Integrand size = 33, antiderivative size = 830 \[ \int \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{(a+b x)^5} \, dx=\frac {6 B^3 d^3 n^3 (c+d x)}{(b c-a d)^4 (a+b x)}-\frac {9 b B^3 d^2 n^3 (c+d x)^2}{8 (b c-a d)^4 (a+b x)^2}+\frac {2 b^2 B^3 d n^3 (c+d x)^3}{9 (b c-a d)^4 (a+b x)^3}-\frac {3 b^3 B^3 n^3 (c+d x)^4}{128 (b c-a d)^4 (a+b x)^4}+\frac {6 B^2 d^3 n^2 (c+d x) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )}{(b c-a d)^4 (a+b x)}-\frac {9 b B^2 d^2 n^2 (c+d x)^2 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )}{4 (b c-a d)^4 (a+b x)^2}+\frac {2 b^2 B^2 d n^2 (c+d x)^3 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )}{3 (b c-a d)^4 (a+b x)^3}-\frac {3 b^3 B^2 n^2 (c+d x)^4 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )}{32 (b c-a d)^4 (a+b x)^4}+\frac {3 B d^3 n (c+d x) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2}{(b c-a d)^4 (a+b x)}-\frac {9 b B d^2 n (c+d x)^2 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2}{4 (b c-a d)^4 (a+b x)^2}+\frac {b^2 B d n (c+d x)^3 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2}{(b c-a d)^4 (a+b x)^3}-\frac {3 b^3 B n (c+d x)^4 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2}{16 (b c-a d)^4 (a+b x)^4}+\frac {d^3 (c+d x) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{(b c-a d)^4 (a+b x)}-\frac {3 b d^2 (c+d x)^2 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{2 (b c-a d)^4 (a+b x)^2}+\frac {b^2 d (c+d x)^3 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{(b c-a d)^4 (a+b x)^3}-\frac {b^3 (c+d x)^4 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{4 (b c-a d)^4 (a+b x)^4} \]
6*B^3*d^3*n^3*(d*x+c)/(-a*d+b*c)^4/(b*x+a)-9/8*b*B^3*d^2*n^3*(d*x+c)^2/(-a *d+b*c)^4/(b*x+a)^2+2/9*b^2*B^3*d*n^3*(d*x+c)^3/(-a*d+b*c)^4/(b*x+a)^3-3/1 28*b^3*B^3*n^3*(d*x+c)^4/(-a*d+b*c)^4/(b*x+a)^4+6*B^2*d^3*n^2*(d*x+c)*(A+B *ln(e*(b*x+a)^n/((d*x+c)^n)))/(-a*d+b*c)^4/(b*x+a)-9/4*b*B^2*d^2*n^2*(d*x+ c)^2*(A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))/(-a*d+b*c)^4/(b*x+a)^2+2/3*b^2*B^2* d*n^2*(d*x+c)^3*(A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))/(-a*d+b*c)^4/(b*x+a)^3-3 /32*b^3*B^2*n^2*(d*x+c)^4*(A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))/(-a*d+b*c)^4/( b*x+a)^4+3*B*d^3*n*(d*x+c)*(A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))^2/(-a*d+b*c)^ 4/(b*x+a)-9/4*b*B*d^2*n*(d*x+c)^2*(A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))^2/(-a* d+b*c)^4/(b*x+a)^2+b^2*B*d*n*(d*x+c)^3*(A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))^2 /(-a*d+b*c)^4/(b*x+a)^3-3/16*b^3*B*n*(d*x+c)^4*(A+B*ln(e*(b*x+a)^n/((d*x+c )^n)))^2/(-a*d+b*c)^4/(b*x+a)^4+d^3*(d*x+c)*(A+B*ln(e*(b*x+a)^n/((d*x+c)^n )))^3/(-a*d+b*c)^4/(b*x+a)-3/2*b*d^2*(d*x+c)^2*(A+B*ln(e*(b*x+a)^n/((d*x+c )^n)))^3/(-a*d+b*c)^4/(b*x+a)^2+b^2*d*(d*x+c)^3*(A+B*ln(e*(b*x+a)^n/((d*x+ c)^n)))^3/(-a*d+b*c)^4/(b*x+a)^3-1/4*b^3*(d*x+c)^4*(A+B*ln(e*(b*x+a)^n/((d *x+c)^n)))^3/(-a*d+b*c)^4/(b*x+a)^4
Time = 1.25 (sec) , antiderivative size = 1370, normalized size of antiderivative = 1.65 \[ \int \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{(a+b x)^5} \, dx=-\frac {-288 B^3 d^4 n^3 (a+b x)^4 \log ^3(a+b x)+288 B^3 d^4 n^3 (a+b x)^4 \log ^3(c+d x)+72 B^2 d^4 n^2 (a+b x)^4 \log ^2(c+d x) \left (12 A+25 B n+12 B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )+72 B^2 d^4 n^2 (a+b x)^4 \log ^2(a+b x) \left (12 A+25 B n+12 B n \log (c+d x)+12 B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )+12 B d^4 n (a+b x)^4 \log (c+d x) \left (72 A^2+300 A B n+415 B^2 n^2+12 B (12 A+25 B n) \log \left (e (a+b x)^n (c+d x)^{-n}\right )+72 B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )\right )+(b c-a d) \left (288 A^3 b^3 c^3-864 a A^3 b^2 c^2 d+864 a^2 A^3 b c d^2-288 a^3 A^3 d^3+216 A^2 b^3 B c^3 n-936 a A^2 b^2 B c^2 d n+1656 a^2 A^2 b B c d^2 n-1800 a^3 A^2 B d^3 n+108 A b^3 B^2 c^3 n^2-660 a A b^2 B^2 c^2 d n^2+1932 a^2 A b B^2 c d^2 n^2-4980 a^3 A B^2 d^3 n^2+27 b^3 B^3 c^3 n^3-229 a b^2 B^3 c^2 d n^3+1067 a^2 b B^3 c d^2 n^3-5845 a^3 B^3 d^3 n^3-288 A^2 b^3 B c^2 d n x+1440 a A^2 b^2 B c d^2 n x-3744 a^2 A^2 b B d^3 n x-336 A b^3 B^2 c^2 d n^2 x+2544 a A b^2 B^2 c d^2 n^2 x-13008 a^2 A b B^2 d^3 n^2 x-148 b^3 B^3 c^2 d n^3 x+1676 a b^2 B^3 c d^2 n^3 x-16468 a^2 b B^3 d^3 n^3 x+432 A^2 b^3 B c d^2 n x^2-3024 a A^2 b^2 B d^3 n x^2+936 A b^3 B^2 c d^2 n^2 x^2-11736 a A b^2 B^2 d^3 n^2 x^2+690 b^3 B^3 c d^2 n^3 x^2-15630 a b^2 B^3 d^3 n^3 x^2-864 A^2 b^3 B d^3 n x^3-3600 A b^3 B^2 d^3 n^2 x^3-4980 b^3 B^3 d^3 n^3 x^3+12 B \left (72 A^2 (b c-a d)^3+B^2 n^2 \left (-415 a^3 d^3+a^2 b d^2 (161 c-1084 d x)+a b^2 d \left (-55 c^2+212 c d x-978 d^2 x^2\right )+b^3 \left (9 c^3-28 c^2 d x+78 c d^2 x^2-300 d^3 x^3\right )\right )+12 A B n \left (-25 a^3 d^3+a^2 b d^2 (23 c-52 d x)+a b^2 d \left (-13 c^2+20 c d x-42 d^2 x^2\right )+b^3 \left (3 c^3-4 c^2 d x+6 c d^2 x^2-12 d^3 x^3\right )\right )\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )+72 B^2 \left (12 A (b c-a d)^3+B n \left (-25 a^3 d^3+a^2 b d^2 (23 c-52 d x)+a b^2 d \left (-13 c^2+20 c d x-42 d^2 x^2\right )+b^3 \left (3 c^3-4 c^2 d x+6 c d^2 x^2-12 d^3 x^3\right )\right )\right ) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )+288 B^3 (b c-a d)^3 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )\right )-12 B d^4 n (a+b x)^4 \log (a+b x) \left (72 A^2+300 A B n+415 B^2 n^2+72 B^2 n^2 \log ^2(c+d x)+12 B (12 A+25 B n) \log \left (e (a+b x)^n (c+d x)^{-n}\right )+72 B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )+12 B n \log (c+d x) \left (12 A+25 B n+12 B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )\right )}{1152 b (b c-a d)^4 (a+b x)^4} \]
-1/1152*(-288*B^3*d^4*n^3*(a + b*x)^4*Log[a + b*x]^3 + 288*B^3*d^4*n^3*(a + b*x)^4*Log[c + d*x]^3 + 72*B^2*d^4*n^2*(a + b*x)^4*Log[c + d*x]^2*(12*A + 25*B*n + 12*B*Log[(e*(a + b*x)^n)/(c + d*x)^n]) + 72*B^2*d^4*n^2*(a + b* x)^4*Log[a + b*x]^2*(12*A + 25*B*n + 12*B*n*Log[c + d*x] + 12*B*Log[(e*(a + b*x)^n)/(c + d*x)^n]) + 12*B*d^4*n*(a + b*x)^4*Log[c + d*x]*(72*A^2 + 30 0*A*B*n + 415*B^2*n^2 + 12*B*(12*A + 25*B*n)*Log[(e*(a + b*x)^n)/(c + d*x) ^n] + 72*B^2*Log[(e*(a + b*x)^n)/(c + d*x)^n]^2) + (b*c - a*d)*(288*A^3*b^ 3*c^3 - 864*a*A^3*b^2*c^2*d + 864*a^2*A^3*b*c*d^2 - 288*a^3*A^3*d^3 + 216* A^2*b^3*B*c^3*n - 936*a*A^2*b^2*B*c^2*d*n + 1656*a^2*A^2*b*B*c*d^2*n - 180 0*a^3*A^2*B*d^3*n + 108*A*b^3*B^2*c^3*n^2 - 660*a*A*b^2*B^2*c^2*d*n^2 + 19 32*a^2*A*b*B^2*c*d^2*n^2 - 4980*a^3*A*B^2*d^3*n^2 + 27*b^3*B^3*c^3*n^3 - 2 29*a*b^2*B^3*c^2*d*n^3 + 1067*a^2*b*B^3*c*d^2*n^3 - 5845*a^3*B^3*d^3*n^3 - 288*A^2*b^3*B*c^2*d*n*x + 1440*a*A^2*b^2*B*c*d^2*n*x - 3744*a^2*A^2*b*B*d ^3*n*x - 336*A*b^3*B^2*c^2*d*n^2*x + 2544*a*A*b^2*B^2*c*d^2*n^2*x - 13008* a^2*A*b*B^2*d^3*n^2*x - 148*b^3*B^3*c^2*d*n^3*x + 1676*a*b^2*B^3*c*d^2*n^3 *x - 16468*a^2*b*B^3*d^3*n^3*x + 432*A^2*b^3*B*c*d^2*n*x^2 - 3024*a*A^2*b^ 2*B*d^3*n*x^2 + 936*A*b^3*B^2*c*d^2*n^2*x^2 - 11736*a*A*b^2*B^2*d^3*n^2*x^ 2 + 690*b^3*B^3*c*d^2*n^3*x^2 - 15630*a*b^2*B^3*d^3*n^3*x^2 - 864*A^2*b^3* B*d^3*n*x^3 - 3600*A*b^3*B^2*d^3*n^2*x^3 - 4980*b^3*B^3*d^3*n^3*x^3 + 12*B *(72*A^2*(b*c - a*d)^3 + B^2*n^2*(-415*a^3*d^3 + a^2*b*d^2*(161*c - 108...
Time = 0.79 (sec) , antiderivative size = 669, normalized size of antiderivative = 0.81, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {2973, 2949, 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.
\(\displaystyle \int \frac {\left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^3}{(a+b x)^5} \, dx\) |
\(\Big \downarrow \) 2973 |
\(\displaystyle \int \frac {\left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^3}{(a+b x)^5}dx\) |
\(\Big \downarrow \) 2949 |
\(\displaystyle \frac {\int \frac {(c+d x)^5 \left (b-\frac {d (a+b x)}{c+d x}\right )^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^3}{(a+b x)^5}d\frac {a+b x}{c+d x}}{(b c-a d)^4}\) |
\(\Big \downarrow \) 2795 |
\(\displaystyle \frac {\int \left (\frac {b^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^3 (c+d x)^5}{(a+b x)^5}-\frac {3 b^2 d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^3 (c+d x)^4}{(a+b x)^4}+\frac {3 b d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^3 (c+d x)^3}{(a+b x)^3}-\frac {d^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^3 (c+d x)^2}{(a+b x)^2}\right )d\frac {a+b x}{c+d x}}{(b c-a d)^4}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {-\frac {3 b^3 B^2 n^2 (c+d x)^4 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{32 (a+b x)^4}-\frac {b^3 (c+d x)^4 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^3}{4 (a+b x)^4}-\frac {3 b^3 B n (c+d x)^4 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{16 (a+b x)^4}+\frac {2 b^2 B^2 d n^2 (c+d x)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{3 (a+b x)^3}+\frac {b^2 d (c+d x)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^3}{(a+b x)^3}+\frac {b^2 B d n (c+d x)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{(a+b x)^3}+\frac {6 B^2 d^3 n^2 (c+d x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{a+b x}-\frac {9 b B^2 d^2 n^2 (c+d x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{4 (a+b x)^2}+\frac {d^3 (c+d x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^3}{a+b x}+\frac {3 B d^3 n (c+d x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{a+b x}-\frac {3 b d^2 (c+d x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^3}{2 (a+b x)^2}-\frac {9 b B d^2 n (c+d x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{4 (a+b x)^2}-\frac {3 b^3 B^3 n^3 (c+d x)^4}{128 (a+b x)^4}+\frac {2 b^2 B^3 d n^3 (c+d x)^3}{9 (a+b x)^3}+\frac {6 B^3 d^3 n^3 (c+d x)}{a+b x}-\frac {9 b B^3 d^2 n^3 (c+d x)^2}{8 (a+b x)^2}}{(b c-a d)^4}\) |
((6*B^3*d^3*n^3*(c + d*x))/(a + b*x) - (9*b*B^3*d^2*n^3*(c + d*x)^2)/(8*(a + b*x)^2) + (2*b^2*B^3*d*n^3*(c + d*x)^3)/(9*(a + b*x)^3) - (3*b^3*B^3*n^ 3*(c + d*x)^4)/(128*(a + b*x)^4) + (6*B^2*d^3*n^2*(c + d*x)*(A + B*Log[e*( (a + b*x)/(c + d*x))^n]))/(a + b*x) - (9*b*B^2*d^2*n^2*(c + d*x)^2*(A + B* Log[e*((a + b*x)/(c + d*x))^n]))/(4*(a + b*x)^2) + (2*b^2*B^2*d*n^2*(c + d *x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(3*(a + b*x)^3) - (3*b^3*B^2 *n^2*(c + d*x)^4*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(32*(a + b*x)^4) + (3*B*d^3*n*(c + d*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(a + b*x) - (9*b*B*d^2*n*(c + d*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(4*( a + b*x)^2) + (b^2*B*d*n*(c + d*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n] )^2)/(a + b*x)^3 - (3*b^3*B*n*(c + d*x)^4*(A + B*Log[e*((a + b*x)/(c + d*x ))^n])^2)/(16*(a + b*x)^4) + (d^3*(c + d*x)*(A + B*Log[e*((a + b*x)/(c + d *x))^n])^3)/(a + b*x) - (3*b*d^2*(c + d*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^3)/(2*(a + b*x)^2) + (b^2*d*(c + d*x)^3*(A + B*Log[e*((a + b*x)/ (c + d*x))^n])^3)/(a + b*x)^3 - (b^3*(c + d*x)^4*(A + B*Log[e*((a + b*x)/( c + d*x))^n])^3)/(4*(a + b*x)^4))/(b*c - a*d)^4
3.2.71.3.1 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_.), x_Symbol] :> Simp[(b*c - a*d)^(m + 1)*(g/b)^m Subst[Int[x^m*((A + B*Log[e*x^n])^p/(b - d*x)^(m + 2)), x], x, (a + b*x)/(c + d*x)], x] /; FreeQ[{a, b, c, d, e, f, g, A, B, n}, x] && Ne Q[b*c - a*d, 0] && IntegersQ[m, p] && EqQ[b*f - a*g, 0] && (GtQ[p, 0] || Lt Q[m, -1])
Int[((A_.) + Log[(e_.)*(u_)^(n_.)*(v_)^(mn_)]*(B_.))^(p_.)*(w_.), x_Symbol] :> Subst[Int[w*(A + B*Log[e*(u/v)^n])^p, x], e*(u/v)^n, e*(u^n/v^n)] /; Fr eeQ[{e, A, B, n, p}, x] && EqQ[n + mn, 0] && LinearQ[{u, v}, x] && !Intege rQ[n]
Leaf count of result is larger than twice the leaf count of optimal. \(8291\) vs. \(2(810)=1620\).
Time = 191.85 (sec) , antiderivative size = 8292, normalized size of antiderivative = 9.99
method | result | size |
parallelrisch | \(\text {Expression too large to display}\) | \(8292\) |
risch | \(\text {Expression too large to display}\) | \(236754\) |
Leaf count of result is larger than twice the leaf count of optimal. 6057 vs. \(2 (810) = 1620\).
Time = 0.53 (sec) , antiderivative size = 6057, normalized size of antiderivative = 7.30 \[ \int \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{(a+b x)^5} \, dx=\text {Too large to display} \]
Timed out. \[ \int \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{(a+b x)^5} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 5280 vs. \(2 (810) = 1620\).
Time = 0.57 (sec) , antiderivative size = 5280, normalized size of antiderivative = 6.36 \[ \int \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{(a+b x)^5} \, dx=\text {Too large to display} \]
-1/4*B^3*log((b*x + a)^n*e/(d*x + c)^n)^3/(b^5*x^4 + 4*a*b^4*x^3 + 6*a^2*b ^3*x^2 + 4*a^3*b^2*x + a^4*b) + 1/1152*(72*(12*d^4*e*n*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) - 12 *d^4*e*n*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) + (12*b^3*d^3*e*n*x^3 - 3*b^3*c^3*e*n + 13*a*b^2*c ^2*d*e*n - 23*a^2*b*c*d^2*e*n + 25*a^3*d^3*e*n - 6*(b^3*c*d^2*e*n - 7*a*b^ 2*d^3*e*n)*x^2 + 4*(b^3*c^2*d*e*n - 5*a*b^2*c*d^2*e*n + 13*a^2*b*d^3*e*n)* 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 + (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)*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)*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)*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)*x))*log((b*x + a)^n*e/(d*x + c)^ n)^2/e - (12*(9*b^4*c^4*e^2*n^2 - 64*a*b^3*c^3*d*e^2*n^2 + 216*a^2*b^2*c^2 *d^2*e^2*n^2 - 576*a^3*b*c*d^3*e^2*n^2 + 415*a^4*d^4*e^2*n^2 - 300*(b^4*c* d^3*e^2*n^2 - a*b^3*d^4*e^2*n^2)*x^3 + 6*(13*b^4*c^2*d^2*e^2*n^2 - 176*a*b ^3*c*d^3*e^2*n^2 + 163*a^2*b^2*d^4*e^2*n^2)*x^2 + 72*(b^4*d^4*e^2*n^2*x^4 + 4*a*b^3*d^4*e^2*n^2*x^3 + 6*a^2*b^2*d^4*e^2*n^2*x^2 + 4*a^3*b*d^4*e^2*n^ 2*x + a^4*d^4*e^2*n^2)*log(b*x + a)^2 + 72*(b^4*d^4*e^2*n^2*x^4 + 4*a*b^3* d^4*e^2*n^2*x^3 + 6*a^2*b^2*d^4*e^2*n^2*x^2 + 4*a^3*b*d^4*e^2*n^2*x + a^4* d^4*e^2*n^2)*log(d*x + c)^2 - 4*(7*b^4*c^3*d*e^2*n^2 - 60*a*b^3*c^2*d^2...
\[ \int \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{(a+b x)^5} \, dx=\int { \frac {{\left (B \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) + A\right )}^{3}}{{\left (b x + a\right )}^{5}} \,d x } \]
Time = 8.17 (sec) , antiderivative size = 4257, normalized size of antiderivative = 5.13 \[ \int \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{(a+b x)^5} \, dx=\text {Too large to display} \]
log((e*(a + b*x)^n)/(c + d*x)^n)*((x*((a*d + b*c)*(a*((9*B^3*a*d^2*n^2)/2 - (3*B^3*b*c*d*n^2)/2) + 13*B^3*a^2*d^2*n^2 + (11*B^3*b^2*c^2*n^2)/2 - 6*A ^2*B*a^2*d^2 - 6*A^2*B*b^2*c^2 - (31*B^3*a*b*c*d*n^2)/2 + 12*A^2*B*a*b*c*d ) + a*c*(b*((9*B^3*a*d^2*n^2)/2 - (3*B^3*b*c*d*n^2)/2) + (27*B^3*a*b*d^2*n ^2)/2 - (9*B^3*b^2*c*d*n^2)/2)) + x^2*((a*d + b*c)*(b*((9*B^3*a*d^2*n^2)/2 - (3*B^3*b*c*d*n^2)/2) + (27*B^3*a*b*d^2*n^2)/2 - (9*B^3*b^2*c*d*n^2)/2) + b*d*(a*((9*B^3*a*d^2*n^2)/2 - (3*B^3*b*c*d*n^2)/2) + 13*B^3*a^2*d^2*n^2 + (11*B^3*b^2*c^2*n^2)/2 - 6*A^2*B*a^2*d^2 - 6*A^2*B*b^2*c^2 - (31*B^3*a*b *c*d*n^2)/2 + 12*A^2*B*a*b*c*d) + 6*B^3*a*b^2*c*d^2*n^2) + x^3*(b*d*(b*((9 *B^3*a*d^2*n^2)/2 - (3*B^3*b*c*d*n^2)/2) + (27*B^3*a*b*d^2*n^2)/2 - (9*B^3 *b^2*c*d*n^2)/2) + 6*B^3*b^2*d^2*n^2*(a*d + b*c)) + a*c*(a*((9*B^3*a*d^2*n ^2)/2 - (3*B^3*b*c*d*n^2)/2) + 13*B^3*a^2*d^2*n^2 + (11*B^3*b^2*c^2*n^2)/2 - 6*A^2*B*a^2*d^2 - 6*A^2*B*b^2*c^2 - (31*B^3*a*b*c*d*n^2)/2 + 12*A^2*B*a *b*c*d) + 6*B^3*b^3*d^3*n^2*x^4)/(8*b*(a*d - b*c)^2*(a + b*x)^5*(c + d*x)) - (d^4*(12*A*B^2 + 25*B^3*n)*(x^3*((a*d + b*c)*(b*(b*((2*a*b*n*(a*d - b*c )^3)/d + (2*b*n*(a*d - b*c)^3*(4*a*d - b*c))/(3*d^2)) + (4*b^2*n*(a*d - b* c)^3*(4*a*d - b*c))/(3*d^2) + (4*a*b^2*n*(a*d - b*c)^3)/d) + (2*b^3*n*(a*d - b*c)^3*(4*a*d - b*c))/d^2 + (6*a*b^3*n*(a*d - b*c)^3)/d) + b*d*(b*(a*(( 2*a*b*n*(a*d - b*c)^3)/d + (2*b*n*(a*d - b*c)^3*(4*a*d - b*c))/(3*d^2)) + (2*b*n*(a*d - b*c)^3*(6*a^2*d^2 + b^2*c^2 - 4*a*b*c*d))/(3*d^3)) + a*(b...