *n + 2*A^2*b^2*d^2)*x^2 - 2*(4*A*B*a*b*c*d - A*B*a^2*d^2)*n + 2*(2*A^2*b^2 *c*d + (4*B^2*b^2*c*d + 3*B^2*a*b*d^2)*n^2 - 2*(2*A*B*b^2*c*d + A*B*a*b*d^ 2)*n)*x)*log((b*x + a)/(d*x + c)))/((b^3*c^3*d^2 - 3*a*b^2*c^2*d^3 + 3*...
Timed out. \[ \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x) (c i+d i x)^3} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 2126 vs. \(2 (394) = 788\).
Time = 0.33 (sec) , antiderivative size = 2126, normalized size of antiderivative = 5.29 \[ \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x) (c i+d i x)^3} \, dx=\text {Too large to display} \]
1/2*B^2*((2*b*d*x + 3*b*c - a*d)/((b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d^4)*g* i^3*x^2 + 2*(b^2*c^3*d - 2*a*b*c^2*d^2 + a^2*c*d^3)*g*i^3*x + (b^2*c^4 - 2 *a*b*c^3*d + a^2*c^2*d^2)*g*i^3) + 2*b^2*log(b*x + a)/((b^3*c^3 - 3*a*b^2* c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*g*i^3) - 2*b^2*log(d*x + c)/((b^3*c^3 - 3 *a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*g*i^3))*log(e*(b*x/(d*x + c) + a/( d*x + c))^n)^2 + A*B*((2*b*d*x + 3*b*c - a*d)/((b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d^4)*g*i^3*x^2 + 2*(b^2*c^3*d - 2*a*b*c^2*d^2 + a^2*c*d^3)*g*i^3*x + (b^2*c^4 - 2*a*b*c^3*d + a^2*c^2*d^2)*g*i^3) + 2*b^2*log(b*x + a)/((b^3*c ^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*g*i^3) - 2*b^2*log(d*x + c)/ ((b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*g*i^3))*log(e*(b*x/(d *x + c) + a/(d*x + c))^n) + 1/12*((45*b^2*c^2 - 48*a*b*c*d + 3*a^2*d^2 + 4 *(b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2)*log(b*x + a)^3 - 4*(b^2*d^2*x^2 + 2 *b^2*c*d*x + b^2*c^2)*log(d*x + c)^3 + 18*(b^2*d^2*x^2 + 2*b^2*c*d*x + b^2 *c^2)*log(b*x + a)^2 + 6*(3*b^2*d^2*x^2 + 6*b^2*c*d*x + 3*b^2*c^2 + 2*(b^2 *d^2*x^2 + 2*b^2*c*d*x + b^2*c^2)*log(b*x + a))*log(d*x + c)^2 + 42*(b^2*c *d - a*b*d^2)*x + 42*(b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2)*log(b*x + a) - 6*(7*b^2*d^2*x^2 + 14*b^2*c*d*x + 7*b^2*c^2 + 2*(b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2)*log(b*x + a)^2 + 6*(b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2)*log(b *x + a))*log(d*x + c))*n^2/(b^3*c^5*g*i^3 - 3*a*b^2*c^4*d*g*i^3 + 3*a^2*b* c^3*d^2*g*i^3 - a^3*c^2*d^3*g*i^3 + (b^3*c^3*d^2*g*i^3 - 3*a*b^2*c^2*d^...
Time = 1.19 (sec) , antiderivative size = 747, normalized size of antiderivative = 1.86 \[ \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x) (c i+d i x)^3} \, dx=\frac {1}{12} \, {\left (\frac {4 \, B^{2} b^{2} n^{2} \log \left (\frac {b x + a}{d x + c}\right )^{3}}{b^{2} c^{2} g i^{3} - 2 \, a b c d g i^{3} + a^{2} d^{2} g i^{3}} - 6 \, {\left (\frac {4 \, {\left (b x + a\right )} B^{2} b d n^{2}}{{\left (b^{2} c^{2} g i^{3} - 2 \, a b c d g i^{3} + a^{2} d^{2} g i^{3}\right )} {\left (d x + c\right )}} - \frac {{\left (b x + a\right )}^{2} B^{2} d^{2} n^{2}}{{\left (b^{2} c^{2} g i^{3} - 2 \, a b c d g i^{3} + a^{2} d^{2} g i^{3}\right )} {\left (d x + c\right )}^{2}} - \frac {2 \, {\left (B^{2} b^{2} n \log \left (e\right ) + A B b^{2} n\right )}}{b^{2} c^{2} g i^{3} - 2 \, a b c d g i^{3} + a^{2} d^{2} g i^{3}}\right )} \log \left (\frac {b x + a}{d x + c}\right )^{2} - 6 \, {\left (\frac {{\left (B^{2} d^{2} n^{2} - 2 \, B^{2} d^{2} n \log \left (e\right ) - 2 \, A B d^{2} n\right )} {\left (b x + a\right )}^{2}}{{\left (b^{2} c^{2} g i^{3} - 2 \, a b c d g i^{3} + a^{2} d^{2} g i^{3}\right )} {\left (d x + c\right )}^{2}} - \frac {8 \, {\left (B^{2} b d n^{2} - B^{2} b d n \log \left (e\right ) - A B b d n\right )} {\left (b x + a\right )}}{{\left (b^{2} c^{2} g i^{3} - 2 \, a b c d g i^{3} + a^{2} d^{2} g i^{3}\right )} {\left (d x + c\right )}}\right )} \log \left (\frac {b x + a}{d x + c}\right ) + \frac {12 \, {\left (B^{2} b^{2} \log \left (e\right )^{2} + 2 \, A B b^{2} \log \left (e\right ) + A^{2} b^{2}\right )} \log \left (\frac {b x + a}{d x + c}\right )}{b^{2} c^{2} g i^{3} - 2 \, a b c d g i^{3} + a^{2} d^{2} g i^{3}} + \frac {3 \, {\left (B^{2} d^{2} n^{2} - 2 \, B^{2} d^{2} n \log \left (e\right ) + 2 \, B^{2} d^{2} \log \left (e\right )^{2} - 2 \, A B d^{2} n + 4 \, A B d^{2} \log \left (e\right ) + 2 \, A^{2} d^{2}\right )} {\left (b x + a\right )}^{2}}{{\left (b^{2} c^{2} g i^{3} - 2 \, a b c d g i^{3} + a^{2} d^{2} g i^{3}\right )} {\left (d x + c\right )}^{2}} - \frac {24 \, {\left (2 \, B^{2} b d n^{2} - 2 \, B^{2} b d n \log \left (e\right ) + B^{2} b d \log \left (e\right )^{2} - 2 \, A B b d n + 2 \, A B b d \log \left (e\right ) + A^{2} b d\right )} {\left (b x + a\right )}}{{\left (b^{2} c^{2} g i^{3} - 2 \, a b c d g i^{3} + a^{2} d^{2} g i^{3}\right )} {\left (d x + c\right )}}\right )} {\left (\frac {b c}{{\left (b c - a d\right )}^{2}} - \frac {a d}{{\left (b c - a d\right )}^{2}}\right )} \]
1/12*(4*B^2*b^2*n^2*log((b*x + a)/(d*x + c))^3/(b^2*c^2*g*i^3 - 2*a*b*c*d* g*i^3 + a^2*d^2*g*i^3) - 6*(4*(b*x + a)*B^2*b*d*n^2/((b^2*c^2*g*i^3 - 2*a* b*c*d*g*i^3 + a^2*d^2*g*i^3)*(d*x + c)) - (b*x + a)^2*B^2*d^2*n^2/((b^2*c^ 2*g*i^3 - 2*a*b*c*d*g*i^3 + a^2*d^2*g*i^3)*(d*x + c)^2) - 2*(B^2*b^2*n*log (e) + A*B*b^2*n)/(b^2*c^2*g*i^3 - 2*a*b*c*d*g*i^3 + a^2*d^2*g*i^3))*log((b *x + a)/(d*x + c))^2 - 6*((B^2*d^2*n^2 - 2*B^2*d^2*n*log(e) - 2*A*B*d^2*n) *(b*x + a)^2/((b^2*c^2*g*i^3 - 2*a*b*c*d*g*i^3 + a^2*d^2*g*i^3)*(d*x + c)^ 2) - 8*(B^2*b*d*n^2 - B^2*b*d*n*log(e) - A*B*b*d*n)*(b*x + a)/((b^2*c^2*g* i^3 - 2*a*b*c*d*g*i^3 + a^2*d^2*g*i^3)*(d*x + c)))*log((b*x + a)/(d*x + c) ) + 12*(B^2*b^2*log(e)^2 + 2*A*B*b^2*log(e) + A^2*b^2)*log((b*x + a)/(d*x + c))/(b^2*c^2*g*i^3 - 2*a*b*c*d*g*i^3 + a^2*d^2*g*i^3) + 3*(B^2*d^2*n^2 - 2*B^2*d^2*n*log(e) + 2*B^2*d^2*log(e)^2 - 2*A*B*d^2*n + 4*A*B*d^2*log(e) + 2*A^2*d^2)*(b*x + a)^2/((b^2*c^2*g*i^3 - 2*a*b*c*d*g*i^3 + a^2*d^2*g*i^3 )*(d*x + c)^2) - 24*(2*B^2*b*d*n^2 - 2*B^2*b*d*n*log(e) + B^2*b*d*log(e)^2 - 2*A*B*b*d*n + 2*A*B*b*d*log(e) + A^2*b*d)*(b*x + a)/((b^2*c^2*g*i^3 - 2 *a*b*c*d*g*i^3 + a^2*d^2*g*i^3)*(d*x + c)))*(b*c/(b*c - a*d)^2 - a*d/(b*c - a*d)^2)
Time = 5.35 (sec) , antiderivative size = 1007, normalized size of antiderivative = 2.50 \[ \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x) (c i+d i x)^3} \, dx={\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )}^2\,\left (\frac {b^2\,\left (3\,B^2\,n-2\,A\,B\right )}{2\,g\,i^3\,n\,\left (a\,d-b\,c\right )\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}+\frac {B^2\,b^2\,\left (\frac {c\,g\,i^3\,n\,\left (a\,d-b\,c\right )}{2\,b}-\frac {g\,i^3\,n\,\left (a\,d-b\,c\right )\,\left (a\,d-2\,b\,c\right )}{2\,b^2}+\frac {d\,g\,i^3\,n\,x\,\left (a\,d-b\,c\right )}{b}\right )}{g\,i^3\,n\,\left (a\,d-b\,c\right )\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )\,\left (g\,c^2\,i^3+2\,g\,c\,d\,i^3\,x+g\,d^2\,i^3\,x^2\right )}\right )-\frac {\frac {2\,A^2\,a\,d-6\,A^2\,b\,c+B^2\,a\,d\,n^2-15\,B^2\,b\,c\,n^2-2\,A\,B\,a\,d\,n+14\,A\,B\,b\,c\,n}{2\,\left (a\,d-b\,c\right )}-\frac {x\,\left (2\,b\,d\,A^2-6\,b\,d\,A\,B\,n+7\,b\,d\,B^2\,n^2\right )}{a\,d-b\,c}}{x^2\,\left (2\,a\,d^3\,g\,i^3-2\,b\,c\,d^2\,g\,i^3\right )+x\,\left (4\,a\,c\,d^2\,g\,i^3-4\,b\,c^2\,d\,g\,i^3\right )-2\,b\,c^3\,g\,i^3+2\,a\,c^2\,d\,g\,i^3}-\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\,\left (\frac {B^2\,n}{x^2\,\left (a\,d^3\,g\,i^3-b\,c\,d^2\,g\,i^3\right )+x\,\left (2\,a\,c\,d^2\,g\,i^3-2\,b\,c^2\,d\,g\,i^3\right )-b\,c^3\,g\,i^3+a\,c^2\,d\,g\,i^3}+\frac {b^2\,\left (3\,B^2\,n-2\,A\,B\right )\,\left (\frac {c\,g\,i^3\,n\,{\left (a\,d-b\,c\right )}^2}{2\,b}-\frac {g\,i^3\,n\,{\left (a\,d-b\,c\right )}^2\,\left (a\,d-2\,b\,c\right )}{2\,b^2}+\frac {d\,g\,i^3\,n\,x\,{\left (a\,d-b\,c\right )}^2}{b}\right )}{g\,i^3\,n\,\left (a\,d-b\,c\right )\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )\,\left (x^2\,\left (a\,d^3\,g\,i^3-b\,c\,d^2\,g\,i^3\right )+x\,\left (2\,a\,c\,d^2\,g\,i^3-2\,b\,c^2\,d\,g\,i^3\right )-b\,c^3\,g\,i^3+a\,c^2\,d\,g\,i^3\right )}\right )-\frac {B^2\,b^2\,{\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )}^3}{3\,g\,i^3\,n\,\left (a\,d-b\,c\right )\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}+\frac {b^2\,\mathrm {atan}\left (\frac {b^2\,\left (\frac {g\,a^3\,d^3\,i^3-g\,a^2\,b\,c\,d^2\,i^3-g\,a\,b^2\,c^2\,d\,i^3+g\,b^3\,c^3\,i^3}{g\,a^2\,d^2\,i^3-2\,g\,a\,b\,c\,d\,i^3+g\,b^2\,c^2\,i^3}+2\,b\,d\,x\right )\,\left (A^2-3\,A\,B\,n+\frac {7\,B^2\,n^2}{2}\right )\,\left (g\,a^2\,d^2\,i^3-2\,g\,a\,b\,c\,d\,i^3+g\,b^2\,c^2\,i^3\right )\,2{}\mathrm {i}}{g\,i^3\,{\left (a\,d-b\,c\right )}^3\,\left (2\,A^2\,b^2-6\,A\,B\,b^2\,n+7\,B^2\,b^2\,n^2\right )}\right )\,\left (A^2-3\,A\,B\,n+\frac {7\,B^2\,n^2}{2}\right )\,2{}\mathrm {i}}{g\,i^3\,{\left (a\,d-b\,c\right )}^3} \]
log(e*((a + b*x)/(c + d*x))^n)^2*((b^2*(3*B^2*n - 2*A*B))/(2*g*i^3*n*(a*d - b*c)*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)) + (B^2*b^2*((c*g*i^3*n*(a*d - b*c) )/(2*b) - (g*i^3*n*(a*d - b*c)*(a*d - 2*b*c))/(2*b^2) + (d*g*i^3*n*x*(a*d - b*c))/b))/(g*i^3*n*(a*d - b*c)*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)*(c^2*g*i^ 3 + d^2*g*i^3*x^2 + 2*c*d*g*i^3*x))) - ((2*A^2*a*d - 6*A^2*b*c + B^2*a*d*n ^2 - 15*B^2*b*c*n^2 - 2*A*B*a*d*n + 14*A*B*b*c*n)/(2*(a*d - b*c)) - (x*(2* A^2*b*d + 7*B^2*b*d*n^2 - 6*A*B*b*d*n))/(a*d - b*c))/(x^2*(2*a*d^3*g*i^3 - 2*b*c*d^2*g*i^3) + x*(4*a*c*d^2*g*i^3 - 4*b*c^2*d*g*i^3) - 2*b*c^3*g*i^3 + 2*a*c^2*d*g*i^3) - log(e*((a + b*x)/(c + d*x))^n)*((B^2*n)/(x^2*(a*d^3*g *i^3 - b*c*d^2*g*i^3) + x*(2*a*c*d^2*g*i^3 - 2*b*c^2*d*g*i^3) - b*c^3*g*i^ 3 + a*c^2*d*g*i^3) + (b^2*(3*B^2*n - 2*A*B)*((c*g*i^3*n*(a*d - b*c)^2)/(2* b) - (g*i^3*n*(a*d - b*c)^2*(a*d - 2*b*c))/(2*b^2) + (d*g*i^3*n*x*(a*d - b *c)^2)/b))/(g*i^3*n*(a*d - b*c)*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)*(x^2*(a*d^ 3*g*i^3 - b*c*d^2*g*i^3) + x*(2*a*c*d^2*g*i^3 - 2*b*c^2*d*g*i^3) - b*c^3*g *i^3 + a*c^2*d*g*i^3))) + (b^2*atan((b^2*((a^3*d^3*g*i^3 + b^3*c^3*g*i^3 - a*b^2*c^2*d*g*i^3 - a^2*b*c*d^2*g*i^3)/(a^2*d^2*g*i^3 + b^2*c^2*g*i^3 - 2 *a*b*c*d*g*i^3) + 2*b*d*x)*(A^2 + (7*B^2*n^2)/2 - 3*A*B*n)*(a^2*d^2*g*i^3 + b^2*c^2*g*i^3 - 2*a*b*c*d*g*i^3)*2i)/(g*i^3*(a*d - b*c)^3*(2*A^2*b^2 + 7 *B^2*b^2*n^2 - 6*A*B*b^2*n)))*(A^2 + (7*B^2*n^2)/2 - 3*A*B*n)*2i)/(g*i^3*( a*d - b*c)^3) - (B^2*b^2*log(e*((a + b*x)/(c + d*x))^n)^3)/(3*g*i^3*n*(...