Integrand size = 31, antiderivative size = 764 \[ \int \frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(a+b x)^4} \, dx=-\frac {2 p^2 r^2}{27 b (a+b x)^3}-\frac {5 d p q r^2}{18 b (b c-a d) (a+b x)^2}+\frac {8 d^2 p q r^2}{9 b (b c-a d)^2 (a+b x)}-\frac {d^2 q^2 r^2}{3 b (b c-a d)^2 (a+b x)}+\frac {2 d^3 p q r^2 \log (a+b x)}{9 b (b c-a d)^3}-\frac {d^3 q^2 r^2 \log (a+b x)}{b (b c-a d)^3}-\frac {d^3 p q r^2 \log ^2(a+b x)}{3 b (b c-a d)^3}-\frac {2 d^3 p q r^2 \log (c+d x)}{9 b (b c-a d)^3}+\frac {d^3 q^2 r^2 \log (c+d x)}{b (b c-a d)^3}+\frac {2 d^3 p q r^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{3 b (b c-a d)^3}+\frac {d^3 q^2 r^2 \log ^2(c+d x)}{3 b (b c-a d)^3}-\frac {2 d^3 q^2 r^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{3 b (b c-a d)^3}-\frac {2 p r \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{9 b (a+b x)^3}-\frac {d q r \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 b (b c-a d) (a+b x)^2}+\frac {2 d^2 q r \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 b (b c-a d)^2 (a+b x)}+\frac {2 d^3 q r \log (a+b x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 b (b c-a d)^3}-\frac {2 d^3 q r \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 b (b c-a d)^3}-\frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 b (a+b x)^3}-\frac {2 d^3 q^2 r^2 \operatorname {PolyLog}\left (2,-\frac {d (a+b x)}{b c-a d}\right )}{3 b (b c-a d)^3}+\frac {2 d^3 p q r^2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )}{3 b (b c-a d)^3} \]
-2/27*p^2*r^2/b/(b*x+a)^3-5/18*d*p*q*r^2/b/(-a*d+b*c)/(b*x+a)^2+8/9*d^2*p* q*r^2/b/(-a*d+b*c)^2/(b*x+a)-1/3*d^2*q^2*r^2/b/(-a*d+b*c)^2/(b*x+a)+2/9*d^ 3*p*q*r^2*ln(b*x+a)/b/(-a*d+b*c)^3-d^3*q^2*r^2*ln(b*x+a)/b/(-a*d+b*c)^3-1/ 3*d^3*p*q*r^2*ln(b*x+a)^2/b/(-a*d+b*c)^3-2/9*d^3*p*q*r^2*ln(d*x+c)/b/(-a*d +b*c)^3+d^3*q^2*r^2*ln(d*x+c)/b/(-a*d+b*c)^3+2/3*d^3*p*q*r^2*ln(-d*(b*x+a) /(-a*d+b*c))*ln(d*x+c)/b/(-a*d+b*c)^3+1/3*d^3*q^2*r^2*ln(d*x+c)^2/b/(-a*d+ b*c)^3-2/3*d^3*q^2*r^2*ln(b*x+a)*ln(b*(d*x+c)/(-a*d+b*c))/b/(-a*d+b*c)^3-2 /9*p*r*ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r)/b/(b*x+a)^3-1/3*d*q*r*ln(e*(f*(b*x+ a)^p*(d*x+c)^q)^r)/b/(-a*d+b*c)/(b*x+a)^2+2/3*d^2*q*r*ln(e*(f*(b*x+a)^p*(d *x+c)^q)^r)/b/(-a*d+b*c)^2/(b*x+a)+2/3*d^3*q*r*ln(b*x+a)*ln(e*(f*(b*x+a)^p *(d*x+c)^q)^r)/b/(-a*d+b*c)^3-2/3*d^3*q*r*ln(d*x+c)*ln(e*(f*(b*x+a)^p*(d*x +c)^q)^r)/b/(-a*d+b*c)^3-1/3*ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r)^2/b/(b*x+a)^3 -2/3*d^3*q^2*r^2*polylog(2,-d*(b*x+a)/(-a*d+b*c))/b/(-a*d+b*c)^3+2/3*d^3*p *q*r^2*polylog(2,b*(d*x+c)/(-a*d+b*c))/b/(-a*d+b*c)^3
Time = 0.94 (sec) , antiderivative size = 1407, normalized size of antiderivative = 1.84 \[ \int \frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(a+b x)^4} \, dx=-\frac {4 b^3 c^3 p^2 r^2-12 a b^2 c^2 d p^2 r^2+12 a^2 b c d^2 p^2 r^2-4 a^3 d^3 p^2 r^2+15 a b^2 c^2 d p q r^2-78 a^2 b c d^2 p q r^2+63 a^3 d^3 p q r^2+18 a^2 b c d^2 q^2 r^2-18 a^3 d^3 q^2 r^2+15 b^3 c^2 d p q r^2 x-126 a b^2 c d^2 p q r^2 x+111 a^2 b d^3 p q r^2 x+36 a b^2 c d^2 q^2 r^2 x-36 a^2 b d^3 q^2 r^2 x-48 b^3 c d^2 p q r^2 x^2+48 a b^2 d^3 p q r^2 x^2+18 b^3 c d^2 q^2 r^2 x^2-18 a b^2 d^3 q^2 r^2 x^2+18 d^3 p q r^2 (a+b x)^3 \log ^2(a+b x)+12 a^3 d^3 p q r^2 \log (c+d x)-54 a^3 d^3 q^2 r^2 \log (c+d x)+36 a^2 b d^3 p q r^2 x \log (c+d x)-162 a^2 b d^3 q^2 r^2 x \log (c+d x)+36 a b^2 d^3 p q r^2 x^2 \log (c+d x)-162 a b^2 d^3 q^2 r^2 x^2 \log (c+d x)+12 b^3 d^3 p q r^2 x^3 \log (c+d x)-54 b^3 d^3 q^2 r^2 x^3 \log (c+d x)-18 a^3 d^3 q^2 r^2 \log ^2(c+d x)-54 a^2 b d^3 q^2 r^2 x \log ^2(c+d x)-54 a b^2 d^3 q^2 r^2 x^2 \log ^2(c+d x)-18 b^3 d^3 q^2 r^2 x^3 \log ^2(c+d x)+12 b^3 c^3 p r \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )-36 a b^2 c^2 d p r \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+36 a^2 b c d^2 p r \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )-12 a^3 d^3 p r \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+18 a b^2 c^2 d q r \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )-72 a^2 b c d^2 q r \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+54 a^3 d^3 q r \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+18 b^3 c^2 d q r x \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )-108 a b^2 c d^2 q r x \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+90 a^2 b d^3 q r x \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )-36 b^3 c d^2 q r x^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+36 a b^2 d^3 q r x^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+36 a^3 d^3 q r \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+108 a^2 b d^3 q r x \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+108 a b^2 d^3 q r x^2 \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+36 b^3 d^3 q r x^3 \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+18 b^3 c^3 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )-54 a b^2 c^2 d \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+54 a^2 b c d^2 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )-18 a^3 d^3 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )-6 d^3 q r (a+b x)^3 \log (a+b x) \left (2 p r-9 q r+6 p r \log (c+d x)-6 (p+q) r \log \left (\frac {b (c+d x)}{b c-a d}\right )+6 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )+36 d^3 q (p+q) r^2 (a+b x)^3 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{-b c+a d}\right )}{54 b (b c-a d)^3 (a+b x)^3} \]
-1/54*(4*b^3*c^3*p^2*r^2 - 12*a*b^2*c^2*d*p^2*r^2 + 12*a^2*b*c*d^2*p^2*r^2 - 4*a^3*d^3*p^2*r^2 + 15*a*b^2*c^2*d*p*q*r^2 - 78*a^2*b*c*d^2*p*q*r^2 + 6 3*a^3*d^3*p*q*r^2 + 18*a^2*b*c*d^2*q^2*r^2 - 18*a^3*d^3*q^2*r^2 + 15*b^3*c ^2*d*p*q*r^2*x - 126*a*b^2*c*d^2*p*q*r^2*x + 111*a^2*b*d^3*p*q*r^2*x + 36* a*b^2*c*d^2*q^2*r^2*x - 36*a^2*b*d^3*q^2*r^2*x - 48*b^3*c*d^2*p*q*r^2*x^2 + 48*a*b^2*d^3*p*q*r^2*x^2 + 18*b^3*c*d^2*q^2*r^2*x^2 - 18*a*b^2*d^3*q^2*r ^2*x^2 + 18*d^3*p*q*r^2*(a + b*x)^3*Log[a + b*x]^2 + 12*a^3*d^3*p*q*r^2*Lo g[c + d*x] - 54*a^3*d^3*q^2*r^2*Log[c + d*x] + 36*a^2*b*d^3*p*q*r^2*x*Log[ c + d*x] - 162*a^2*b*d^3*q^2*r^2*x*Log[c + d*x] + 36*a*b^2*d^3*p*q*r^2*x^2 *Log[c + d*x] - 162*a*b^2*d^3*q^2*r^2*x^2*Log[c + d*x] + 12*b^3*d^3*p*q*r^ 2*x^3*Log[c + d*x] - 54*b^3*d^3*q^2*r^2*x^3*Log[c + d*x] - 18*a^3*d^3*q^2* r^2*Log[c + d*x]^2 - 54*a^2*b*d^3*q^2*r^2*x*Log[c + d*x]^2 - 54*a*b^2*d^3* q^2*r^2*x^2*Log[c + d*x]^2 - 18*b^3*d^3*q^2*r^2*x^3*Log[c + d*x]^2 + 12*b^ 3*c^3*p*r*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r] - 36*a*b^2*c^2*d*p*r*Log[e* (f*(a + b*x)^p*(c + d*x)^q)^r] + 36*a^2*b*c*d^2*p*r*Log[e*(f*(a + b*x)^p*( c + d*x)^q)^r] - 12*a^3*d^3*p*r*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r] + 18* a*b^2*c^2*d*q*r*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r] - 72*a^2*b*c*d^2*q*r* Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r] + 54*a^3*d^3*q*r*Log[e*(f*(a + b*x)^p *(c + d*x)^q)^r] + 18*b^3*c^2*d*q*r*x*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r] - 108*a*b^2*c*d^2*q*r*x*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r] + 90*a^2*...
Time = 1.18 (sec) , antiderivative size = 690, normalized size of antiderivative = 0.90, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.226, Rules used = {2984, 2981, 17, 54, 2009, 2994, 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.
| \(\displaystyle \int \frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(a+b x)^4} \, dx\) |
| \(\Big \downarrow \) 2984 |
| \(\displaystyle \frac {2}{3} p r \int \frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(a+b x)^4}dx+\frac {2 d q r \int \frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(a+b x)^3 (c+d x)}dx}{3 b}-\frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 b (a+b x)^3}\) |
| \(\Big \downarrow \) 2981 |
| \(\displaystyle \frac {2}{3} p r \left (\frac {d q r \int \frac {1}{(a+b x)^3 (c+d x)}dx}{3 b}+\frac {1}{3} p r \int \frac {1}{(a+b x)^4}dx-\frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 b (a+b x)^3}\right )+\frac {2 d q r \int \frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(a+b x)^3 (c+d x)}dx}{3 b}-\frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 b (a+b x)^3}\) |
| \(\Big \downarrow \) 17 |
| \(\displaystyle \frac {2}{3} p r \left (\frac {d q r \int \frac {1}{(a+b x)^3 (c+d x)}dx}{3 b}-\frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 b (a+b x)^3}-\frac {p r}{9 b (a+b x)^3}\right )+\frac {2 d q r \int \frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(a+b x)^3 (c+d x)}dx}{3 b}-\frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 b (a+b x)^3}\) |
| \(\Big \downarrow \) 54 |
| \(\displaystyle \frac {2}{3} p r \left (\frac {d q r \int \left (-\frac {d^3}{(b c-a d)^3 (c+d x)}+\frac {b d^2}{(b c-a d)^3 (a+b x)}-\frac {b d}{(b c-a d)^2 (a+b x)^2}+\frac {b}{(b c-a d) (a+b x)^3}\right )dx}{3 b}-\frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 b (a+b x)^3}-\frac {p r}{9 b (a+b x)^3}\right )+\frac {2 d q r \int \frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(a+b x)^3 (c+d x)}dx}{3 b}-\frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 b (a+b x)^3}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 d q r \int \frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(a+b x)^3 (c+d x)}dx}{3 b}+\frac {2}{3} p r \left (\frac {d q r \left (\frac {d^2 \log (a+b x)}{(b c-a d)^3}-\frac {d^2 \log (c+d x)}{(b c-a d)^3}+\frac {d}{(a+b x) (b c-a d)^2}-\frac {1}{2 (a+b x)^2 (b c-a d)}\right )}{3 b}-\frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 b (a+b x)^3}-\frac {p r}{9 b (a+b x)^3}\right )-\frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 b (a+b x)^3}\) |
| \(\Big \downarrow \) 2994 |
| \(\displaystyle \frac {2 d q r \int \left (-\frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) d^3}{(b c-a d)^3 (c+d x)}+\frac {b \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) d^2}{(b c-a d)^3 (a+b x)}-\frac {b \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) d}{(b c-a d)^2 (a+b x)^2}+\frac {b \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(b c-a d) (a+b x)^3}\right )dx}{3 b}+\frac {2}{3} p r \left (\frac {d q r \left (\frac {d^2 \log (a+b x)}{(b c-a d)^3}-\frac {d^2 \log (c+d x)}{(b c-a d)^3}+\frac {d}{(a+b x) (b c-a d)^2}-\frac {1}{2 (a+b x)^2 (b c-a d)}\right )}{3 b}-\frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 b (a+b x)^3}-\frac {p r}{9 b (a+b x)^3}\right )-\frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 b (a+b x)^3}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 d q r \left (\frac {d^2 \log (a+b x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(b c-a d)^3}-\frac {d^2 \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(b c-a d)^3}+\frac {d^2 p r \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )}{(b c-a d)^3}-\frac {d^2 p r \log ^2(a+b x)}{2 (b c-a d)^3}+\frac {d^2 p r \log (c+d x) \log \left (-\frac {d (a+b x)}{b c-a d}\right )}{(b c-a d)^3}-\frac {d^2 q r \operatorname {PolyLog}\left (2,-\frac {d (a+b x)}{b c-a d}\right )}{(b c-a d)^3}+\frac {d^2 q r \log ^2(c+d x)}{2 (b c-a d)^3}-\frac {3 d^2 q r \log (a+b x)}{2 (b c-a d)^3}+\frac {3 d^2 q r \log (c+d x)}{2 (b c-a d)^3}-\frac {d^2 q r \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{(b c-a d)^3}+\frac {d \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(a+b x) (b c-a d)^2}-\frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 (a+b x)^2 (b c-a d)}+\frac {d p r}{(a+b x) (b c-a d)^2}-\frac {p r}{4 (a+b x)^2 (b c-a d)}-\frac {d q r}{2 (a+b x) (b c-a d)^2}\right )}{3 b}+\frac {2}{3} p r \left (\frac {d q r \left (\frac {d^2 \log (a+b x)}{(b c-a d)^3}-\frac {d^2 \log (c+d x)}{(b c-a d)^3}+\frac {d}{(a+b x) (b c-a d)^2}-\frac {1}{2 (a+b x)^2 (b c-a d)}\right )}{3 b}-\frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 b (a+b x)^3}-\frac {p r}{9 b (a+b x)^3}\right )-\frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 b (a+b x)^3}\) |
-1/3*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^2/(b*(a + b*x)^3) + (2*p*r*(-1/9 *(p*r)/(b*(a + b*x)^3) + (d*q*r*(-1/2*1/((b*c - a*d)*(a + b*x)^2) + d/((b* c - a*d)^2*(a + b*x)) + (d^2*Log[a + b*x])/(b*c - a*d)^3 - (d^2*Log[c + d* x])/(b*c - a*d)^3))/(3*b) - Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]/(3*b*(a + b*x)^3)))/3 + (2*d*q*r*(-1/4*(p*r)/((b*c - a*d)*(a + b*x)^2) + (d*p*r)/(( b*c - a*d)^2*(a + b*x)) - (d*q*r)/(2*(b*c - a*d)^2*(a + b*x)) - (3*d^2*q*r *Log[a + b*x])/(2*(b*c - a*d)^3) - (d^2*p*r*Log[a + b*x]^2)/(2*(b*c - a*d) ^3) + (3*d^2*q*r*Log[c + d*x])/(2*(b*c - a*d)^3) + (d^2*p*r*Log[-((d*(a + b*x))/(b*c - a*d))]*Log[c + d*x])/(b*c - a*d)^3 + (d^2*q*r*Log[c + d*x]^2) /(2*(b*c - a*d)^3) - (d^2*q*r*Log[a + b*x]*Log[(b*(c + d*x))/(b*c - a*d)]) /(b*c - a*d)^3 - Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]/(2*(b*c - a*d)*(a + b*x)^2) + (d*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/((b*c - a*d)^2*(a + b*x )) + (d^2*Log[a + b*x]*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(b*c - a*d)^3 - (d^2*Log[c + d*x]*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(b*c - a*d)^3 - (d^2*q*r*PolyLog[2, -((d*(a + b*x))/(b*c - a*d))])/(b*c - a*d)^3 + (d^2*p *r*PolyLog[2, (b*(c + d*x))/(b*c - a*d)])/(b*c - a*d)^3))/(3*b)
3.1.23.3.1 Defintions of rubi rules used
Int[(c_.)*((a_.) + (b_.)*(x_))^(m_.), x_Symbol] :> Simp[c*((a + b*x)^(m + 1 )/(b*(m + 1))), x] /; FreeQ[{a, b, c, m}, x] && NeQ[m, -1]
Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[E xpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[m, 0] && IntegerQ[n] && !(IGtQ[n, 0] && LtQ[m + n + 2, 0])
Int[Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.)) ^(r_.)]*((g_.) + (h_.)*(x_))^(m_.), x_Symbol] :> Simp[(g + h*x)^(m + 1)*(Lo g[e*(f*(a + b*x)^p*(c + d*x)^q)^r]/(h*(m + 1))), x] + (-Simp[b*p*(r/(h*(m + 1))) Int[(g + h*x)^(m + 1)/(a + b*x), x], x] - Simp[d*q*(r/(h*(m + 1))) Int[(g + h*x)^(m + 1)/(c + d*x), x], x]) /; FreeQ[{a, b, c, d, e, f, g, h , m, p, q, r}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1]
Int[Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.)) ^(r_.)]^(s_)*((g_.) + (h_.)*(x_))^(m_.), x_Symbol] :> Simp[(g + h*x)^(m + 1 )*(Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^s/(h*(m + 1))), x] + (-Simp[b*p*r*( s/(h*(m + 1))) Int[(g + h*x)^(m + 1)*(Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r ]^(s - 1)/(a + b*x)), x], x] - Simp[d*q*r*(s/(h*(m + 1))) Int[(g + h*x)^( m + 1)*(Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^(s - 1)/(c + d*x)), x], x]) /; FreeQ[{a, b, c, d, e, f, g, h, m, p, q, r, s}, x] && NeQ[b*c - a*d, 0] && IGtQ[s, 0] && NeQ[m, -1]
Int[Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.)) ^(r_.)]^(s_.)*(RFx_), x_Symbol] :> With[{u = ExpandIntegrand[Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^s, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c , d, e, f, p, q, r, s}, x] && RationalFunctionQ[RFx, x] && IGtQ[s, 0]
\[\int \frac {{\ln \left (e \left (f \left (b x +a \right )^{p} \left (d x +c \right )^{q}\right )^{r}\right )}^{2}}{\left (b x +a \right )^{4}}d x\]
\[ \int \frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(a+b x)^4} \, dx=\int { \frac {\log \left (\left ({\left (b x + a\right )}^{p} {\left (d x + c\right )}^{q} f\right )^{r} e\right )^{2}}{{\left (b x + a\right )}^{4}} \,d x } \]
integral(log(((b*x + a)^p*(d*x + c)^q*f)^r*e)^2/(b^4*x^4 + 4*a*b^3*x^3 + 6 *a^2*b^2*x^2 + 4*a^3*b*x + a^4), x)
\[ \int \frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(a+b x)^4} \, dx=\int \frac {\log {\left (e \left (f \left (a + b x\right )^{p} \left (c + d x\right )^{q}\right )^{r} \right )}^{2}}{\left (a + b x\right )^{4}}\, dx \]
Time = 0.29 (sec) , antiderivative size = 1252, normalized size of antiderivative = 1.64 \[ \int \frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(a+b x)^4} \, dx=\text {Too large to display} \]
1/9*(6*d^3*f*q*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) - 6*d^3*f*q*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) + (6*b^2*d^2*f*q*x^2 + a*b*c*d*f*(4*p - 3*q) - a^2*d^2*f*(2*p - 9 *q) - 2*b^2*c^2*f*p - 3*(b^2*c*d*f*q - 5*a*b*d^2*f*q)*x)/(a^3*b^2*c^2 - 2* a^4*b*c*d + a^5*d^2 + (b^5*c^2 - 2*a*b^4*c*d + a^2*b^3*d^2)*x^3 + 3*(a*b^4 *c^2 - 2*a^2*b^3*c*d + a^3*b^2*d^2)*x^2 + 3*(a^2*b^3*c^2 - 2*a^3*b^2*c*d + a^4*b*d^2)*x))*r*log(((b*x + a)^p*(d*x + c)^q*f)^r*e)/(b*f) - 1/54*(36*(p *q + q^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)))*d^3*f^2/(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a ^3*d^3) + 6*(2*p*q - 9*q^2)*d^3*f^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) + (4*b^3*c^3*f^2*p^2 - 3*(4*p^2 - 5*p*q)*a*b^2* c^2*d*f^2 + 6*(2*p^2 - 13*p*q + 3*q^2)*a^2*b*c*d^2*f^2 - (4*p^2 - 63*p*q + 18*q^2)*a^3*d^3*f^2 - 6*((8*p*q - 3*q^2)*b^3*c*d^2*f^2 - (8*p*q - 3*q^2)* a*b^2*d^3*f^2)*x^2 + 18*(b^3*d^3*f^2*p*q*x^3 + 3*a*b^2*d^3*f^2*p*q*x^2 + 3 *a^2*b*d^3*f^2*p*q*x + a^3*d^3*f^2*p*q)*log(b*x + a)^2 - 36*(b^3*d^3*f^2*p *q*x^3 + 3*a*b^2*d^3*f^2*p*q*x^2 + 3*a^2*b*d^3*f^2*p*q*x + a^3*d^3*f^2*p*q )*log(b*x + a)*log(d*x + c) - 18*(b^3*d^3*f^2*q^2*x^3 + 3*a*b^2*d^3*f^2*q^ 2*x^2 + 3*a^2*b*d^3*f^2*q^2*x + a^3*d^3*f^2*q^2)*log(d*x + c)^2 + 3*(5*b^3 *c^2*d*f^2*p*q - 6*(7*p*q - 2*q^2)*a*b^2*c*d^2*f^2 + (37*p*q - 12*q^2)*a^2 *b*d^3*f^2)*x - 6*((2*p*q - 9*q^2)*b^3*d^3*f^2*x^3 + 3*(2*p*q - 9*q^2)*...
\[ \int \frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(a+b x)^4} \, dx=\int { \frac {\log \left (\left ({\left (b x + a\right )}^{p} {\left (d x + c\right )}^{q} f\right )^{r} e\right )^{2}}{{\left (b x + a\right )}^{4}} \,d x } \]
Timed out. \[ \int \frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(a+b x)^4} \, dx=\int \frac {{\ln \left (e\,{\left (f\,{\left (a+b\,x\right )}^p\,{\left (c+d\,x\right )}^q\right )}^r\right )}^2}{{\left (a+b\,x\right )}^4} \,d x \]