Integrand size = 31, antiderivative size = 884 \[ \int \frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(a+b x)^5} \, dx=-\frac {p^2 r^2}{32 b (a+b x)^4}-\frac {7 d p q r^2}{72 b (b c-a d) (a+b x)^3}+\frac {3 d^2 p q r^2}{16 b (b c-a d)^2 (a+b x)^2}-\frac {d^2 q^2 r^2}{12 b (b c-a d)^2 (a+b x)^2}-\frac {5 d^3 p q r^2}{8 b (b c-a d)^3 (a+b x)}+\frac {5 d^3 q^2 r^2}{12 b (b c-a d)^3 (a+b x)}-\frac {d^4 p q r^2 \log (a+b x)}{8 b (b c-a d)^4}+\frac {11 d^4 q^2 r^2 \log (a+b x)}{12 b (b c-a d)^4}+\frac {d^4 p q r^2 \log ^2(a+b x)}{4 b (b c-a d)^4}+\frac {d^4 p q r^2 \log (c+d x)}{8 b (b c-a d)^4}-\frac {11 d^4 q^2 r^2 \log (c+d x)}{12 b (b c-a d)^4}-\frac {d^4 p q r^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{2 b (b c-a d)^4}-\frac {d^4 q^2 r^2 \log ^2(c+d x)}{4 b (b c-a d)^4}+\frac {d^4 q^2 r^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{2 b (b c-a d)^4}-\frac {p r \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{8 b (a+b x)^4}-\frac {d q r \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{6 b (b c-a d) (a+b x)^3}+\frac {d^2 q r \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{4 b (b c-a d)^2 (a+b x)^2}-\frac {d^3 q r \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 b (b c-a d)^3 (a+b x)}-\frac {d^4 q r \log (a+b x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 b (b c-a d)^4}+\frac {d^4 q r \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 b (b c-a d)^4}-\frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{4 b (a+b x)^4}+\frac {d^4 q^2 r^2 \operatorname {PolyLog}\left (2,-\frac {d (a+b x)}{b c-a d}\right )}{2 b (b c-a d)^4}-\frac {d^4 p q r^2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )}{2 b (b c-a d)^4} \]
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Time = 0.54 (sec) , antiderivative size = 884, normalized size of antiderivative = 1.00, number of steps used = 32, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.419, Rules used = {2584, 2581, 32, 46, 2594, 36, 31, 2580, 2437, 2338, 2441, 2440, 2438} \[ \int \frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(a+b x)^5} \, dx=\frac {p q r^2 \log ^2(a+b x) d^4}{4 b (b c-a d)^4}-\frac {q^2 r^2 \log ^2(c+d x) d^4}{4 b (b c-a d)^4}+\frac {11 q^2 r^2 \log (a+b x) d^4}{12 b (b c-a d)^4}-\frac {p q r^2 \log (a+b x) d^4}{8 b (b c-a d)^4}-\frac {11 q^2 r^2 \log (c+d x) d^4}{12 b (b c-a d)^4}+\frac {p q r^2 \log (c+d x) d^4}{8 b (b c-a d)^4}-\frac {p q r^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x) d^4}{2 b (b c-a d)^4}+\frac {q^2 r^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right ) d^4}{2 b (b c-a d)^4}-\frac {q r \log (a+b x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) d^4}{2 b (b c-a d)^4}+\frac {q r \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) d^4}{2 b (b c-a d)^4}+\frac {q^2 r^2 \operatorname {PolyLog}\left (2,-\frac {d (a+b x)}{b c-a d}\right ) d^4}{2 b (b c-a d)^4}-\frac {p q r^2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right ) d^4}{2 b (b c-a d)^4}-\frac {q r \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) d^3}{2 b (b c-a d)^3 (a+b x)}+\frac {5 q^2 r^2 d^3}{12 b (b c-a d)^3 (a+b x)}-\frac {5 p q r^2 d^3}{8 b (b c-a d)^3 (a+b x)}+\frac {q r \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) d^2}{4 b (b c-a d)^2 (a+b x)^2}-\frac {q^2 r^2 d^2}{12 b (b c-a d)^2 (a+b x)^2}+\frac {3 p q r^2 d^2}{16 b (b c-a d)^2 (a+b x)^2}-\frac {q r \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) d}{6 b (b c-a d) (a+b x)^3}-\frac {7 p q r^2 d}{72 b (b c-a d) (a+b x)^3}-\frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{4 b (a+b x)^4}-\frac {p r \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{8 b (a+b x)^4}-\frac {p^2 r^2}{32 b (a+b x)^4} \]
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Rule 31
Rule 32
Rule 36
Rule 46
Rule 2338
Rule 2437
Rule 2438
Rule 2440
Rule 2441
Rule 2580
Rule 2581
Rule 2584
Rule 2594
Rubi steps \begin{align*} \text {integral}& = -\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) \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 {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 {\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 q r) \int \left (\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}-\frac {b d \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(b c-a d)^2 (a+b x)^3}+\frac {b d^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(b c-a d)^3 (a+b x)^2}-\frac {b d^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(b c-a d)^4 (a+b x)}+\frac {d^4 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(b c-a d)^4 (c+d x)}\right ) \, dx}{2 b}+\frac {1}{8} \left (p^2 r^2\right ) \int \frac {1}{(a+b x)^5} \, dx+\frac {\left (d p q r^2\right ) \int \frac {1}{(a+b x)^4 (c+d x)} \, dx}{8 b} \\ & = -\frac {p^2 r^2}{32 b (a+b x)^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 {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{4 b (a+b x)^4}-\frac {\left (d^4 q r\right ) \int \frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{a+b x} \, dx}{2 (b c-a d)^4}+\frac {\left (d^5 q r\right ) \int \frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{c+d x} \, dx}{2 b (b c-a d)^4}+\frac {\left (d^3 q r\right ) \int \frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(a+b x)^2} \, dx}{2 (b c-a d)^3}-\frac {\left (d^2 q r\right ) \int \frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(a+b x)^3} \, dx}{2 (b c-a d)^2}+\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} \, dx}{2 (b c-a d)}+\frac {\left (d p q r^2\right ) \int \left (\frac {b}{(b c-a d) (a+b x)^4}-\frac {b d}{(b c-a d)^2 (a+b x)^3}+\frac {b d^2}{(b c-a d)^3 (a+b x)^2}-\frac {b d^3}{(b c-a d)^4 (a+b x)}+\frac {d^4}{(b c-a d)^4 (c+d x)}\right ) \, dx}{8 b} \\ & = -\frac {p^2 r^2}{32 b (a+b x)^4}-\frac {d p q r^2}{24 b (b c-a d) (a+b x)^3}+\frac {d^2 p q r^2}{16 b (b c-a d)^2 (a+b x)^2}-\frac {d^3 p q r^2}{8 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 {d^4 p q r^2 \log (c+d x)}{8 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 {\left (d^4 p q r^2\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{2 (b c-a d)^4}-\frac {\left (d^4 p q r^2\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{2 (b c-a d)^4}+\frac {\left (d^3 p q r^2\right ) \int \frac {1}{(a+b x)^2} \, dx}{2 (b c-a d)^3}-\frac {\left (d^2 p q r^2\right ) \int \frac {1}{(a+b x)^3} \, dx}{4 (b c-a d)^2}+\frac {\left (d p q r^2\right ) \int \frac {1}{(a+b x)^4} \, dx}{6 (b c-a d)}+\frac {\left (d^5 q^2 r^2\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{2 b (b c-a d)^4}-\frac {\left (d^5 q^2 r^2\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{2 b (b c-a d)^4}+\frac {\left (d^4 q^2 r^2\right ) \int \frac {1}{(a+b x) (c+d x)} \, dx}{2 b (b c-a d)^3}-\frac {\left (d^3 q^2 r^2\right ) \int \frac {1}{(a+b x)^2 (c+d x)} \, dx}{4 b (b c-a d)^2}+\frac {\left (d^2 q^2 r^2\right ) \int \frac {1}{(a+b x)^3 (c+d x)} \, dx}{6 b (b c-a d)} \\ & = -\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 {5 d^3 p q r^2}{8 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 {d^4 p q r^2 \log (c+d x)}{8 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 (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 {\left (d^4 p q r^2\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{2 b (b c-a d)^4}+\frac {\left (d^5 p q r^2\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{2 b (b c-a d)^4}+\frac {\left (d^4 q^2 r^2\right ) \int \frac {1}{a+b x} \, dx}{2 (b c-a d)^4}-\frac {\left (d^4 q^2 r^2\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{2 (b c-a d)^4}-\frac {\left (d^4 q^2 r^2\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{2 b (b c-a d)^4}-\frac {\left (d^5 q^2 r^2\right ) \int \frac {1}{c+d x} \, dx}{2 b (b c-a d)^4}-\frac {\left (d^3 q^2 r^2\right ) \int \left (\frac {b}{(b c-a d) (a+b x)^2}-\frac {b d}{(b c-a d)^2 (a+b x)}+\frac {d^2}{(b c-a d)^2 (c+d x)}\right ) \, dx}{4 b (b c-a d)^2}+\frac {\left (d^2 q^2 r^2\right ) \int \left (\frac {b}{(b c-a d) (a+b x)^3}-\frac {b d}{(b c-a d)^2 (a+b x)^2}+\frac {b d^2}{(b c-a d)^3 (a+b x)}-\frac {d^3}{(b c-a d)^3 (c+d x)}\right ) \, dx}{6 b (b c-a d)} \\ & = -\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 {\left (d^4 p q r^2\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{2 b (b c-a d)^4}-\frac {\left (d^4 q^2 r^2\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{2 b (b c-a d)^4} \\ & = -\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 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{2 b (b c-a d)^4}-\frac {d^4 p q r^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{2 b (b c-a d)^4} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(2003\) vs. \(2(884)=1768\).
Time = 1.32 (sec) , antiderivative size = 2003, normalized size of antiderivative = 2.27 \[ \int \frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(a+b x)^5} \, dx=\text {Result too large to show} \]
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\[\int \frac {{\ln \left (e \left (f \left (b x +a \right )^{p} \left (d x +c \right )^{q}\right )^{r}\right )}^{2}}{\left (b x +a \right )^{5}}d x\]
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\[ \int \frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(a+b x)^5} \, dx=\int { \frac {\log \left (\left ({\left (b x + a\right )}^{p} {\left (d x + c\right )}^{q} f\right )^{r} e\right )^{2}}{{\left (b x + a\right )}^{5}} \,d x } \]
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\[ \int \frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(a+b x)^5} \, dx=\int \frac {\log {\left (e \left (f \left (a + b x\right )^{p} \left (c + d x\right )^{q}\right )^{r} \right )}^{2}}{\left (a + b x\right )^{5}}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 1816 vs. \(2 (836) = 1672\).
Time = 0.35 (sec) , antiderivative size = 1816, normalized size of antiderivative = 2.05 \[ \int \frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(a+b x)^5} \, dx=\text {Too large to display} \]
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\[ \int \frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(a+b x)^5} \, dx=\int { \frac {\log \left (\left ({\left (b x + a\right )}^{p} {\left (d x + c\right )}^{q} f\right )^{r} e\right )^{2}}{{\left (b x + a\right )}^{5}} \,d x } \]
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Timed out. \[ \int \frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(a+b x)^5} \, dx=\int \frac {{\ln \left (e\,{\left (f\,{\left (a+b\,x\right )}^p\,{\left (c+d\,x\right )}^q\right )}^r\right )}^2}{{\left (a+b\,x\right )}^5} \,d x \]
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