Integrand size = 31, antiderivative size = 686 \[ \int (a+b x)^2 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx=-\frac {a (b c-a d) p q r^2 x}{3 d}+\frac {2 (b c-a d)^2 p q r^2 x}{9 d^2}+\frac {5 (b c-a d)^2 q^2 r^2 x}{9 d^2}+\frac {2 (b c-a d)^2 q (p+q) r^2 x}{3 d^2}-\frac {b (b c-a d) p q r^2 x^2}{6 d}-\frac {(b c-a d) p q r^2 (a+b x)^2}{9 b d}-\frac {5 (b c-a d) q^2 r^2 (a+b x)^2}{18 b d}+\frac {2 p^2 r^2 (a+b x)^3}{27 b}+\frac {4 p q r^2 (a+b x)^3}{27 b}+\frac {2 q^2 r^2 (a+b x)^3}{27 b}-\frac {2 (b c-a d)^3 p q r^2 \log (c+d x)}{9 b d^3}-\frac {11 (b c-a d)^3 q^2 r^2 \log (c+d x)}{9 b d^3}-\frac {2 (b c-a d)^3 p q r^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{3 b d^3}-\frac {(b c-a d)^3 q^2 r^2 \log ^2(c+d x)}{3 b d^3}-\frac {2 (b c-a d)^2 q r (a+b x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 b d^2}+\frac {(b c-a d) q r (a+b x)^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 b d}-\frac {2 p r (a+b x)^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{9 b}-\frac {2 q r (a+b x)^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{9 b}+\frac {2 (b c-a d)^3 q r \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 b d^3}+\frac {(a+b x)^3 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 b}-\frac {2 (b c-a d)^3 p q r^2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )}{3 b d^3} \]
[Out]
Time = 0.38 (sec) , antiderivative size = 686, normalized size of antiderivative = 1.00, number of steps used = 24, 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} \[ \int (a+b x)^2 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx=\frac {2 q r (b c-a d)^3 \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 b d^3}-\frac {2 p q r^2 (b c-a d)^3 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )}{3 b d^3}-\frac {2 p q r^2 (b c-a d)^3 \log (c+d x)}{9 b d^3}-\frac {2 p q r^2 (b c-a d)^3 \log (c+d x) \log \left (-\frac {d (a+b x)}{b c-a d}\right )}{3 b d^3}-\frac {q^2 r^2 (b c-a d)^3 \log ^2(c+d x)}{3 b d^3}-\frac {11 q^2 r^2 (b c-a d)^3 \log (c+d x)}{9 b d^3}-\frac {2 q r (a+b x) (b c-a d)^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 b d^2}+\frac {2 p q r^2 x (b c-a d)^2}{9 d^2}+\frac {2 q r^2 x (p+q) (b c-a d)^2}{3 d^2}+\frac {5 q^2 r^2 x (b c-a d)^2}{9 d^2}+\frac {(a+b x)^3 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 b}+\frac {q r (a+b x)^2 (b c-a d) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 b d}-\frac {2 p r (a+b x)^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{9 b}-\frac {2 q r (a+b x)^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{9 b}-\frac {b p q r^2 x^2 (b c-a d)}{6 d}-\frac {p q r^2 (a+b x)^2 (b c-a d)}{9 b d}-\frac {a p q r^2 x (b c-a d)}{3 d}-\frac {5 q^2 r^2 (a+b x)^2 (b c-a d)}{18 b d}+\frac {2 p^2 r^2 (a+b x)^3}{27 b}+\frac {4 p q r^2 (a+b x)^3}{27 b}+\frac {2 q^2 r^2 (a+b x)^3}{27 b} \]
[In]
[Out]
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)^3 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 b}-\frac {1}{3} (2 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 {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 {(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 d q r) \int \left (\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 (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}+\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 c+a d)^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{d^3 (c+d x)}\right ) \, dx}{3 b}+\frac {1}{9} \left (2 p^2 r^2\right ) \int (a+b x)^2 \, dx+\frac {\left (2 d p q r^2\right ) \int \frac {(a+b x)^3}{c+d x} \, dx}{9 b} \\ & = \frac {2 p^2 r^2 (a+b x)^3}{27 b}-\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 {(a+b x)^3 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 b}-\frac {1}{3} (2 q 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 (b c-a d) q r) \int (a+b x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx}{3 d}-\frac {\left (2 (b c-a d)^2 q r\right ) \int \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx}{3 d^2}+\frac {\left (2 (b c-a d)^3 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}{3 b d^2}+\frac {\left (2 d p q 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}{9 b} \\ & = \frac {2 (b c-a d)^2 p q r^2 x}{9 d^2}-\frac {(b c-a d) p q r^2 (a+b x)^2}{9 b d}+\frac {2 p^2 r^2 (a+b x)^3}{27 b}+\frac {2 p q 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 {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 {1}{9} \left (2 p q r^2\right ) \int (a+b x)^2 \, dx-\frac {\left ((b c-a d) p q r^2\right ) \int (a+b x) \, dx}{3 d}-\frac {\left (2 (b c-a d)^3 p q r^2\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{3 d^3}+\frac {\left (2 d q^2 r^2\right ) \int \frac {(a+b x)^3}{c+d x} \, dx}{9 b}-\frac {\left ((b c-a d) q^2 r^2\right ) \int \frac {(a+b x)^2}{c+d x} \, dx}{3 b}-\frac {\left (2 (b c-a d)^3 q^2 r^2\right ) \int \frac {1}{c+d x} \, dx}{3 b d^2}-\frac {\left (2 (b c-a d)^3 q^2 r^2\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{3 b d^2}+\frac {\left (2 (b c-a d)^2 q (p+q) r^2\right ) \int 1 \, dx}{3 d^2} \\ & = -\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 {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 {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 (b c-a d)^3 p q r^2 \log (c+d x)}{9 b d^3}-\frac {2 (b c-a d)^3 q^2 r^2 \log (c+d x)}{3 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 {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 {\left (2 (b c-a d)^3 p q r^2\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{3 b d^2}+\frac {\left (2 d 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}{9 b}-\frac {\left ((b c-a d) 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}{3 b}-\frac {\left (2 (b c-a d)^3 q^2 r^2\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{3 b d^3} \\ & = -\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 {\left (2 (b c-a d)^3 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 )}{3 b d^3} \\ & = -\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 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{3 b d^3} \\ \end{align*}
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 ) \]
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\[\int \left (b x +a \right )^{2} {\ln \left (e \left (f \left (b x +a \right )^{p} \left (d x +c \right )^{q}\right )^{r}\right )}^{2}d x\]
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\[ \int (a+b x)^2 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx=\int { {\left (b x + a\right )}^{2} \log \left (\left ({\left (b x + a\right )}^{p} {\left (d x + c\right )}^{q} f\right )^{r} e\right )^{2} \,d x } \]
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\[ \int (a+b x)^2 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx=\int \left (a + b x\right )^{2} \log {\left (e \left (f \left (a + b x\right )^{p} \left (c + d x\right )^{q}\right )^{r} \right )}^{2}\, dx \]
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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}} \]
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\[ \int (a+b x)^2 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx=\int { {\left (b x + a\right )}^{2} \log \left (\left ({\left (b x + a\right )}^{p} {\left (d x + c\right )}^{q} f\right )^{r} e\right )^{2} \,d x } \]
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Timed out. \[ \int (a+b x)^2 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx=\int {\ln \left (e\,{\left (f\,{\left (a+b\,x\right )}^p\,{\left (c+d\,x\right )}^q\right )}^r\right )}^2\,{\left (a+b\,x\right )}^2 \,d x \]
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