Integrand size = 31, antiderivative size = 920 \[ \int (a+b x)^4 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx=-\frac {a (b c-a d)^3 p q r^2 x}{5 d^3}+\frac {2 (b c-a d)^4 p q r^2 x}{25 d^4}+\frac {77 (b c-a d)^4 q^2 r^2 x}{150 d^4}+\frac {2 (b c-a d)^4 q (p+q) r^2 x}{5 d^4}-\frac {b (b c-a d)^3 p q r^2 x^2}{10 d^3}-\frac {(b c-a d)^3 p q r^2 (a+b x)^2}{25 b d^3}-\frac {77 (b c-a d)^3 q^2 r^2 (a+b x)^2}{300 b d^3}+\frac {16 (b c-a d)^2 p q r^2 (a+b x)^3}{225 b d^2}+\frac {47 (b c-a d)^2 q^2 r^2 (a+b x)^3}{450 b d^2}-\frac {9 (b c-a d) p q r^2 (a+b x)^4}{200 b d}-\frac {9 (b c-a d) q^2 r^2 (a+b x)^4}{200 b d}+\frac {2 p^2 r^2 (a+b x)^5}{125 b}+\frac {4 p q r^2 (a+b x)^5}{125 b}+\frac {2 q^2 r^2 (a+b x)^5}{125 b}-\frac {2 (b c-a d)^5 p q r^2 \log (c+d x)}{25 b d^5}-\frac {137 (b c-a d)^5 q^2 r^2 \log (c+d x)}{150 b d^5}-\frac {2 (b c-a d)^5 p q r^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{5 b d^5}-\frac {(b c-a d)^5 q^2 r^2 \log ^2(c+d x)}{5 b d^5}-\frac {2 (b c-a d)^4 q r (a+b x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{5 b d^4}+\frac {(b c-a d)^3 q r (a+b x)^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{5 b d^3}-\frac {2 (b c-a d)^2 q r (a+b x)^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{15 b d^2}+\frac {(b c-a d) q r (a+b x)^4 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{10 b d}-\frac {2 p r (a+b x)^5 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{25 b}-\frac {2 q r (a+b x)^5 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{25 b}+\frac {2 (b c-a d)^5 q r \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{5 b d^5}+\frac {(a+b x)^5 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{5 b}-\frac {2 (b c-a d)^5 p q r^2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )}{5 b d^5} \]
[Out]
Time = 0.59 (sec) , antiderivative size = 920, normalized size of antiderivative = 1.00, number of steps used = 32, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.452, Rules used = {2584, 2581, 32, 45, 2594, 2579, 31, 8, 2580, 2441, 2440, 2438, 2437, 2338} \[ \int (a+b x)^4 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx=-\frac {q^2 r^2 \log ^2(c+d x) (b c-a d)^5}{5 b d^5}-\frac {137 q^2 r^2 \log (c+d x) (b c-a d)^5}{150 b d^5}-\frac {2 p q r^2 \log (c+d x) (b c-a d)^5}{25 b d^5}-\frac {2 p q r^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x) (b c-a d)^5}{5 b d^5}+\frac {2 q r \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) (b c-a d)^5}{5 b d^5}-\frac {2 p q r^2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right ) (b c-a d)^5}{5 b d^5}+\frac {77 q^2 r^2 x (b c-a d)^4}{150 d^4}+\frac {2 p q r^2 x (b c-a d)^4}{25 d^4}+\frac {2 q (p+q) r^2 x (b c-a d)^4}{5 d^4}-\frac {2 q r (a+b x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) (b c-a d)^4}{5 b d^4}-\frac {b p q r^2 x^2 (b c-a d)^3}{10 d^3}-\frac {77 q^2 r^2 (a+b x)^2 (b c-a d)^3}{300 b d^3}-\frac {p q r^2 (a+b x)^2 (b c-a d)^3}{25 b d^3}-\frac {a p q r^2 x (b c-a d)^3}{5 d^3}+\frac {q r (a+b x)^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) (b c-a d)^3}{5 b d^3}+\frac {47 q^2 r^2 (a+b x)^3 (b c-a d)^2}{450 b d^2}+\frac {16 p q r^2 (a+b x)^3 (b c-a d)^2}{225 b d^2}-\frac {2 q r (a+b x)^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) (b c-a d)^2}{15 b d^2}-\frac {9 q^2 r^2 (a+b x)^4 (b c-a d)}{200 b d}-\frac {9 p q r^2 (a+b x)^4 (b c-a d)}{200 b d}+\frac {q r (a+b x)^4 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) (b c-a d)}{10 b d}+\frac {2 p^2 r^2 (a+b x)^5}{125 b}+\frac {2 q^2 r^2 (a+b x)^5}{125 b}+\frac {4 p q r^2 (a+b x)^5}{125 b}+\frac {(a+b x)^5 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{5 b}-\frac {2 p r (a+b x)^5 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{25 b}-\frac {2 q r (a+b x)^5 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{25 b} \]
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Rule 8
Rule 31
Rule 32
Rule 45
Rule 2338
Rule 2437
Rule 2438
Rule 2440
Rule 2441
Rule 2579
Rule 2580
Rule 2581
Rule 2584
Rule 2594
Rubi steps \begin{align*} \text {integral}& = \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 {1}{5} (2 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 {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 {(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 d q r) \int \left (\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 (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}+\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) (a+b x)^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{d^2}+\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 c+a d)^5 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{d^5 (c+d x)}\right ) \, dx}{5 b}+\frac {1}{25} \left (2 p^2 r^2\right ) \int (a+b x)^4 \, dx+\frac {\left (2 d p q r^2\right ) \int \frac {(a+b x)^5}{c+d x} \, dx}{25 b} \\ & = \frac {2 p^2 r^2 (a+b x)^5}{125 b}-\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 {(a+b x)^5 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{5 b}-\frac {1}{5} (2 q 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 (b c-a d) q r) \int (a+b x)^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx}{5 d}-\frac {\left (2 (b c-a d)^2 q r\right ) \int (a+b x)^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx}{5 d^2}+\frac {\left (2 (b c-a d)^3 q r\right ) \int (a+b x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx}{5 d^3}-\frac {\left (2 (b c-a d)^4 q r\right ) \int \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx}{5 d^4}+\frac {\left (2 (b c-a 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}{5 b d^4}+\frac {\left (2 d p q r^2\right ) \int \left (\frac {b (b c-a d)^4}{d^5}-\frac {b (b c-a d)^3 (a+b x)}{d^4}+\frac {b (b c-a d)^2 (a+b x)^2}{d^3}-\frac {b (b c-a d) (a+b x)^3}{d^2}+\frac {b (a+b x)^4}{d}+\frac {(-b c+a d)^5}{d^5 (c+d x)}\right ) \, dx}{25 b} \\ & = \frac {2 (b c-a d)^4 p q r^2 x}{25 d^4}-\frac {(b c-a d)^3 p q r^2 (a+b x)^2}{25 b d^3}+\frac {2 (b c-a d)^2 p q r^2 (a+b x)^3}{75 b d^2}-\frac {(b c-a d) p q r^2 (a+b x)^4}{50 b d}+\frac {2 p^2 r^2 (a+b x)^5}{125 b}+\frac {2 p q 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 {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 {1}{25} \left (2 p q r^2\right ) \int (a+b x)^4 \, dx-\frac {\left ((b c-a d) p q r^2\right ) \int (a+b x)^3 \, dx}{10 d}+\frac {\left (2 (b c-a d)^2 p q r^2\right ) \int (a+b x)^2 \, dx}{15 d^2}-\frac {\left ((b c-a d)^3 p q r^2\right ) \int (a+b x) \, dx}{5 d^3}-\frac {\left (2 (b c-a d)^5 p q r^2\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{5 d^5}+\frac {\left (2 d q^2 r^2\right ) \int \frac {(a+b x)^5}{c+d x} \, dx}{25 b}-\frac {\left ((b c-a d) q^2 r^2\right ) \int \frac {(a+b x)^4}{c+d x} \, dx}{10 b}+\frac {\left (2 (b c-a d)^2 q^2 r^2\right ) \int \frac {(a+b x)^3}{c+d x} \, dx}{15 b d}-\frac {\left ((b c-a d)^3 q^2 r^2\right ) \int \frac {(a+b x)^2}{c+d x} \, dx}{5 b d^2}-\frac {\left (2 (b c-a d)^5 q^2 r^2\right ) \int \frac {1}{c+d x} \, dx}{5 b d^4}-\frac {\left (2 (b c-a d)^5 q^2 r^2\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{5 b d^4}+\frac {\left (2 (b c-a d)^4 q (p+q) r^2\right ) \int 1 \, dx}{5 d^4} \\ & = -\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 {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 {16 (b c-a d)^2 p q r^2 (a+b x)^3}{225 b d^2}-\frac {9 (b c-a d) p q 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 (b c-a d)^5 p q r^2 \log (c+d x)}{25 b d^5}-\frac {2 (b c-a d)^5 q^2 r^2 \log (c+d x)}{5 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 {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 {\left (2 (b c-a 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}{5 b d^4}+\frac {\left (2 d q^2 r^2\right ) \int \left (\frac {b (b c-a d)^4}{d^5}-\frac {b (b c-a d)^3 (a+b x)}{d^4}+\frac {b (b c-a d)^2 (a+b x)^2}{d^3}-\frac {b (b c-a d) (a+b x)^3}{d^2}+\frac {b (a+b x)^4}{d}+\frac {(-b c+a d)^5}{d^5 (c+d x)}\right ) \, dx}{25 b}-\frac {\left ((b c-a d) q^2 r^2\right ) \int \left (-\frac {b (b c-a d)^3}{d^4}+\frac {b (b c-a d)^2 (a+b x)}{d^3}-\frac {b (b c-a d) (a+b x)^2}{d^2}+\frac {b (a+b x)^3}{d}+\frac {(-b c+a d)^4}{d^4 (c+d x)}\right ) \, dx}{10 b}+\frac {\left (2 (b c-a d)^2 q^2 r^2\right ) \int \left (\frac {b (b c-a d)^2}{d^3}-\frac {b (b c-a d) (a+b x)}{d^2}+\frac {b (a+b x)^2}{d}+\frac {(-b c+a d)^3}{d^3 (c+d x)}\right ) \, dx}{15 b d}-\frac {\left ((b c-a d)^3 q^2 r^2\right ) \int \left (-\frac {b (b c-a d)}{d^2}+\frac {b (a+b x)}{d}+\frac {(-b c+a d)^2}{d^2 (c+d x)}\right ) \, dx}{5 b d^2}-\frac {\left (2 (b c-a d)^5 q^2 r^2\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{5 b d^5} \\ & = -\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 {\left (2 (b c-a d)^5 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 )}{5 b d^5} \\ & = -\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 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{5 b d^5} \\ \end{align*}
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 \[ \int (a+b x)^4 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx=\text {Result too large to show} \]
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\[\int \left (b x +a \right )^{4} {\ln \left (e \left (f \left (b x +a \right )^{p} \left (d x +c \right )^{q}\right )^{r}\right )}^{2}d x\]
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\[ \int (a+b x)^4 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx=\int { {\left (b x + a\right )}^{4} \log \left (\left ({\left (b x + a\right )}^{p} {\left (d x + c\right )}^{q} f\right )^{r} e\right )^{2} \,d x } \]
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Timed out. \[ \int (a+b x)^4 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx=\text {Timed out} \]
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Time = 0.24 (sec) , antiderivative size = 1421, normalized size of antiderivative = 1.54 \[ \int (a+b x)^4 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx=\text {Too large to display} \]
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\[ \int (a+b x)^4 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx=\int { {\left (b x + a\right )}^{4} \log \left (\left ({\left (b x + a\right )}^{p} {\left (d x + c\right )}^{q} f\right )^{r} e\right )^{2} \,d x } \]
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Timed out. \[ \int (a+b x)^4 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx=\int {\ln \left (e\,{\left (f\,{\left (a+b\,x\right )}^p\,{\left (c+d\,x\right )}^q\right )}^r\right )}^2\,{\left (a+b\,x\right )}^4 \,d x \]
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