Integrand size = 31, antiderivative size = 686 \[ \int (a+b x)^2 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx=-\frac {a (b c-a d) p q r^2 x}{3 d}+\frac {2 (b c-a d)^2 p q r^2 x}{9 d^2}+\frac {5 (b c-a d)^2 q^2 r^2 x}{9 d^2}+\frac {2 (b c-a d)^2 q (p+q) r^2 x}{3 d^2}-\frac {b (b c-a d) p q r^2 x^2}{6 d}-\frac {(b c-a d) p q r^2 (a+b x)^2}{9 b d}-\frac {5 (b c-a d) q^2 r^2 (a+b x)^2}{18 b d}+\frac {2 p^2 r^2 (a+b x)^3}{27 b}+\frac {4 p q r^2 (a+b x)^3}{27 b}+\frac {2 q^2 r^2 (a+b x)^3}{27 b}-\frac {2 (b c-a d)^3 p q r^2 \log (c+d x)}{9 b d^3}-\frac {11 (b c-a d)^3 q^2 r^2 \log (c+d x)}{9 b d^3}-\frac {2 (b c-a d)^3 p q r^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{3 b d^3}-\frac {(b c-a d)^3 q^2 r^2 \log ^2(c+d x)}{3 b d^3}-\frac {2 (b c-a d)^2 q r (a+b x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 b d^2}+\frac {(b c-a d) q r (a+b x)^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 b d}-\frac {2 p r (a+b x)^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{9 b}-\frac {2 q r (a+b x)^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{9 b}+\frac {2 (b c-a d)^3 q r \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 b d^3}+\frac {(a+b x)^3 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 b}-\frac {2 (b c-a d)^3 p q r^2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )}{3 b d^3} \]
-1/3*a*(-a*d+b*c)*p*q*r^2*x/d+2/9*(-a*d+b*c)^2*p*q*r^2*x/d^2+5/9*(-a*d+b*c )^2*q^2*r^2*x/d^2+2/3*(-a*d+b*c)^2*q*(p+q)*r^2*x/d^2-1/6*b*(-a*d+b*c)*p*q* r^2*x^2/d-1/9*(-a*d+b*c)*p*q*r^2*(b*x+a)^2/b/d-5/18*(-a*d+b*c)*q^2*r^2*(b* x+a)^2/b/d+2/27*p^2*r^2*(b*x+a)^3/b+4/27*p*q*r^2*(b*x+a)^3/b+2/27*q^2*r^2* (b*x+a)^3/b-2/9*(-a*d+b*c)^3*p*q*r^2*ln(d*x+c)/b/d^3-11/9*(-a*d+b*c)^3*q^2 *r^2*ln(d*x+c)/b/d^3-2/3*(-a*d+b*c)^3*p*q*r^2*ln(-d*(b*x+a)/(-a*d+b*c))*ln (d*x+c)/b/d^3-1/3*(-a*d+b*c)^3*q^2*r^2*ln(d*x+c)^2/b/d^3-2/3*(-a*d+b*c)^2* q*r*(b*x+a)*ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r)/b/d^2+1/3*(-a*d+b*c)*q*r*(b*x+ a)^2*ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r)/b/d-2/9*p*r*(b*x+a)^3*ln(e*(f*(b*x+a) ^p*(d*x+c)^q)^r)/b-2/9*q*r*(b*x+a)^3*ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r)/b+2/3 *(-a*d+b*c)^3*q*r*ln(d*x+c)*ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r)/b/d^3+1/3*(b*x +a)^3*ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r)^2/b-2/3*(-a*d+b*c)^3*p*q*r^2*polylog (2,b*(d*x+c)/(-a*d+b*c))/b/d^3
Time = 0.61 (sec) , antiderivative size = 1211, normalized size of antiderivative = 1.77 \[ \int (a+b x)^2 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx=\frac {1}{54} \left (\frac {108 a^3 p q r^2}{b}+\frac {36 a b c^2 p q r^2}{d^2}-\frac {108 a^2 c p q r^2}{d}+12 a^2 p^2 r^2 x+102 a^2 p q r^2 x+\frac {48 b^2 c^2 p q r^2 x}{d^2}-\frac {126 a b c p q r^2 x}{d}+108 a^2 q^2 r^2 x+\frac {66 b^2 c^2 q^2 r^2 x}{d^2}-\frac {162 a b c q^2 r^2 x}{d}+12 a b p^2 r^2 x^2+39 a b p q r^2 x^2-\frac {15 b^2 c p q r^2 x^2}{d}+27 a b q^2 r^2 x^2-\frac {15 b^2 c q^2 r^2 x^2}{d}+4 b^2 p^2 r^2 x^3+8 b^2 p q r^2 x^3+4 b^2 q^2 r^2 x^3-\frac {18 a^3 p^2 r^2 \log ^2(a+b x)}{b}+\frac {108 a^3 p q r^2 \log (c+d x)}{b}-\frac {12 b^2 c^3 p q r^2 \log (c+d x)}{d^3}+\frac {36 a b c^2 p q r^2 \log (c+d x)}{d^2}-\frac {36 a^2 c p q r^2 \log (c+d x)}{d}-\frac {66 b^2 c^3 q^2 r^2 \log (c+d x)}{d^3}+\frac {162 a b c^2 q^2 r^2 \log (c+d x)}{d^2}-\frac {108 a^2 c q^2 r^2 \log (c+d x)}{d}-\frac {18 b^2 c^3 q^2 r^2 \log ^2(c+d x)}{d^3}+\frac {54 a b c^2 q^2 r^2 \log ^2(c+d x)}{d^2}-\frac {54 a^2 c q^2 r^2 \log ^2(c+d x)}{d}-\frac {108 a^3 p r \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{b}-36 a^2 p r x \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )-108 a^2 q r x \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )-\frac {36 b^2 c^2 q r x \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{d^2}+\frac {108 a b c q r x \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{d}-36 a b p r x^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )-54 a b q r x^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+\frac {18 b^2 c q r x^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{d}-12 b^2 p r x^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )-12 b^2 q r x^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+\frac {36 b^2 c^3 q r \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{d^3}-\frac {108 a b c^2 q r \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{d^2}+\frac {108 a^2 c q r \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{d}+54 a^2 x \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+54 a b x^2 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+18 b^2 x^3 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+\frac {6 p r \log (a+b x) \left (a d \left (a^2 d^2 (16 p-11 q)-6 b^2 c^2 q+15 a b c d q\right ) r-6 b c \left (b^2 c^2-3 a b c d+3 a^2 d^2\right ) q r \log (c+d x)+6 (b c-a d)^3 q r \log \left (\frac {b (c+d x)}{b c-a d}\right )+6 a^3 d^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )}{b d^3}+\frac {36 (b c-a d)^3 p q r^2 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{-b c+a d}\right )}{b d^3}\right ) \]
((108*a^3*p*q*r^2)/b + (36*a*b*c^2*p*q*r^2)/d^2 - (108*a^2*c*p*q*r^2)/d + 12*a^2*p^2*r^2*x + 102*a^2*p*q*r^2*x + (48*b^2*c^2*p*q*r^2*x)/d^2 - (126*a *b*c*p*q*r^2*x)/d + 108*a^2*q^2*r^2*x + (66*b^2*c^2*q^2*r^2*x)/d^2 - (162* a*b*c*q^2*r^2*x)/d + 12*a*b*p^2*r^2*x^2 + 39*a*b*p*q*r^2*x^2 - (15*b^2*c*p *q*r^2*x^2)/d + 27*a*b*q^2*r^2*x^2 - (15*b^2*c*q^2*r^2*x^2)/d + 4*b^2*p^2* r^2*x^3 + 8*b^2*p*q*r^2*x^3 + 4*b^2*q^2*r^2*x^3 - (18*a^3*p^2*r^2*Log[a + b*x]^2)/b + (108*a^3*p*q*r^2*Log[c + d*x])/b - (12*b^2*c^3*p*q*r^2*Log[c + d*x])/d^3 + (36*a*b*c^2*p*q*r^2*Log[c + d*x])/d^2 - (36*a^2*c*p*q*r^2*Log [c + d*x])/d - (66*b^2*c^3*q^2*r^2*Log[c + d*x])/d^3 + (162*a*b*c^2*q^2*r^ 2*Log[c + d*x])/d^2 - (108*a^2*c*q^2*r^2*Log[c + d*x])/d - (18*b^2*c^3*q^2 *r^2*Log[c + d*x]^2)/d^3 + (54*a*b*c^2*q^2*r^2*Log[c + d*x]^2)/d^2 - (54*a ^2*c*q^2*r^2*Log[c + d*x]^2)/d - (108*a^3*p*r*Log[e*(f*(a + b*x)^p*(c + d* x)^q)^r])/b - 36*a^2*p*r*x*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r] - 108*a^2* q*r*x*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r] - (36*b^2*c^2*q*r*x*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/d^2 + (108*a*b*c*q*r*x*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/d - 36*a*b*p*r*x^2*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r] - 54 *a*b*q*r*x^2*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r] + (18*b^2*c*q*r*x^2*Log[ e*(f*(a + b*x)^p*(c + d*x)^q)^r])/d - 12*b^2*p*r*x^3*Log[e*(f*(a + b*x)^p* (c + d*x)^q)^r] - 12*b^2*q*r*x^3*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r] + (3 6*b^2*c^3*q*r*Log[c + d*x]*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/d^3 - ...
Time = 1.01 (sec) , antiderivative size = 601, normalized size of antiderivative = 0.88, 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, 49, 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.
| \(\displaystyle \int (a+b x)^2 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx\) |
| \(\Big \downarrow \) 2984 |
| \(\displaystyle -\frac {2}{3} p r \int (a+b x)^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )dx-\frac {2 d q r \int \frac {(a+b x)^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{c+d x}dx}{3 b}+\frac {(a+b x)^3 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 b}\) |
| \(\Big \downarrow \) 2981 |
| \(\displaystyle -\frac {2}{3} p r \left (-\frac {d q r \int \frac {(a+b x)^3}{c+d x}dx}{3 b}-\frac {1}{3} p r \int (a+b x)^2dx+\frac {(a+b x)^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 b}\right )-\frac {2 d q r \int \frac {(a+b x)^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{c+d x}dx}{3 b}+\frac {(a+b x)^3 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 b}\) |
| \(\Big \downarrow \) 17 |
| \(\displaystyle -\frac {2}{3} p r \left (-\frac {d q r \int \frac {(a+b x)^3}{c+d x}dx}{3 b}+\frac {(a+b x)^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 b}-\frac {p r (a+b x)^3}{9 b}\right )-\frac {2 d q r \int \frac {(a+b x)^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{c+d x}dx}{3 b}+\frac {(a+b x)^3 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 b}\) |
| \(\Big \downarrow \) 49 |
| \(\displaystyle -\frac {2}{3} p r \left (-\frac {d q r \int \left (\frac {(a d-b c)^3}{d^3 (c+d x)}+\frac {b (b c-a d)^2}{d^3}+\frac {b (a+b x)^2}{d}-\frac {b (b c-a d) (a+b x)}{d^2}\right )dx}{3 b}+\frac {(a+b x)^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 b}-\frac {p r (a+b x)^3}{9 b}\right )-\frac {2 d q r \int \frac {(a+b x)^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{c+d x}dx}{3 b}+\frac {(a+b x)^3 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 b}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 d q r \int \frac {(a+b x)^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{c+d x}dx}{3 b}-\frac {2}{3} p r \left (-\frac {d q r \left (-\frac {(b c-a d)^3 \log (c+d x)}{d^4}+\frac {b x (b c-a d)^2}{d^3}-\frac {(a+b x)^2 (b c-a d)}{2 d^2}+\frac {(a+b x)^3}{3 d}\right )}{3 b}+\frac {(a+b x)^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 b}-\frac {p r (a+b x)^3}{9 b}\right )+\frac {(a+b x)^3 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 b}\) |
| \(\Big \downarrow \) 2994 |
| \(\displaystyle -\frac {2 d q r \int \left (\frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) (a d-b c)^3}{d^3 (c+d x)}+\frac {b (b c-a d)^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{d^3}+\frac {b (a+b x)^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{d}-\frac {b (b c-a d) (a+b x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{d^2}\right )dx}{3 b}-\frac {2}{3} p r \left (-\frac {d q r \left (-\frac {(b c-a d)^3 \log (c+d x)}{d^4}+\frac {b x (b c-a d)^2}{d^3}-\frac {(a+b x)^2 (b c-a d)}{2 d^2}+\frac {(a+b x)^3}{3 d}\right )}{3 b}+\frac {(a+b x)^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 b}-\frac {p r (a+b x)^3}{9 b}\right )+\frac {(a+b x)^3 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 b}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 d q r \left (-\frac {(b c-a d)^3 \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{d^4}+\frac {p r (b c-a d)^3 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )}{d^4}+\frac {p r (b c-a d)^3 \log (c+d x) \log \left (-\frac {d (a+b x)}{b c-a d}\right )}{d^4}+\frac {q r (b c-a d)^3 \log ^2(c+d x)}{2 d^4}+\frac {11 q r (b c-a d)^3 \log (c+d x)}{6 d^4}+\frac {(a+b x) (b c-a d)^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{d^3}-\frac {b r x (p+q) (b c-a d)^2}{d^3}-\frac {5 b q r x (b c-a d)^2}{6 d^3}-\frac {(a+b x)^2 (b c-a d) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 d^2}+\frac {p r (a+b x)^2 (b c-a d)}{4 d^2}+\frac {5 q r (a+b x)^2 (b c-a d)}{12 d^2}+\frac {(a+b x)^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 d}-\frac {p r (a+b x)^3}{9 d}-\frac {q r (a+b x)^3}{9 d}\right )}{3 b}-\frac {2}{3} p r \left (-\frac {d q r \left (-\frac {(b c-a d)^3 \log (c+d x)}{d^4}+\frac {b x (b c-a d)^2}{d^3}-\frac {(a+b x)^2 (b c-a d)}{2 d^2}+\frac {(a+b x)^3}{3 d}\right )}{3 b}+\frac {(a+b x)^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 b}-\frac {p r (a+b x)^3}{9 b}\right )+\frac {(a+b x)^3 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 b}\) |
((a + b*x)^3*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^2)/(3*b) - (2*p*r*(-1/9* (p*r*(a + b*x)^3)/b - (d*q*r*((b*(b*c - a*d)^2*x)/d^3 - ((b*c - a*d)*(a + b*x)^2)/(2*d^2) + (a + b*x)^3/(3*d) - ((b*c - a*d)^3*Log[c + d*x])/d^4))/( 3*b) + ((a + b*x)^3*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(3*b)))/3 - (2*d *q*r*((-5*b*(b*c - a*d)^2*q*r*x)/(6*d^3) - (b*(b*c - a*d)^2*(p + q)*r*x)/d ^3 + ((b*c - a*d)*p*r*(a + b*x)^2)/(4*d^2) + (5*(b*c - a*d)*q*r*(a + b*x)^ 2)/(12*d^2) - (p*r*(a + b*x)^3)/(9*d) - (q*r*(a + b*x)^3)/(9*d) + (11*(b*c - a*d)^3*q*r*Log[c + d*x])/(6*d^4) + ((b*c - a*d)^3*p*r*Log[-((d*(a + b*x ))/(b*c - a*d))]*Log[c + d*x])/d^4 + ((b*c - a*d)^3*q*r*Log[c + d*x]^2)/(2 *d^4) + ((b*c - a*d)^2*(a + b*x)*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/d^3 - ((b*c - a*d)*(a + b*x)^2*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(2*d^2) + ((a + b*x)^3*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(3*d) - ((b*c - a*d)^ 3*Log[c + d*x]*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/d^4 + ((b*c - a*d)^3* p*r*PolyLog[2, (b*(c + d*x))/(b*c - a*d)])/d^4))/(3*b)
3.1.18.3.1 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 [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0] && IGtQ[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]
\[\int \left (b x +a \right )^{2} {\ln \left (e \left (f \left (b x +a \right )^{p} \left (d x +c \right )^{q}\right )^{r}\right )}^{2}d x\]
\[ \int (a+b x)^2 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx=\int { {\left (b x + a\right )}^{2} \log \left (\left ({\left (b x + a\right )}^{p} {\left (d x + c\right )}^{q} f\right )^{r} e\right )^{2} \,d x } \]
\[ \int (a+b x)^2 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx=\int \left (a + b x\right )^{2} \log {\left (e \left (f \left (a + b x\right )^{p} \left (c + d x\right )^{q}\right )^{r} \right )}^{2}\, dx \]
Time = 0.23 (sec) , antiderivative size = 769, normalized size of antiderivative = 1.12 \[ \int (a+b x)^2 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx=\frac {1}{3} \, {\left (b^{2} x^{3} + 3 \, a b x^{2} + 3 \, a^{2} x\right )} \log \left (\left ({\left (b x + a\right )}^{p} {\left (d x + c\right )}^{q} f\right )^{r} e\right )^{2} + \frac {{\left (\frac {6 \, a^{3} f p \log \left (b x + a\right )}{b} - \frac {2 \, b^{2} d^{2} f {\left (p + q\right )} x^{3} + 3 \, {\left (a b d^{2} f {\left (2 \, p + 3 \, q\right )} - b^{2} c d f q\right )} x^{2} + 6 \, {\left (a^{2} d^{2} f {\left (p + 3 \, q\right )} + b^{2} c^{2} f q - 3 \, a b c d f q\right )} x}{d^{2}} + \frac {6 \, {\left (b^{2} c^{3} f q - 3 \, a b c^{2} d f q + 3 \, a^{2} c d^{2} f q\right )} \log \left (d x + c\right )}{d^{3}}\right )} r \log \left (\left ({\left (b x + a\right )}^{p} {\left (d x + c\right )}^{q} f\right )^{r} e\right )}{9 \, f} - \frac {r^{2} {\left (\frac {6 \, {\left ({\left (2 \, p q + 11 \, q^{2}\right )} b^{2} c^{3} f^{2} - 3 \, {\left (2 \, p q + 9 \, q^{2}\right )} a b c^{2} d f^{2} + 6 \, {\left (p q + 3 \, q^{2}\right )} a^{2} c d^{2} f^{2}\right )} \log \left (d x + c\right )}{d^{3}} - \frac {36 \, {\left (b^{3} c^{3} f^{2} p q - 3 \, a b^{2} c^{2} d f^{2} p q + 3 \, a^{2} b c d^{2} f^{2} p q - a^{3} d^{3} f^{2} p q\right )} {\left (\log \left (b x + a\right ) \log \left (\frac {b d x + a d}{b c - a d} + 1\right ) + {\rm Li}_2\left (-\frac {b d x + a d}{b c - a d}\right )\right )}}{b d^{3}} - \frac {4 \, {\left (p^{2} + 2 \, p q + q^{2}\right )} b^{3} d^{3} f^{2} x^{3} - 18 \, a^{3} d^{3} f^{2} p^{2} \log \left (b x + a\right )^{2} - 3 \, {\left (5 \, {\left (p q + q^{2}\right )} b^{3} c d^{2} f^{2} - {\left (4 \, p^{2} + 13 \, p q + 9 \, q^{2}\right )} a b^{2} d^{3} f^{2}\right )} x^{2} - 36 \, {\left (b^{3} c^{3} f^{2} p q - 3 \, a b^{2} c^{2} d f^{2} p q + 3 \, a^{2} b c d^{2} f^{2} p q\right )} \log \left (b x + a\right ) \log \left (d x + c\right ) - 18 \, {\left (b^{3} c^{3} f^{2} q^{2} - 3 \, a b^{2} c^{2} d f^{2} q^{2} + 3 \, a^{2} b c d^{2} f^{2} q^{2}\right )} \log \left (d x + c\right )^{2} + 6 \, {\left ({\left (8 \, p q + 11 \, q^{2}\right )} b^{3} c^{2} d f^{2} - 3 \, {\left (7 \, p q + 9 \, q^{2}\right )} a b^{2} c d^{2} f^{2} + {\left (2 \, p^{2} + 17 \, p q + 18 \, q^{2}\right )} a^{2} b d^{3} f^{2}\right )} x - 6 \, {\left (6 \, a b^{2} c^{2} d f^{2} p q - 15 \, a^{2} b c d^{2} f^{2} p q + {\left (2 \, p^{2} + 11 \, p q\right )} a^{3} d^{3} f^{2}\right )} \log \left (b x + a\right )}{b d^{3}}\right )}}{54 \, f^{2}} \]
1/3*(b^2*x^3 + 3*a*b*x^2 + 3*a^2*x)*log(((b*x + a)^p*(d*x + c)^q*f)^r*e)^2 + 1/9*(6*a^3*f*p*log(b*x + a)/b - (2*b^2*d^2*f*(p + q)*x^3 + 3*(a*b*d^2*f *(2*p + 3*q) - b^2*c*d*f*q)*x^2 + 6*(a^2*d^2*f*(p + 3*q) + b^2*c^2*f*q - 3 *a*b*c*d*f*q)*x)/d^2 + 6*(b^2*c^3*f*q - 3*a*b*c^2*d*f*q + 3*a^2*c*d^2*f*q) *log(d*x + c)/d^3)*r*log(((b*x + a)^p*(d*x + c)^q*f)^r*e)/f - 1/54*r^2*(6* ((2*p*q + 11*q^2)*b^2*c^3*f^2 - 3*(2*p*q + 9*q^2)*a*b*c^2*d*f^2 + 6*(p*q + 3*q^2)*a^2*c*d^2*f^2)*log(d*x + c)/d^3 - 36*(b^3*c^3*f^2*p*q - 3*a*b^2*c^ 2*d*f^2*p*q + 3*a^2*b*c*d^2*f^2*p*q - a^3*d^3*f^2*p*q)*(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)))/(b*d^3) - (4*(p^2 + 2*p*q + q^2)*b^3*d^3*f^2*x^3 - 18*a^3*d^3*f^2*p^2*log(b*x + a )^2 - 3*(5*(p*q + q^2)*b^3*c*d^2*f^2 - (4*p^2 + 13*p*q + 9*q^2)*a*b^2*d^3* f^2)*x^2 - 36*(b^3*c^3*f^2*p*q - 3*a*b^2*c^2*d*f^2*p*q + 3*a^2*b*c*d^2*f^2 *p*q)*log(b*x + a)*log(d*x + c) - 18*(b^3*c^3*f^2*q^2 - 3*a*b^2*c^2*d*f^2* q^2 + 3*a^2*b*c*d^2*f^2*q^2)*log(d*x + c)^2 + 6*((8*p*q + 11*q^2)*b^3*c^2* d*f^2 - 3*(7*p*q + 9*q^2)*a*b^2*c*d^2*f^2 + (2*p^2 + 17*p*q + 18*q^2)*a^2* b*d^3*f^2)*x - 6*(6*a*b^2*c^2*d*f^2*p*q - 15*a^2*b*c*d^2*f^2*p*q + (2*p^2 + 11*p*q)*a^3*d^3*f^2)*log(b*x + a))/(b*d^3))/f^2
\[ \int (a+b x)^2 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx=\int { {\left (b x + a\right )}^{2} \log \left (\left ({\left (b x + a\right )}^{p} {\left (d x + c\right )}^{q} f\right )^{r} e\right )^{2} \,d x } \]
Timed out. \[ \int (a+b x)^2 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx=\int {\ln \left (e\,{\left (f\,{\left (a+b\,x\right )}^p\,{\left (c+d\,x\right )}^q\right )}^r\right )}^2\,{\left (a+b\,x\right )}^2 \,d x \]