Integrand size = 42, antiderivative size = 908 \[ \int (a g+b g x)^2 (c i+d i x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \, dx=-\frac {7 B^2 (b c-a d)^5 g^2 i^3 x}{180 b^3 d^2}-\frac {7 B^2 (b c-a d)^4 g^2 i^3 (c+d x)^2}{360 b^2 d^3}-\frac {B^2 (b c-a d)^3 g^2 i^3 (c+d x)^3}{60 b d^3}+\frac {B^2 (b c-a d)^2 g^2 i^3 (c+d x)^4}{60 d^3}+\frac {B^2 (b c-a d)^6 g^2 i^3 \log \left (\frac {a+b x}{c+d x}\right )}{36 b^4 d^3}-\frac {B (b c-a d)^4 g^2 i^3 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{60 b^4 d}-\frac {B (b c-a d)^3 g^2 i^3 (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{30 b^4}-\frac {B (b c-a d)^4 g^2 i^3 (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{10 b^2 d^3}+\frac {B (b c-a d)^3 g^2 i^3 (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{45 b d^3}+\frac {7 B (b c-a d)^2 g^2 i^3 (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{60 d^3}-\frac {b B (b c-a d) g^2 i^3 (c+d x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{15 d^3}+\frac {(b c-a d)^3 g^2 i^3 (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{60 b^4}+\frac {(b c-a d)^2 g^2 i^3 (a+b x)^3 (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{20 b^3}+\frac {(b c-a d) g^2 i^3 (a+b x)^3 (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{10 b^2}+\frac {g^2 i^3 (a+b x)^3 (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{6 b}+\frac {B (b c-a d)^5 g^2 i^3 (a+b x) \left (2 A+B+2 B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{60 b^4 d^2}+\frac {B (b c-a d)^6 g^2 i^3 \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (2 A+3 B+2 B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{60 b^4 d^3}+\frac {11 B^2 (b c-a d)^6 g^2 i^3 \log (c+d x)}{180 b^4 d^3}+\frac {B^2 (b c-a d)^6 g^2 i^3 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{30 b^4 d^3} \]
[Out]
Time = 0.82 (sec) , antiderivative size = 908, normalized size of antiderivative = 1.00, number of steps used = 20, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.262, Rules used = {2562, 2383, 2381, 2384, 2354, 2438, 2373, 45, 2382, 12, 907} \[ \int (a g+b g x)^2 (c i+d i x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \, dx=\frac {B^2 g^2 i^3 \log \left (\frac {a+b x}{c+d x}\right ) (b c-a d)^6}{36 b^4 d^3}+\frac {B g^2 i^3 \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (2 A+3 B+2 B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) (b c-a d)^6}{60 b^4 d^3}+\frac {11 B^2 g^2 i^3 \log (c+d x) (b c-a d)^6}{180 b^4 d^3}+\frac {B^2 g^2 i^3 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right ) (b c-a d)^6}{30 b^4 d^3}-\frac {7 B^2 g^2 i^3 x (b c-a d)^5}{180 b^3 d^2}+\frac {B g^2 i^3 (a+b x) \left (2 A+B+2 B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) (b c-a d)^5}{60 b^4 d^2}-\frac {7 B^2 g^2 i^3 (c+d x)^2 (b c-a d)^4}{360 b^2 d^3}-\frac {B g^2 i^3 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) (b c-a d)^4}{60 b^4 d}-\frac {B g^2 i^3 (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) (b c-a d)^4}{10 b^2 d^3}-\frac {B^2 g^2 i^3 (c+d x)^3 (b c-a d)^3}{60 b d^3}+\frac {g^2 i^3 (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 (b c-a d)^3}{60 b^4}-\frac {B g^2 i^3 (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) (b c-a d)^3}{30 b^4}+\frac {B g^2 i^3 (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) (b c-a d)^3}{45 b d^3}+\frac {B^2 g^2 i^3 (c+d x)^4 (b c-a d)^2}{60 d^3}+\frac {g^2 i^3 (a+b x)^3 (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 (b c-a d)^2}{20 b^3}+\frac {7 B g^2 i^3 (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) (b c-a d)^2}{60 d^3}+\frac {g^2 i^3 (a+b x)^3 (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 (b c-a d)}{10 b^2}-\frac {b B g^2 i^3 (c+d x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) (b c-a d)}{15 d^3}+\frac {g^2 i^3 (a+b x)^3 (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{6 b} \]
[In]
[Out]
Rule 12
Rule 45
Rule 907
Rule 2354
Rule 2373
Rule 2381
Rule 2382
Rule 2383
Rule 2384
Rule 2438
Rule 2562
Rubi steps \begin{align*} \text {integral}& = \left ((b c-a d)^6 g^2 i^3\right ) \text {Subst}\left (\int \frac {x^2 (A+B \log (e x))^2}{(b-d x)^7} \, dx,x,\frac {a+b x}{c+d x}\right ) \\ & = \frac {g^2 i^3 (a+b x)^3 (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{6 b}+\frac {\left ((b c-a d)^6 g^2 i^3\right ) \text {Subst}\left (\int \frac {x^2 (A+B \log (e x))^2}{(b-d x)^6} \, dx,x,\frac {a+b x}{c+d x}\right )}{2 b}-\frac {\left (B (b c-a d)^6 g^2 i^3\right ) \text {Subst}\left (\int \frac {x^2 (A+B \log (e x))}{(b-d x)^6} \, dx,x,\frac {a+b x}{c+d x}\right )}{3 b} \\ & = -\frac {B (b c-a d)^3 g^2 i^3 (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{9 b d^3}+\frac {B (b c-a d)^2 g^2 i^3 (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{6 d^3}-\frac {b B (b c-a d) g^2 i^3 (c+d x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{15 d^3}+\frac {(b c-a d) g^2 i^3 (a+b x)^3 (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{10 b^2}+\frac {g^2 i^3 (a+b x)^3 (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{6 b}+\frac {\left ((b c-a d)^6 g^2 i^3\right ) \text {Subst}\left (\int \frac {x^2 (A+B \log (e x))^2}{(b-d x)^5} \, dx,x,\frac {a+b x}{c+d x}\right )}{5 b^2}-\frac {\left (B (b c-a d)^6 g^2 i^3\right ) \text {Subst}\left (\int \frac {x^2 (A+B \log (e x))}{(b-d x)^5} \, dx,x,\frac {a+b x}{c+d x}\right )}{5 b^2}+\frac {\left (B^2 (b c-a d)^6 g^2 i^3\right ) \text {Subst}\left (\int \frac {b^2-5 b d x+10 d^2 x^2}{30 d^3 x (b-d x)^5} \, dx,x,\frac {a+b x}{c+d x}\right )}{3 b} \\ & = -\frac {B (b c-a d)^4 g^2 i^3 (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{10 b^2 d^3}+\frac {B (b c-a d)^3 g^2 i^3 (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{45 b d^3}+\frac {7 B (b c-a d)^2 g^2 i^3 (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{60 d^3}-\frac {b B (b c-a d) g^2 i^3 (c+d x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{15 d^3}+\frac {(b c-a d)^2 g^2 i^3 (a+b x)^3 (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{20 b^3}+\frac {(b c-a d) g^2 i^3 (a+b x)^3 (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{10 b^2}+\frac {g^2 i^3 (a+b x)^3 (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{6 b}+\frac {\left ((b c-a d)^6 g^2 i^3\right ) \text {Subst}\left (\int \frac {x^2 (A+B \log (e x))^2}{(b-d x)^4} \, dx,x,\frac {a+b x}{c+d x}\right )}{20 b^3}-\frac {\left (B (b c-a d)^6 g^2 i^3\right ) \text {Subst}\left (\int \frac {x^2 (A+B \log (e x))}{(b-d x)^4} \, dx,x,\frac {a+b x}{c+d x}\right )}{10 b^3}+\frac {\left (B^2 (b c-a d)^6 g^2 i^3\right ) \text {Subst}\left (\int \frac {b^2-4 b d x+6 d^2 x^2}{12 d^3 x (b-d x)^4} \, dx,x,\frac {a+b x}{c+d x}\right )}{5 b^2}+\frac {\left (B^2 (b c-a d)^6 g^2 i^3\right ) \text {Subst}\left (\int \frac {b^2-5 b d x+10 d^2 x^2}{x (b-d x)^5} \, dx,x,\frac {a+b x}{c+d x}\right )}{90 b d^3} \\ & = -\frac {B (b c-a d)^3 g^2 i^3 (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{30 b^4}-\frac {B (b c-a d)^4 g^2 i^3 (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{10 b^2 d^3}+\frac {B (b c-a d)^3 g^2 i^3 (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{45 b d^3}+\frac {7 B (b c-a d)^2 g^2 i^3 (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{60 d^3}-\frac {b B (b c-a d) g^2 i^3 (c+d x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{15 d^3}+\frac {(b c-a d)^3 g^2 i^3 (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{60 b^4}+\frac {(b c-a d)^2 g^2 i^3 (a+b x)^3 (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{20 b^3}+\frac {(b c-a d) g^2 i^3 (a+b x)^3 (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{10 b^2}+\frac {g^2 i^3 (a+b x)^3 (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{6 b}-\frac {\left (B (b c-a d)^6 g^2 i^3\right ) \text {Subst}\left (\int \frac {x^2 (A+B \log (e x))}{(b-d x)^3} \, dx,x,\frac {a+b x}{c+d x}\right )}{30 b^4}+\frac {\left (B^2 (b c-a d)^6 g^2 i^3\right ) \text {Subst}\left (\int \frac {x^2}{(b-d x)^3} \, dx,x,\frac {a+b x}{c+d x}\right )}{30 b^4}+\frac {\left (B^2 (b c-a d)^6 g^2 i^3\right ) \text {Subst}\left (\int \frac {b^2-4 b d x+6 d^2 x^2}{x (b-d x)^4} \, dx,x,\frac {a+b x}{c+d x}\right )}{60 b^2 d^3}+\frac {\left (B^2 (b c-a d)^6 g^2 i^3\right ) \text {Subst}\left (\int \left (\frac {1}{b^3 x}+\frac {6 b d}{(b-d x)^5}-\frac {9 d}{(b-d x)^4}+\frac {d}{b (b-d x)^3}+\frac {d}{b^2 (b-d x)^2}+\frac {d}{b^3 (b-d x)}\right ) \, dx,x,\frac {a+b x}{c+d x}\right )}{90 b d^3} \\ & = \frac {B^2 (b c-a d)^5 g^2 i^3 x}{90 b^3 d^2}+\frac {B^2 (b c-a d)^4 g^2 i^3 (c+d x)^2}{180 b^2 d^3}-\frac {B^2 (b c-a d)^3 g^2 i^3 (c+d x)^3}{30 b d^3}+\frac {B^2 (b c-a d)^2 g^2 i^3 (c+d x)^4}{60 d^3}+\frac {B^2 (b c-a d)^6 g^2 i^3 \log \left (\frac {a+b x}{c+d x}\right )}{90 b^4 d^3}-\frac {B (b c-a d)^4 g^2 i^3 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{60 b^4 d}-\frac {B (b c-a d)^3 g^2 i^3 (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{30 b^4}-\frac {B (b c-a d)^4 g^2 i^3 (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{10 b^2 d^3}+\frac {B (b c-a d)^3 g^2 i^3 (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{45 b d^3}+\frac {7 B (b c-a d)^2 g^2 i^3 (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{60 d^3}-\frac {b B (b c-a d) g^2 i^3 (c+d x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{15 d^3}+\frac {(b c-a d)^3 g^2 i^3 (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{60 b^4}+\frac {(b c-a d)^2 g^2 i^3 (a+b x)^3 (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{20 b^3}+\frac {(b c-a d) g^2 i^3 (a+b x)^3 (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{10 b^2}+\frac {g^2 i^3 (a+b x)^3 (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{6 b}+\frac {B^2 (b c-a d)^6 g^2 i^3 \log (c+d x)}{90 b^4 d^3}+\frac {\left (B^2 (b c-a d)^6 g^2 i^3\right ) \text {Subst}\left (\int \left (\frac {b^2}{d^2 (b-d x)^3}-\frac {2 b}{d^2 (b-d x)^2}+\frac {1}{d^2 (b-d x)}\right ) \, dx,x,\frac {a+b x}{c+d x}\right )}{30 b^4}+\frac {\left (B^2 (b c-a d)^6 g^2 i^3\right ) \text {Subst}\left (\int \left (\frac {1}{b^2 x}+\frac {3 b d}{(b-d x)^4}-\frac {5 d}{(b-d x)^3}+\frac {d}{b (b-d x)^2}+\frac {d}{b^2 (b-d x)}\right ) \, dx,x,\frac {a+b x}{c+d x}\right )}{60 b^2 d^3}+\frac {\left (B (b c-a d)^6 g^2 i^3\right ) \text {Subst}\left (\int \frac {x (2 A+B+2 B \log (e x))}{(b-d x)^2} \, dx,x,\frac {a+b x}{c+d x}\right )}{60 b^4 d} \\ & = -\frac {7 B^2 (b c-a d)^5 g^2 i^3 x}{180 b^3 d^2}-\frac {7 B^2 (b c-a d)^4 g^2 i^3 (c+d x)^2}{360 b^2 d^3}-\frac {B^2 (b c-a d)^3 g^2 i^3 (c+d x)^3}{60 b d^3}+\frac {B^2 (b c-a d)^2 g^2 i^3 (c+d x)^4}{60 d^3}+\frac {B^2 (b c-a d)^6 g^2 i^3 \log \left (\frac {a+b x}{c+d x}\right )}{36 b^4 d^3}-\frac {B (b c-a d)^4 g^2 i^3 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{60 b^4 d}-\frac {B (b c-a d)^3 g^2 i^3 (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{30 b^4}-\frac {B (b c-a d)^4 g^2 i^3 (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{10 b^2 d^3}+\frac {B (b c-a d)^3 g^2 i^3 (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{45 b d^3}+\frac {7 B (b c-a d)^2 g^2 i^3 (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{60 d^3}-\frac {b B (b c-a d) g^2 i^3 (c+d x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{15 d^3}+\frac {(b c-a d)^3 g^2 i^3 (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{60 b^4}+\frac {(b c-a d)^2 g^2 i^3 (a+b x)^3 (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{20 b^3}+\frac {(b c-a d) g^2 i^3 (a+b x)^3 (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{10 b^2}+\frac {g^2 i^3 (a+b x)^3 (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{6 b}+\frac {B (b c-a d)^5 g^2 i^3 (a+b x) \left (2 A+B+2 B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{60 b^4 d^2}+\frac {11 B^2 (b c-a d)^6 g^2 i^3 \log (c+d x)}{180 b^4 d^3}-\frac {\left (B (b c-a d)^6 g^2 i^3\right ) \text {Subst}\left (\int \frac {2 A+3 B+2 B \log (e x)}{b-d x} \, dx,x,\frac {a+b x}{c+d x}\right )}{60 b^4 d^2} \\ & = -\frac {7 B^2 (b c-a d)^5 g^2 i^3 x}{180 b^3 d^2}-\frac {7 B^2 (b c-a d)^4 g^2 i^3 (c+d x)^2}{360 b^2 d^3}-\frac {B^2 (b c-a d)^3 g^2 i^3 (c+d x)^3}{60 b d^3}+\frac {B^2 (b c-a d)^2 g^2 i^3 (c+d x)^4}{60 d^3}+\frac {B^2 (b c-a d)^6 g^2 i^3 \log \left (\frac {a+b x}{c+d x}\right )}{36 b^4 d^3}-\frac {B (b c-a d)^4 g^2 i^3 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{60 b^4 d}-\frac {B (b c-a d)^3 g^2 i^3 (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{30 b^4}-\frac {B (b c-a d)^4 g^2 i^3 (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{10 b^2 d^3}+\frac {B (b c-a d)^3 g^2 i^3 (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{45 b d^3}+\frac {7 B (b c-a d)^2 g^2 i^3 (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{60 d^3}-\frac {b B (b c-a d) g^2 i^3 (c+d x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{15 d^3}+\frac {(b c-a d)^3 g^2 i^3 (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{60 b^4}+\frac {(b c-a d)^2 g^2 i^3 (a+b x)^3 (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{20 b^3}+\frac {(b c-a d) g^2 i^3 (a+b x)^3 (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{10 b^2}+\frac {g^2 i^3 (a+b x)^3 (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{6 b}+\frac {B (b c-a d)^5 g^2 i^3 (a+b x) \left (2 A+B+2 B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{60 b^4 d^2}+\frac {B (b c-a d)^6 g^2 i^3 \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (2 A+3 B+2 B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{60 b^4 d^3}+\frac {11 B^2 (b c-a d)^6 g^2 i^3 \log (c+d x)}{180 b^4 d^3}-\frac {\left (B^2 (b c-a d)^6 g^2 i^3\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {d x}{b}\right )}{x} \, dx,x,\frac {a+b x}{c+d x}\right )}{30 b^4 d^3} \\ & = -\frac {7 B^2 (b c-a d)^5 g^2 i^3 x}{180 b^3 d^2}-\frac {7 B^2 (b c-a d)^4 g^2 i^3 (c+d x)^2}{360 b^2 d^3}-\frac {B^2 (b c-a d)^3 g^2 i^3 (c+d x)^3}{60 b d^3}+\frac {B^2 (b c-a d)^2 g^2 i^3 (c+d x)^4}{60 d^3}+\frac {B^2 (b c-a d)^6 g^2 i^3 \log \left (\frac {a+b x}{c+d x}\right )}{36 b^4 d^3}-\frac {B (b c-a d)^4 g^2 i^3 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{60 b^4 d}-\frac {B (b c-a d)^3 g^2 i^3 (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{30 b^4}-\frac {B (b c-a d)^4 g^2 i^3 (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{10 b^2 d^3}+\frac {B (b c-a d)^3 g^2 i^3 (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{45 b d^3}+\frac {7 B (b c-a d)^2 g^2 i^3 (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{60 d^3}-\frac {b B (b c-a d) g^2 i^3 (c+d x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{15 d^3}+\frac {(b c-a d)^3 g^2 i^3 (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{60 b^4}+\frac {(b c-a d)^2 g^2 i^3 (a+b x)^3 (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{20 b^3}+\frac {(b c-a d) g^2 i^3 (a+b x)^3 (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{10 b^2}+\frac {g^2 i^3 (a+b x)^3 (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{6 b}+\frac {B (b c-a d)^5 g^2 i^3 (a+b x) \left (2 A+B+2 B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{60 b^4 d^2}+\frac {B (b c-a d)^6 g^2 i^3 \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (2 A+3 B+2 B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{60 b^4 d^3}+\frac {11 B^2 (b c-a d)^6 g^2 i^3 \log (c+d x)}{180 b^4 d^3}+\frac {B^2 (b c-a d)^6 g^2 i^3 \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right )}{30 b^4 d^3} \\ \end{align*}
Time = 0.80 (sec) , antiderivative size = 1555, normalized size of antiderivative = 1.71 \[ \int (a g+b g x)^2 (c i+d i x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \, dx=\frac {g^2 i^3 \left (15 (b c-a d)^2 (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2-24 b (b c-a d) (c+d x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2+10 b^2 (c+d x)^6 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2-\frac {5 B (b c-a d)^3 \left (6 A b d (b c-a d)^2 x-3 B (b c-a d)^2 (b d x+(b c-a d) \log (a+b x))-B (b c-a d) \left (2 b d (b c-a d) x+b^2 (c+d x)^2+2 (b c-a d)^2 \log (a+b x)\right )+6 B d (b c-a d)^2 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )+3 b^2 (b c-a d) (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+2 b^3 (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+6 (b c-a d)^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )-6 B (b c-a d)^3 \log (c+d x)-3 B (b c-a d)^3 \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{-b c+a d}\right )\right )\right )}{b^4}+\frac {2 B (b c-a d)^2 \left (24 A b d (b c-a d)^3 x-12 B (b c-a d)^3 (b d x+(b c-a d) \log (a+b x))-4 B (b c-a d)^2 \left (2 b d (b c-a d) x+b^2 (c+d x)^2+2 (b c-a d)^2 \log (a+b x)\right )-B (b c-a d) \left (6 b d (b c-a d)^2 x+3 b^2 (b c-a d) (c+d x)^2+2 b^3 (c+d x)^3+6 (b c-a d)^3 \log (a+b x)\right )+24 B d (b c-a d)^3 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )+12 b^2 (b c-a d)^2 (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+8 b^3 (b c-a d) (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+6 b^4 (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+24 (b c-a d)^4 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )-24 B (b c-a d)^4 \log (c+d x)-12 B (b c-a d)^4 \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{-b c+a d}\right )\right )\right )}{b^4}-\frac {B (b c-a d) \left (120 A b d (b c-a d)^4 x-60 B (b c-a d)^4 (b d x+(b c-a d) \log (a+b x))-20 B (b c-a d)^3 \left (2 b d (b c-a d) x+b^2 (c+d x)^2+2 (b c-a d)^2 \log (a+b x)\right )-5 B (b c-a d)^2 \left (6 b d (b c-a d)^2 x+3 b^2 (b c-a d) (c+d x)^2+2 b^3 (c+d x)^3+6 (b c-a d)^3 \log (a+b x)\right )-2 B (b c-a d) \left (12 b d (b c-a d)^3 x+6 b^2 (b c-a d)^2 (c+d x)^2+4 b^3 (b c-a d) (c+d x)^3+3 b^4 (c+d x)^4+12 (b c-a d)^4 \log (a+b x)\right )+120 B d (b c-a d)^4 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )+60 b^2 (b c-a d)^3 (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+40 b^3 (b c-a d)^2 (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+30 b^4 (b c-a d) (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+24 b^5 (c+d x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+120 (b c-a d)^5 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )-120 B (b c-a d)^5 \log (c+d x)-60 B (b c-a d)^5 \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{-b c+a d}\right )\right )\right )}{6 b^4}\right )}{60 d^3} \]
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\[\int \left (b g x +a g \right )^{2} \left (d i x +c i \right )^{3} \left (A +B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )\right )^{2}d x\]
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\[ \int (a g+b g x)^2 (c i+d i x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \, dx=\int { {\left (b g x + a g\right )}^{2} {\left (d i x + c i\right )}^{3} {\left (B \log \left (\frac {{\left (b x + a\right )} e}{d x + c}\right ) + A\right )}^{2} \,d x } \]
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Timed out. \[ \int (a g+b g x)^2 (c i+d i x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 5196 vs. \(2 (869) = 1738\).
Time = 0.36 (sec) , antiderivative size = 5196, normalized size of antiderivative = 5.72 \[ \int (a g+b g x)^2 (c i+d i x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \, dx=\text {Too large to display} \]
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\[ \int (a g+b g x)^2 (c i+d i x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \, dx=\int { {\left (b g x + a g\right )}^{2} {\left (d i x + c i\right )}^{3} {\left (B \log \left (\frac {{\left (b x + a\right )} e}{d x + c}\right ) + A\right )}^{2} \,d x } \]
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Timed out. \[ \int (a g+b g x)^2 (c i+d i x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \, dx=\int {\left (a\,g+b\,g\,x\right )}^2\,{\left (c\,i+d\,i\,x\right )}^3\,{\left (A+B\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )\right )}^2 \,d x \]
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