Integrand size = 31, antiderivative size = 920 \[ \int (a+b x)^4 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx=-\frac {a (b c-a d)^3 p q r^2 x}{5 d^3}+\frac {2 (b c-a d)^4 p q r^2 x}{25 d^4}+\frac {77 (b c-a d)^4 q^2 r^2 x}{150 d^4}+\frac {2 (b c-a d)^4 q (p+q) r^2 x}{5 d^4}-\frac {b (b c-a d)^3 p q r^2 x^2}{10 d^3}-\frac {(b c-a d)^3 p q r^2 (a+b x)^2}{25 b d^3}-\frac {77 (b c-a d)^3 q^2 r^2 (a+b x)^2}{300 b d^3}+\frac {16 (b c-a d)^2 p q r^2 (a+b x)^3}{225 b d^2}+\frac {47 (b c-a d)^2 q^2 r^2 (a+b x)^3}{450 b d^2}-\frac {9 (b c-a d) p q r^2 (a+b x)^4}{200 b d}-\frac {9 (b c-a d) q^2 r^2 (a+b x)^4}{200 b d}+\frac {2 p^2 r^2 (a+b x)^5}{125 b}+\frac {4 p q r^2 (a+b x)^5}{125 b}+\frac {2 q^2 r^2 (a+b x)^5}{125 b}-\frac {2 (b c-a d)^5 p q r^2 \log (c+d x)}{25 b d^5}-\frac {137 (b c-a d)^5 q^2 r^2 \log (c+d x)}{150 b d^5}-\frac {2 (b c-a d)^5 p q r^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{5 b d^5}-\frac {(b c-a d)^5 q^2 r^2 \log ^2(c+d x)}{5 b d^5}-\frac {2 (b c-a d)^4 q r (a+b x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{5 b d^4}+\frac {(b c-a d)^3 q r (a+b x)^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{5 b d^3}-\frac {2 (b c-a d)^2 q r (a+b x)^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{15 b d^2}+\frac {(b c-a d) q r (a+b x)^4 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{10 b d}-\frac {2 p r (a+b x)^5 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{25 b}-\frac {2 q r (a+b x)^5 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{25 b}+\frac {2 (b c-a d)^5 q r \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{5 b d^5}+\frac {(a+b x)^5 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{5 b}-\frac {2 (b c-a d)^5 p q r^2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )}{5 b d^5} \]
1/5*(b*x+a)^5*ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r)^2/b+2/125*p^2*r^2*(b*x+a)^5/ b+2/125*q^2*r^2*(b*x+a)^5/b-2/5*(-a*d+b*c)^5*p*q*r^2*ln(-d*(b*x+a)/(-a*d+b *c))*ln(d*x+c)/b/d^5-137/150*(-a*d+b*c)^5*q^2*r^2*ln(d*x+c)/b/d^5-1/5*(-a* d+b*c)^5*q^2*r^2*ln(d*x+c)^2/b/d^5+2/25*(-a*d+b*c)^4*p*q*r^2*x/d^4+2/5*(-a *d+b*c)^4*q*(p+q)*r^2*x/d^4-77/300*(-a*d+b*c)^3*q^2*r^2*(b*x+a)^2/b/d^3+47 /450*(-a*d+b*c)^2*q^2*r^2*(b*x+a)^3/b/d^2-9/200*(-a*d+b*c)*q^2*r^2*(b*x+a) ^4/b/d-2/25*p*r*(b*x+a)^5*ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r)/b-2/25*q*r*(b*x+ a)^5*ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r)/b+77/150*(-a*d+b*c)^4*q^2*r^2*x/d^4+4 /125*p*q*r^2*(b*x+a)^5/b-2/25*(-a*d+b*c)^5*p*q*r^2*ln(d*x+c)/b/d^5-2/5*(-a *d+b*c)^4*q*r*(b*x+a)*ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r)/b/d^4+1/5*(-a*d+b*c) ^3*q*r*(b*x+a)^2*ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r)/b/d^3-2/15*(-a*d+b*c)^2*q *r*(b*x+a)^3*ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r)/b/d^2+1/10*(-a*d+b*c)*q*r*(b* x+a)^4*ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r)/b/d+2/5*(-a*d+b*c)^5*q*r*ln(d*x+c)* ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r)/b/d^5-9/200*(-a*d+b*c)*p*q*r^2*(b*x+a)^4/b /d-2/5*(-a*d+b*c)^5*p*q*r^2*polylog(2,b*(d*x+c)/(-a*d+b*c))/b/d^5-1/5*a*(- a*d+b*c)^3*p*q*r^2*x/d^3-1/10*b*(-a*d+b*c)^3*p*q*r^2*x^2/d^3-1/25*(-a*d+b* c)^3*p*q*r^2*(b*x+a)^2/b/d^3+16/225*(-a*d+b*c)^2*p*q*r^2*(b*x+a)^3/b/d^2
Leaf count is larger than twice the leaf count of optimal. \(2508\) vs. \(2(920)=1840\).
Time = 1.54 (sec) , antiderivative size = 2508, normalized size of antiderivative = 2.73 \[ \int (a+b x)^4 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx=\text {Result too large to show} \]
(2*a^5*p*q*r^2)/b + (2*a*b^3*c^4*p*q*r^2)/(5*d^4) - (2*a^2*b^2*c^3*p*q*r^2 )/d^3 + (4*a^3*b*c^2*p*q*r^2)/d^2 - (4*a^4*c*p*q*r^2)/d + (2*a^4*p^2*r^2*x )/25 + (197*a^4*p*q*r^2*x)/150 + (12*b^4*c^4*p*q*r^2*x)/(25*d^4) - (11*a*b ^3*c^3*p*q*r^2*x)/(5*d^3) + (59*a^2*b^2*c^2*p*q*r^2*x)/(15*d^2) - (101*a^3 *b*c*p*q*r^2*x)/(30*d) + 2*a^4*q^2*r^2*x + (137*b^4*c^4*q^2*r^2*x)/(150*d^ 4) - (25*a*b^3*c^3*q^2*r^2*x)/(6*d^3) + (22*a^2*b^2*c^2*q^2*r^2*x)/(3*d^2) - (6*a^3*b*c*q^2*r^2*x)/d + (4*a^3*b*p^2*r^2*x^2)/25 + (283*a^3*b*p*q*r^2 *x^2)/300 - (7*b^4*c^3*p*q*r^2*x^2)/(50*d^3) + (19*a*b^3*c^2*p*q*r^2*x^2)/ (30*d^2) - (67*a^2*b^2*c*p*q*r^2*x^2)/(60*d) + a^3*b*q^2*r^2*x^2 - (77*b^4 *c^3*q^2*r^2*x^2)/(300*d^3) + (13*a*b^3*c^2*q^2*r^2*x^2)/(12*d^2) - (5*a^2 *b^2*c*q^2*r^2*x^2)/(3*d) + (4*a^2*b^2*p^2*r^2*x^3)/25 + (257*a^2*b^2*p*q* r^2*x^3)/450 + (16*b^4*c^2*p*q*r^2*x^3)/(225*d^2) - (29*a*b^3*c*p*q*r^2*x^ 3)/(90*d) + (4*a^2*b^2*q^2*r^2*x^3)/9 + (47*b^4*c^2*q^2*r^2*x^3)/(450*d^2) - (7*a*b^3*c*q^2*r^2*x^3)/(18*d) + (2*a*b^3*p^2*r^2*x^4)/25 + (41*a*b^3*p *q*r^2*x^4)/200 - (9*b^4*c*p*q*r^2*x^4)/(200*d) + (a*b^3*q^2*r^2*x^4)/8 - (9*b^4*c*q^2*r^2*x^4)/(200*d) + (2*b^4*p^2*r^2*x^5)/125 + (4*b^4*p*q*r^2*x ^5)/125 + (2*b^4*q^2*r^2*x^5)/125 - (a^5*p^2*r^2*Log[a + b*x]^2)/(5*b) + ( 2*a^5*p*q*r^2*Log[c + d*x])/b - (2*b^4*c^5*p*q*r^2*Log[c + d*x])/(25*d^5) + (2*a*b^3*c^4*p*q*r^2*Log[c + d*x])/(5*d^4) - (4*a^2*b^2*c^3*p*q*r^2*Log[ c + d*x])/(5*d^3) + (4*a^3*b*c^2*p*q*r^2*Log[c + d*x])/(5*d^2) - (2*a^4...
Time = 1.36 (sec) , antiderivative size = 843, normalized size of antiderivative = 0.92, 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.
| \(\displaystyle \int (a+b x)^4 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx\) |
| \(\Big \downarrow \) 2984 |
| \(\displaystyle -\frac {2}{5} p r \int (a+b x)^4 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )dx-\frac {2 d q r \int \frac {(a+b x)^5 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{c+d x}dx}{5 b}+\frac {(a+b x)^5 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{5 b}\) |
| \(\Big \downarrow \) 2981 |
| \(\displaystyle -\frac {2}{5} p r \left (-\frac {d q r \int \frac {(a+b x)^5}{c+d x}dx}{5 b}-\frac {1}{5} p r \int (a+b x)^4dx+\frac {(a+b x)^5 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{5 b}\right )-\frac {2 d q r \int \frac {(a+b x)^5 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{c+d x}dx}{5 b}+\frac {(a+b x)^5 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{5 b}\) |
| \(\Big \downarrow \) 17 |
| \(\displaystyle -\frac {2}{5} p r \left (-\frac {d q r \int \frac {(a+b x)^5}{c+d x}dx}{5 b}+\frac {(a+b x)^5 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{5 b}-\frac {p r (a+b x)^5}{25 b}\right )-\frac {2 d q r \int \frac {(a+b x)^5 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{c+d x}dx}{5 b}+\frac {(a+b x)^5 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{5 b}\) |
| \(\Big \downarrow \) 49 |
| \(\displaystyle -\frac {2}{5} p r \left (-\frac {d q r \int \left (\frac {(a d-b c)^5}{d^5 (c+d x)}+\frac {b (b c-a d)^4}{d^5}+\frac {b (a+b x)^4}{d}-\frac {b (b c-a d) (a+b x)^3}{d^2}+\frac {b (b c-a d)^2 (a+b x)^2}{d^3}-\frac {b (b c-a d)^3 (a+b x)}{d^4}\right )dx}{5 b}+\frac {(a+b x)^5 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{5 b}-\frac {p r (a+b x)^5}{25 b}\right )-\frac {2 d q r \int \frac {(a+b x)^5 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{c+d x}dx}{5 b}+\frac {(a+b x)^5 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{5 b}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 d q r \int \frac {(a+b x)^5 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{c+d x}dx}{5 b}-\frac {2}{5} p r \left (-\frac {d q r \left (-\frac {(b c-a d)^5 \log (c+d x)}{d^6}+\frac {b x (b c-a d)^4}{d^5}-\frac {(a+b x)^2 (b c-a d)^3}{2 d^4}+\frac {(a+b x)^3 (b c-a d)^2}{3 d^3}-\frac {(a+b x)^4 (b c-a d)}{4 d^2}+\frac {(a+b x)^5}{5 d}\right )}{5 b}+\frac {(a+b x)^5 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{5 b}-\frac {p r (a+b x)^5}{25 b}\right )+\frac {(a+b x)^5 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{5 b}\) |
| \(\Big \downarrow \) 2994 |
| \(\displaystyle -\frac {2 d q r \int \left (\frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) (a d-b c)^5}{d^5 (c+d x)}+\frac {b (b c-a d)^4 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{d^5}+\frac {b (a+b x)^4 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{d}-\frac {b (b c-a d) (a+b x)^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{d^2}+\frac {b (b c-a d)^2 (a+b x)^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{d^3}-\frac {b (b c-a d)^3 (a+b x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{d^4}\right )dx}{5 b}-\frac {2}{5} p r \left (-\frac {d q r \left (-\frac {(b c-a d)^5 \log (c+d x)}{d^6}+\frac {b x (b c-a d)^4}{d^5}-\frac {(a+b x)^2 (b c-a d)^3}{2 d^4}+\frac {(a+b x)^3 (b c-a d)^2}{3 d^3}-\frac {(a+b x)^4 (b c-a d)}{4 d^2}+\frac {(a+b x)^5}{5 d}\right )}{5 b}+\frac {(a+b x)^5 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{5 b}-\frac {p r (a+b x)^5}{25 b}\right )+\frac {(a+b x)^5 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{5 b}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) (a+b x)^5}{5 b}-\frac {2}{5} p r \left (-\frac {p r (a+b x)^5}{25 b}+\frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) (a+b x)^5}{5 b}-\frac {d q r \left (-\frac {\log (c+d x) (b c-a d)^5}{d^6}+\frac {b x (b c-a d)^4}{d^5}-\frac {(a+b x)^2 (b c-a d)^3}{2 d^4}+\frac {(a+b x)^3 (b c-a d)^2}{3 d^3}-\frac {(a+b x)^4 (b c-a d)}{4 d^2}+\frac {(a+b x)^5}{5 d}\right )}{5 b}\right )-\frac {2 d q r \left (\frac {q r \log ^2(c+d x) (b c-a d)^5}{2 d^6}+\frac {137 q r \log (c+d x) (b c-a d)^5}{60 d^6}+\frac {p r \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x) (b c-a d)^5}{d^6}-\frac {\log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) (b c-a d)^5}{d^6}+\frac {p r \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right ) (b c-a d)^5}{d^6}-\frac {77 b q r x (b c-a d)^4}{60 d^5}-\frac {b (p+q) r x (b c-a d)^4}{d^5}+\frac {(a+b x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) (b c-a d)^4}{d^5}+\frac {p r (a+b x)^2 (b c-a d)^3}{4 d^4}+\frac {77 q r (a+b x)^2 (b c-a d)^3}{120 d^4}-\frac {(a+b x)^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) (b c-a d)^3}{2 d^4}-\frac {p r (a+b x)^3 (b c-a d)^2}{9 d^3}-\frac {47 q r (a+b x)^3 (b c-a d)^2}{180 d^3}+\frac {(a+b x)^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) (b c-a d)^2}{3 d^3}+\frac {p r (a+b x)^4 (b c-a d)}{16 d^2}+\frac {9 q r (a+b x)^4 (b c-a d)}{80 d^2}-\frac {(a+b x)^4 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) (b c-a d)}{4 d^2}-\frac {p r (a+b x)^5}{25 d}-\frac {q r (a+b x)^5}{25 d}+\frac {(a+b x)^5 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{5 d}\right )}{5 b}\) |
((a + b*x)^5*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^2)/(5*b) - (2*p*r*(-1/25 *(p*r*(a + b*x)^5)/b - (d*q*r*((b*(b*c - a*d)^4*x)/d^5 - ((b*c - a*d)^3*(a + b*x)^2)/(2*d^4) + ((b*c - a*d)^2*(a + b*x)^3)/(3*d^3) - ((b*c - a*d)*(a + b*x)^4)/(4*d^2) + (a + b*x)^5/(5*d) - ((b*c - a*d)^5*Log[c + d*x])/d^6) )/(5*b) + ((a + b*x)^5*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(5*b)))/5 - ( 2*d*q*r*((-77*b*(b*c - a*d)^4*q*r*x)/(60*d^5) - (b*(b*c - a*d)^4*(p + q)*r *x)/d^5 + ((b*c - a*d)^3*p*r*(a + b*x)^2)/(4*d^4) + (77*(b*c - a*d)^3*q*r* (a + b*x)^2)/(120*d^4) - ((b*c - a*d)^2*p*r*(a + b*x)^3)/(9*d^3) - (47*(b* c - a*d)^2*q*r*(a + b*x)^3)/(180*d^3) + ((b*c - a*d)*p*r*(a + b*x)^4)/(16* d^2) + (9*(b*c - a*d)*q*r*(a + b*x)^4)/(80*d^2) - (p*r*(a + b*x)^5)/(25*d) - (q*r*(a + b*x)^5)/(25*d) + (137*(b*c - a*d)^5*q*r*Log[c + d*x])/(60*d^6 ) + ((b*c - a*d)^5*p*r*Log[-((d*(a + b*x))/(b*c - a*d))]*Log[c + d*x])/d^6 + ((b*c - a*d)^5*q*r*Log[c + d*x]^2)/(2*d^6) + ((b*c - a*d)^4*(a + b*x)*L og[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/d^5 - ((b*c - a*d)^3*(a + b*x)^2*Log[ e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(2*d^4) + ((b*c - a*d)^2*(a + b*x)^3*Log [e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(3*d^3) - ((b*c - a*d)*(a + b*x)^4*Log[ e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(4*d^2) + ((a + b*x)^5*Log[e*(f*(a + b*x )^p*(c + d*x)^q)^r])/(5*d) - ((b*c - a*d)^5*Log[c + d*x]*Log[e*(f*(a + b*x )^p*(c + d*x)^q)^r])/d^6 + ((b*c - a*d)^5*p*r*PolyLog[2, (b*(c + d*x))/(b* c - a*d)])/d^6))/(5*b)
3.1.16.3.1 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]
\[\int \left (b x +a \right )^{4} {\ln \left (e \left (f \left (b x +a \right )^{p} \left (d x +c \right )^{q}\right )^{r}\right )}^{2}d x\]
\[ \int (a+b x)^4 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx=\int { {\left (b x + a\right )}^{4} \log \left (\left ({\left (b x + a\right )}^{p} {\left (d x + c\right )}^{q} f\right )^{r} e\right )^{2} \,d x } \]
integral((b^4*x^4 + 4*a*b^3*x^3 + 6*a^2*b^2*x^2 + 4*a^3*b*x + a^4)*log(((b *x + a)^p*(d*x + c)^q*f)^r*e)^2, x)
Timed out. \[ \int (a+b x)^4 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx=\text {Timed out} \]
Time = 0.24 (sec) , antiderivative size = 1421, normalized size of antiderivative = 1.54 \[ \int (a+b x)^4 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx=\text {Too large to display} \]
1/5*(b^4*x^5 + 5*a*b^3*x^4 + 10*a^2*b^2*x^3 + 10*a^3*b*x^2 + 5*a^4*x)*log( ((b*x + a)^p*(d*x + c)^q*f)^r*e)^2 + 1/150*(60*a^5*f*p*log(b*x + a)/b - (1 2*b^4*d^4*f*(p + q)*x^5 + 15*(a*b^3*d^4*f*(4*p + 5*q) - b^4*c*d^3*f*q)*x^4 + 20*(2*a^2*b^2*d^4*f*(3*p + 5*q) + b^4*c^2*d^2*f*q - 5*a*b^3*c*d^3*f*q)* x^3 + 30*(2*a^3*b*d^4*f*(2*p + 5*q) - b^4*c^3*d*f*q + 5*a*b^3*c^2*d^2*f*q - 10*a^2*b^2*c*d^3*f*q)*x^2 + 60*(a^4*d^4*f*(p + 5*q) + b^4*c^4*f*q - 5*a* b^3*c^3*d*f*q + 10*a^2*b^2*c^2*d^2*f*q - 10*a^3*b*c*d^3*f*q)*x)/d^4 + 60*( b^4*c^5*f*q - 5*a*b^3*c^4*d*f*q + 10*a^2*b^2*c^3*d^2*f*q - 10*a^3*b*c^2*d^ 3*f*q + 5*a^4*c*d^4*f*q)*log(d*x + c)/d^5)*r*log(((b*x + a)^p*(d*x + c)^q* f)^r*e)/f - 1/9000*r^2*(60*((12*p*q + 137*q^2)*b^4*c^5*f^2 - 5*(12*p*q + 1 25*q^2)*a*b^3*c^4*d*f^2 + 20*(6*p*q + 55*q^2)*a^2*b^2*c^3*d^2*f^2 - 60*(2* p*q + 15*q^2)*a^3*b*c^2*d^3*f^2 + 60*(p*q + 5*q^2)*a^4*c*d^4*f^2)*log(d*x + c)/d^5 - 3600*(b^5*c^5*f^2*p*q - 5*a*b^4*c^4*d*f^2*p*q + 10*a^2*b^3*c^3* d^2*f^2*p*q - 10*a^3*b^2*c^2*d^3*f^2*p*q + 5*a^4*b*c*d^4*f^2*p*q - a^5*d^5 *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^5) - (144*(p^2 + 2*p*q + q^2)*b^5*d^5*f^2*x^5 - 1800*a^5*d^5*f^2*p^2*log(b*x + a)^2 - 45*(9*(p*q + q^2)*b^5*c*d^4*f^2 - ( 16*p^2 + 41*p*q + 25*q^2)*a*b^4*d^5*f^2)*x^4 + 20*((32*p*q + 47*q^2)*b^5*c ^2*d^3*f^2 - 5*(29*p*q + 35*q^2)*a*b^4*c*d^4*f^2 + (72*p^2 + 257*p*q + 200 *q^2)*a^2*b^3*d^5*f^2)*x^3 - 30*(7*(6*p*q + 11*q^2)*b^5*c^3*d^2*f^2 - 5...
\[ \int (a+b x)^4 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx=\int { {\left (b x + a\right )}^{4} \log \left (\left ({\left (b x + a\right )}^{p} {\left (d x + c\right )}^{q} f\right )^{r} e\right )^{2} \,d x } \]
Timed out. \[ \int (a+b x)^4 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx=\int {\ln \left (e\,{\left (f\,{\left (a+b\,x\right )}^p\,{\left (c+d\,x\right )}^q\right )}^r\right )}^2\,{\left (a+b\,x\right )}^4 \,d x \]