\(\int \frac {\log ^2(e (f (a+b x)^p (c+d x)^q)^r)}{(a+b x)^5} \, dx\) [24]

Optimal result

Integrand size = 31, antiderivative size = 884 \[ \int \frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(a+b x)^5} \, dx=-\frac {p^2 r^2}{32 b (a+b x)^4}-\frac {7 d p q r^2}{72 b (b c-a d) (a+b x)^3}+\frac {3 d^2 p q r^2}{16 b (b c-a d)^2 (a+b x)^2}-\frac {d^2 q^2 r^2}{12 b (b c-a d)^2 (a+b x)^2}-\frac {5 d^3 p q r^2}{8 b (b c-a d)^3 (a+b x)}+\frac {5 d^3 q^2 r^2}{12 b (b c-a d)^3 (a+b x)}-\frac {d^4 p q r^2 \log (a+b x)}{8 b (b c-a d)^4}+\frac {11 d^4 q^2 r^2 \log (a+b x)}{12 b (b c-a d)^4}+\frac {d^4 p q r^2 \log ^2(a+b x)}{4 b (b c-a d)^4}+\frac {d^4 p q r^2 \log (c+d x)}{8 b (b c-a d)^4}-\frac {11 d^4 q^2 r^2 \log (c+d x)}{12 b (b c-a d)^4}-\frac {d^4 p q r^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{2 b (b c-a d)^4}-\frac {d^4 q^2 r^2 \log ^2(c+d x)}{4 b (b c-a d)^4}+\frac {d^4 q^2 r^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{2 b (b c-a d)^4}-\frac {p r \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{8 b (a+b x)^4}-\frac {d q r \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{6 b (b c-a d) (a+b x)^3}+\frac {d^2 q r \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{4 b (b c-a d)^2 (a+b x)^2}-\frac {d^3 q r \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 b (b c-a d)^3 (a+b x)}-\frac {d^4 q r \log (a+b x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 b (b c-a d)^4}+\frac {d^4 q r \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 b (b c-a d)^4}-\frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{4 b (a+b x)^4}+\frac {d^4 q^2 r^2 \operatorname {PolyLog}\left (2,-\frac {d (a+b x)}{b c-a d}\right )}{2 b (b c-a d)^4}-\frac {d^4 p q r^2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )}{2 b (b c-a d)^4} \] Output:

-1/32*p^2*r^2/b/(b*x+a)^4-7/72*d*p*q*r^2/b/(-a*d+b*c)/(b*x+a)^3+3/16*d^2*p 
*q*r^2/b/(-a*d+b*c)^2/(b*x+a)^2-1/12*d^2*q^2*r^2/b/(-a*d+b*c)^2/(b*x+a)^2- 
5/8*d^3*p*q*r^2/b/(-a*d+b*c)^3/(b*x+a)+5/12*d^3*q^2*r^2/b/(-a*d+b*c)^3/(b* 
x+a)-1/8*d^4*p*q*r^2*ln(b*x+a)/b/(-a*d+b*c)^4+11/12*d^4*q^2*r^2*ln(b*x+a)/ 
b/(-a*d+b*c)^4+1/4*d^4*p*q*r^2*ln(b*x+a)^2/b/(-a*d+b*c)^4+1/8*d^4*p*q*r^2* 
ln(d*x+c)/b/(-a*d+b*c)^4-11/12*d^4*q^2*r^2*ln(d*x+c)/b/(-a*d+b*c)^4-1/2*d^ 
4*p*q*r^2*ln(-d*(b*x+a)/(-a*d+b*c))*ln(d*x+c)/b/(-a*d+b*c)^4-1/4*d^4*q^2*r 
^2*ln(d*x+c)^2/b/(-a*d+b*c)^4+1/2*d^4*q^2*r^2*ln(b*x+a)*ln(b*(d*x+c)/(-a*d 
+b*c))/b/(-a*d+b*c)^4-1/8*p*r*ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r)/b/(b*x+a)^4- 
1/6*d*q*r*ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r)/b/(-a*d+b*c)/(b*x+a)^3+1/4*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)^2-1/2*d^3*q*r*ln 
(e*(f*(b*x+a)^p*(d*x+c)^q)^r)/b/(-a*d+b*c)^3/(b*x+a)-1/2*d^4*q*r*ln(b*x+a) 
*ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r)/b/(-a*d+b*c)^4+1/2*d^4*q*r*ln(d*x+c)*ln(e 
*(f*(b*x+a)^p*(d*x+c)^q)^r)/b/(-a*d+b*c)^4-1/4*ln(e*(f*(b*x+a)^p*(d*x+c)^q 
)^r)^2/b/(b*x+a)^4+1/2*d^4*q^2*r^2*polylog(2,-d*(b*x+a)/(-a*d+b*c))/b/(-a* 
d+b*c)^4-1/2*d^4*p*q*r^2*polylog(2,b*(d*x+c)/(-a*d+b*c))/b/(-a*d+b*c)^4
 

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(2003\) vs. \(2(884)=1768\).

Time = 2.35 (sec) , antiderivative size = 2003, normalized size of antiderivative = 2.27 \[ \int \frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(a+b x)^5} \, dx=\text {Result too large to show} \] Input:

Integrate[Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^2/(a + b*x)^5,x]
 

Output:

(-9*b^4*c^4*p^2*r^2 + 36*a*b^3*c^3*d*p^2*r^2 - 54*a^2*b^2*c^2*d^2*p^2*r^2 
+ 36*a^3*b*c*d^3*p^2*r^2 - 9*a^4*d^4*p^2*r^2 - 28*a*b^3*c^3*d*p*q*r^2 + 13 
8*a^2*b^2*c^2*d^2*p*q*r^2 - 372*a^3*b*c*d^3*p*q*r^2 + 262*a^4*d^4*p*q*r^2 
- 24*a^2*b^2*c^2*d^2*q^2*r^2 + 168*a^3*b*c*d^3*q^2*r^2 - 144*a^4*d^4*q^2*r 
^2 - 28*b^4*c^3*d*p*q*r^2*x + 192*a*b^3*c^2*d^2*p*q*r^2*x - 840*a^2*b^2*c* 
d^3*p*q*r^2*x + 676*a^3*b*d^4*p*q*r^2*x - 48*a*b^3*c^2*d^2*q^2*r^2*x + 456 
*a^2*b^2*c*d^3*q^2*r^2*x - 408*a^3*b*d^4*q^2*r^2*x + 54*b^4*c^2*d^2*p*q*r^ 
2*x^2 - 648*a*b^3*c*d^3*p*q*r^2*x^2 + 594*a^2*b^2*d^4*p*q*r^2*x^2 - 24*b^4 
*c^2*d^2*q^2*r^2*x^2 + 408*a*b^3*c*d^3*q^2*r^2*x^2 - 384*a^2*b^2*d^4*q^2*r 
^2*x^2 - 180*b^4*c*d^3*p*q*r^2*x^3 + 180*a*b^3*d^4*p*q*r^2*x^3 + 120*b^4*c 
*d^3*q^2*r^2*x^3 - 120*a*b^3*d^4*q^2*r^2*x^3 + 72*d^4*p*q*r^2*(a + b*x)^4* 
Log[a + b*x]^2 + 36*a^4*d^4*p*q*r^2*Log[c + d*x] - 264*a^4*d^4*q^2*r^2*Log 
[c + d*x] + 144*a^3*b*d^4*p*q*r^2*x*Log[c + d*x] - 1056*a^3*b*d^4*q^2*r^2* 
x*Log[c + d*x] + 216*a^2*b^2*d^4*p*q*r^2*x^2*Log[c + d*x] - 1584*a^2*b^2*d 
^4*q^2*r^2*x^2*Log[c + d*x] + 144*a*b^3*d^4*p*q*r^2*x^3*Log[c + d*x] - 105 
6*a*b^3*d^4*q^2*r^2*x^3*Log[c + d*x] + 36*b^4*d^4*p*q*r^2*x^4*Log[c + d*x] 
 - 264*b^4*d^4*q^2*r^2*x^4*Log[c + d*x] - 72*a^4*d^4*q^2*r^2*Log[c + d*x]^ 
2 - 288*a^3*b*d^4*q^2*r^2*x*Log[c + d*x]^2 - 432*a^2*b^2*d^4*q^2*r^2*x^2*L 
og[c + d*x]^2 - 288*a*b^3*d^4*q^2*r^2*x^3*Log[c + d*x]^2 - 72*b^4*d^4*q^2* 
r^2*x^4*Log[c + d*x]^2 + 12*d^4*q*r*(a + b*x)^4*Log[a + b*x]*(-3*p*r + ...
 

Rubi [A] (verified)

Time = 1.45 (sec) , antiderivative size = 814, 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, 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.

\(\displaystyle \int \frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(a+b x)^5} \, dx\)

\(\Big \downarrow \) 2984

\(\displaystyle \frac {1}{2} p r \int \frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(a+b x)^5}dx+\frac {d q r \int \frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(a+b x)^4 (c+d x)}dx}{2 b}-\frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{4 b (a+b x)^4}\)

\(\Big \downarrow \) 2981

\(\displaystyle \frac {1}{2} p r \left (\frac {d q r \int \frac {1}{(a+b x)^4 (c+d x)}dx}{4 b}+\frac {1}{4} p r \int \frac {1}{(a+b x)^5}dx-\frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{4 b (a+b x)^4}\right )+\frac {d q r \int \frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(a+b x)^4 (c+d x)}dx}{2 b}-\frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{4 b (a+b x)^4}\)

\(\Big \downarrow \) 17

\(\displaystyle \frac {1}{2} p r \left (\frac {d q r \int \frac {1}{(a+b x)^4 (c+d x)}dx}{4 b}-\frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{4 b (a+b x)^4}-\frac {p r}{16 b (a+b x)^4}\right )+\frac {d q r \int \frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(a+b x)^4 (c+d x)}dx}{2 b}-\frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{4 b (a+b x)^4}\)

\(\Big \downarrow \) 54

\(\displaystyle \frac {1}{2} p r \left (\frac {d q r \int \left (\frac {d^4}{(b c-a d)^4 (c+d x)}-\frac {b d^3}{(b c-a d)^4 (a+b x)}+\frac {b d^2}{(b c-a d)^3 (a+b x)^2}-\frac {b d}{(b c-a d)^2 (a+b x)^3}+\frac {b}{(b c-a d) (a+b x)^4}\right )dx}{4 b}-\frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{4 b (a+b x)^4}-\frac {p r}{16 b (a+b x)^4}\right )+\frac {d q r \int \frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(a+b x)^4 (c+d x)}dx}{2 b}-\frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{4 b (a+b x)^4}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {d q r \int \frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(a+b x)^4 (c+d x)}dx}{2 b}+\frac {1}{2} p r \left (\frac {d q r \left (-\frac {d^3 \log (a+b x)}{(b c-a d)^4}+\frac {d^3 \log (c+d x)}{(b c-a d)^4}-\frac {d^2}{(a+b x) (b c-a d)^3}+\frac {d}{2 (a+b x)^2 (b c-a d)^2}-\frac {1}{3 (a+b x)^3 (b c-a d)}\right )}{4 b}-\frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{4 b (a+b x)^4}-\frac {p r}{16 b (a+b x)^4}\right )-\frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{4 b (a+b x)^4}\)

\(\Big \downarrow \) 2994

\(\displaystyle \frac {d q r \int \left (\frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) d^4}{(b c-a d)^4 (c+d x)}-\frac {b \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) d^3}{(b c-a d)^4 (a+b x)}+\frac {b \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) d^2}{(b c-a d)^3 (a+b x)^2}-\frac {b \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) d}{(b c-a d)^2 (a+b x)^3}+\frac {b \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(b c-a d) (a+b x)^4}\right )dx}{2 b}+\frac {1}{2} p r \left (\frac {d q r \left (-\frac {d^3 \log (a+b x)}{(b c-a d)^4}+\frac {d^3 \log (c+d x)}{(b c-a d)^4}-\frac {d^2}{(a+b x) (b c-a d)^3}+\frac {d}{2 (a+b x)^2 (b c-a d)^2}-\frac {1}{3 (a+b x)^3 (b c-a d)}\right )}{4 b}-\frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{4 b (a+b x)^4}-\frac {p r}{16 b (a+b x)^4}\right )-\frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{4 b (a+b x)^4}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{4 b (a+b x)^4}+\frac {1}{2} p r \left (\frac {d q \left (-\frac {\log (a+b x) d^3}{(b c-a d)^4}+\frac {\log (c+d x) d^3}{(b c-a d)^4}-\frac {d^2}{(b c-a d)^3 (a+b x)}+\frac {d}{2 (b c-a d)^2 (a+b x)^2}-\frac {1}{3 (b c-a d) (a+b x)^3}\right ) r}{4 b}-\frac {p r}{16 b (a+b x)^4}-\frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{4 b (a+b x)^4}\right )+\frac {d q r \left (\frac {p r \log ^2(a+b x) d^3}{2 (b c-a d)^4}-\frac {q r \log ^2(c+d x) d^3}{2 (b c-a d)^4}+\frac {11 q r \log (a+b x) d^3}{6 (b c-a d)^4}-\frac {11 q r \log (c+d x) d^3}{6 (b c-a d)^4}-\frac {p r \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x) d^3}{(b c-a d)^4}+\frac {q r \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right ) d^3}{(b c-a d)^4}-\frac {\log (a+b x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) d^3}{(b c-a d)^4}+\frac {\log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) d^3}{(b c-a d)^4}+\frac {q r \operatorname {PolyLog}\left (2,-\frac {d (a+b x)}{b c-a d}\right ) d^3}{(b c-a d)^4}-\frac {p r \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right ) d^3}{(b c-a d)^4}-\frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) d^2}{(b c-a d)^3 (a+b x)}-\frac {p r d^2}{(b c-a d)^3 (a+b x)}+\frac {5 q r d^2}{6 (b c-a d)^3 (a+b x)}+\frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) d}{2 (b c-a d)^2 (a+b x)^2}+\frac {p r d}{4 (b c-a d)^2 (a+b x)^2}-\frac {q r d}{6 (b c-a d)^2 (a+b x)^2}-\frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 (b c-a d) (a+b x)^3}-\frac {p r}{9 (b c-a d) (a+b x)^3}\right )}{2 b}\)

Input:

Int[Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^2/(a + b*x)^5,x]
 

Output:

-1/4*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^2/(b*(a + b*x)^4) + (p*r*(-1/16* 
(p*r)/(b*(a + b*x)^4) + (d*q*r*(-1/3*1/((b*c - a*d)*(a + b*x)^3) + d/(2*(b 
*c - a*d)^2*(a + b*x)^2) - d^2/((b*c - a*d)^3*(a + b*x)) - (d^3*Log[a + b* 
x])/(b*c - a*d)^4 + (d^3*Log[c + d*x])/(b*c - a*d)^4))/(4*b) - Log[e*(f*(a 
 + b*x)^p*(c + d*x)^q)^r]/(4*b*(a + b*x)^4)))/2 + (d*q*r*(-1/9*(p*r)/((b*c 
 - a*d)*(a + b*x)^3) + (d*p*r)/(4*(b*c - a*d)^2*(a + b*x)^2) - (d*q*r)/(6* 
(b*c - a*d)^2*(a + b*x)^2) - (d^2*p*r)/((b*c - a*d)^3*(a + b*x)) + (5*d^2* 
q*r)/(6*(b*c - a*d)^3*(a + b*x)) + (11*d^3*q*r*Log[a + b*x])/(6*(b*c - a*d 
)^4) + (d^3*p*r*Log[a + b*x]^2)/(2*(b*c - a*d)^4) - (11*d^3*q*r*Log[c + d* 
x])/(6*(b*c - a*d)^4) - (d^3*p*r*Log[-((d*(a + b*x))/(b*c - a*d))]*Log[c + 
 d*x])/(b*c - a*d)^4 - (d^3*q*r*Log[c + d*x]^2)/(2*(b*c - a*d)^4) + (d^3*q 
*r*Log[a + b*x]*Log[(b*(c + d*x))/(b*c - a*d)])/(b*c - a*d)^4 - Log[e*(f*( 
a + b*x)^p*(c + d*x)^q)^r]/(3*(b*c - a*d)*(a + b*x)^3) + (d*Log[e*(f*(a + 
b*x)^p*(c + d*x)^q)^r])/(2*(b*c - a*d)^2*(a + b*x)^2) - (d^2*Log[e*(f*(a + 
 b*x)^p*(c + d*x)^q)^r])/((b*c - a*d)^3*(a + b*x)) - (d^3*Log[a + b*x]*Log 
[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(b*c - a*d)^4 + (d^3*Log[c + d*x]*Log[e 
*(f*(a + b*x)^p*(c + d*x)^q)^r])/(b*c - a*d)^4 + (d^3*q*r*PolyLog[2, -((d* 
(a + b*x))/(b*c - a*d))])/(b*c - a*d)^4 - (d^3*p*r*PolyLog[2, (b*(c + d*x) 
)/(b*c - a*d)])/(b*c - a*d)^4))/(2*b)
 

Defintions of rubi rules used

rule 17

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]
 

rule 54

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])
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2981

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]
 

rule 2984

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]
 

rule 2994

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]
 

Maple [F]

\[\int \frac {{\ln \left (e \left (f \left (b x +a \right )^{p} \left (d x +c \right )^{q}\right )^{r}\right )}^{2}}{\left (b x +a \right )^{5}}d x\]

Input:

int(ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r)^2/(b*x+a)^5,x)
 

Output:

int(ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r)^2/(b*x+a)^5,x)
 

Fricas [F]

\[ \int \frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(a+b x)^5} \, dx=\int { \frac {\log \left (\left ({\left (b x + a\right )}^{p} {\left (d x + c\right )}^{q} f\right )^{r} e\right )^{2}}{{\left (b x + a\right )}^{5}} \,d x } \] Input:

integrate(log(e*(f*(b*x+a)^p*(d*x+c)^q)^r)^2/(b*x+a)^5,x, algorithm="frica 
s")
 

Output:

integral(log(((b*x + a)^p*(d*x + c)^q*f)^r*e)^2/(b^5*x^5 + 5*a*b^4*x^4 + 1 
0*a^2*b^3*x^3 + 10*a^3*b^2*x^2 + 5*a^4*b*x + a^5), x)
 

Sympy [F]

\[ \int \frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(a+b x)^5} \, dx=\int \frac {\log {\left (e \left (f \left (a + b x\right )^{p} \left (c + d x\right )^{q}\right )^{r} \right )}^{2}}{\left (a + b x\right )^{5}}\, dx \] Input:

integrate(ln(e*(f*(b*x+a)**p*(d*x+c)**q)**r)**2/(b*x+a)**5,x)
 

Output:

Integral(log(e*(f*(a + b*x)**p*(c + d*x)**q)**r)**2/(a + b*x)**5, x)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1816 vs. \(2 (836) = 1672\).

Time = 0.19 (sec) , antiderivative size = 1816, normalized size of antiderivative = 2.05 \[ \int \frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(a+b x)^5} \, dx=\text {Too large to display} \] Input:

integrate(log(e*(f*(b*x+a)^p*(d*x+c)^q)^r)^2/(b*x+a)^5,x, algorithm="maxim 
a")
 

Output:

-1/24*(12*d^4*f*q*log(b*x + a)/(b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^ 
2 - 4*a^3*b*c*d^3 + a^4*d^4) - 12*d^4*f*q*log(d*x + c)/(b^4*c^4 - 4*a*b^3* 
c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4) + (12*b^3*d^3*f*q*x^3 
 - a*b^2*c^2*d*f*(9*p - 4*q) + a^2*b*c*d^2*f*(9*p - 14*q) - a^3*d^3*f*(3*p 
 - 22*q) + 3*b^3*c^3*f*p - 6*(b^3*c*d^2*f*q - 7*a*b^2*d^3*f*q)*x^2 + 4*(b^ 
3*c^2*d*f*q - 5*a*b^2*c*d^2*f*q + 13*a^2*b*d^3*f*q)*x)/(a^4*b^3*c^3 - 3*a^ 
5*b^2*c^2*d + 3*a^6*b*c*d^2 - a^7*d^3 + (b^7*c^3 - 3*a*b^6*c^2*d + 3*a^2*b 
^5*c*d^2 - a^3*b^4*d^3)*x^4 + 4*(a*b^6*c^3 - 3*a^2*b^5*c^2*d + 3*a^3*b^4*c 
*d^2 - a^4*b^3*d^3)*x^3 + 6*(a^2*b^5*c^3 - 3*a^3*b^4*c^2*d + 3*a^4*b^3*c*d 
^2 - a^5*b^2*d^3)*x^2 + 4*(a^3*b^4*c^3 - 3*a^4*b^3*c^2*d + 3*a^5*b^2*c*d^2 
 - a^6*b*d^3)*x))*r*log(((b*x + a)^p*(d*x + c)^q*f)^r*e)/(b*f) + 1/288*(14 
4*(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^4*f^2/(b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2 
*d^2 - 4*a^3*b*c*d^3 + a^4*d^4) + 12*(3*p*q - 22*q^2)*d^4*f^2*log(d*x + c) 
/(b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4) - 
 (9*b^4*c^4*f^2*p^2 - 4*(9*p^2 - 7*p*q)*a*b^3*c^3*d*f^2 + 6*(9*p^2 - 23*p* 
q + 4*q^2)*a^2*b^2*c^2*d^2*f^2 - 12*(3*p^2 - 31*p*q + 14*q^2)*a^3*b*c*d^3* 
f^2 + (9*p^2 - 262*p*q + 144*q^2)*a^4*d^4*f^2 + 60*((3*p*q - 2*q^2)*b^4*c* 
d^3*f^2 - (3*p*q - 2*q^2)*a*b^3*d^4*f^2)*x^3 - 6*((9*p*q - 4*q^2)*b^4*c^2* 
d^2*f^2 - 4*(27*p*q - 17*q^2)*a*b^3*c*d^3*f^2 + (99*p*q - 64*q^2)*a^2*b...
 

Giac [F]

\[ \int \frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(a+b x)^5} \, dx=\int { \frac {\log \left (\left ({\left (b x + a\right )}^{p} {\left (d x + c\right )}^{q} f\right )^{r} e\right )^{2}}{{\left (b x + a\right )}^{5}} \,d x } \] Input:

integrate(log(e*(f*(b*x+a)^p*(d*x+c)^q)^r)^2/(b*x+a)^5,x, algorithm="giac" 
)
 

Output:

integrate(log(((b*x + a)^p*(d*x + c)^q*f)^r*e)^2/(b*x + a)^5, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(a+b x)^5} \, dx=\int \frac {{\ln \left (e\,{\left (f\,{\left (a+b\,x\right )}^p\,{\left (c+d\,x\right )}^q\right )}^r\right )}^2}{{\left (a+b\,x\right )}^5} \,d x \] Input:

int(log(e*(f*(a + b*x)^p*(c + d*x)^q)^r)^2/(a + b*x)^5,x)
 

Output:

int(log(e*(f*(a + b*x)^p*(c + d*x)^q)^r)^2/(a + b*x)^5, x)
 

Reduce [F]

\[ \int \frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(a+b x)^5} \, dx=\text {too large to display} \] Input:

int(log(e*(f*(b*x+a)^p*(d*x+c)^q)^r)^2/(b*x+a)^5,x)
 

Output:

( - 48*int((log(f**r*(c + d*x)**(q*r)*(a + b*x)**(p*r)*e)*x)/(a**5*c + a** 
5*d*x + 5*a**4*b*c*x + 5*a**4*b*d*x**2 + 10*a**3*b**2*c*x**2 + 10*a**3*b** 
2*d*x**3 + 10*a**2*b**3*c*x**3 + 10*a**2*b**3*d*x**4 + 5*a*b**4*c*x**4 + 5 
*a*b**4*d*x**5 + b**5*c*x**5 + b**5*d*x**6),x)*a**10*b*d**6*q*r + 240*int( 
(log(f**r*(c + d*x)**(q*r)*(a + b*x)**(p*r)*e)*x)/(a**5*c + a**5*d*x + 5*a 
**4*b*c*x + 5*a**4*b*d*x**2 + 10*a**3*b**2*c*x**2 + 10*a**3*b**2*d*x**3 + 
10*a**2*b**3*c*x**3 + 10*a**2*b**3*d*x**4 + 5*a*b**4*c*x**4 + 5*a*b**4*d*x 
**5 + b**5*c*x**5 + b**5*d*x**6),x)*a**9*b**2*c*d**5*q*r - 192*int((log(f* 
*r*(c + d*x)**(q*r)*(a + b*x)**(p*r)*e)*x)/(a**5*c + a**5*d*x + 5*a**4*b*c 
*x + 5*a**4*b*d*x**2 + 10*a**3*b**2*c*x**2 + 10*a**3*b**2*d*x**3 + 10*a**2 
*b**3*c*x**3 + 10*a**2*b**3*d*x**4 + 5*a*b**4*c*x**4 + 5*a*b**4*d*x**5 + b 
**5*c*x**5 + b**5*d*x**6),x)*a**9*b**2*d**6*q*r*x - 480*int((log(f**r*(c + 
 d*x)**(q*r)*(a + b*x)**(p*r)*e)*x)/(a**5*c + a**5*d*x + 5*a**4*b*c*x + 5* 
a**4*b*d*x**2 + 10*a**3*b**2*c*x**2 + 10*a**3*b**2*d*x**3 + 10*a**2*b**3*c 
*x**3 + 10*a**2*b**3*d*x**4 + 5*a*b**4*c*x**4 + 5*a*b**4*d*x**5 + b**5*c*x 
**5 + b**5*d*x**6),x)*a**8*b**3*c**2*d**4*q*r + 960*int((log(f**r*(c + d*x 
)**(q*r)*(a + b*x)**(p*r)*e)*x)/(a**5*c + a**5*d*x + 5*a**4*b*c*x + 5*a**4 
*b*d*x**2 + 10*a**3*b**2*c*x**2 + 10*a**3*b**2*d*x**3 + 10*a**2*b**3*c*x** 
3 + 10*a**2*b**3*d*x**4 + 5*a*b**4*c*x**4 + 5*a*b**4*d*x**5 + b**5*c*x**5 
+ b**5*d*x**6),x)*a**8*b**3*c*d**5*q*r*x - 288*int((log(f**r*(c + d*x)*...