Integrand size = 43, antiderivative size = 487 \[ \int (a g+b g x)^2 (c i+d i x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx=-\frac {B^2 (b c-a d)^3 g^2 i n^2 x}{3 b d^2}+\frac {B^2 (b c-a d)^2 g^2 i n^2 (c+d x)^2}{12 d^3}-\frac {B (b c-a d)^2 g^2 i n (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{12 b^2 d}-\frac {B (b c-a d) g^2 i n (a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{6 b^2}+\frac {(b c-a d) g^2 i (a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{12 b^2}+\frac {g^2 i (a+b x)^3 (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{4 b}+\frac {B (b c-a d)^3 g^2 i n (a+b x) \left (2 A+B n+2 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{12 b^2 d^2}+\frac {B (b c-a d)^4 g^2 i n \left (2 A+3 B n+2 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log \left (\frac {b c-a d}{b (c+d x)}\right )}{12 b^2 d^3}+\frac {B^2 (b c-a d)^4 g^2 i n^2 \log (c+d x)}{6 b^2 d^3}+\frac {B^2 (b c-a d)^4 g^2 i n^2 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{6 b^2 d^3} \]
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Time = 0.39 (sec) , antiderivative size = 487, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.186, Rules used = {2561, 2383, 2381, 2384, 2354, 2438, 2373, 45} \[ \int (a g+b g x)^2 (c i+d i x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx=\frac {B g^2 i n (b c-a d)^4 \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (2 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+2 A+3 B n\right )}{12 b^2 d^3}+\frac {B g^2 i n (a+b x) (b c-a d)^3 \left (2 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+2 A+B n\right )}{12 b^2 d^2}-\frac {B g^2 i n (a+b x)^2 (b c-a d)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{12 b^2 d}+\frac {g^2 i (a+b x)^3 (b c-a d) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{12 b^2}-\frac {B g^2 i n (a+b x)^3 (b c-a d) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{6 b^2}+\frac {g^2 i (a+b x)^3 (c+d x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{4 b}+\frac {B^2 g^2 i n^2 (b c-a d)^4 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{6 b^2 d^3}+\frac {B^2 g^2 i n^2 (b c-a d)^4 \log (c+d x)}{6 b^2 d^3}+\frac {B^2 g^2 i n^2 (c+d x)^2 (b c-a d)^2}{12 d^3}-\frac {B^2 g^2 i n^2 x (b c-a d)^3}{3 b d^2} \]
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Rule 45
Rule 2354
Rule 2373
Rule 2381
Rule 2383
Rule 2384
Rule 2438
Rule 2561
Rubi steps \begin{align*} \text {integral}& = \left ((b c-a d)^4 g^2 i\right ) \text {Subst}\left (\int \frac {x^2 \left (A+B \log \left (e x^n\right )\right )^2}{(b-d x)^5} \, dx,x,\frac {a+b x}{c+d x}\right ) \\ & = \frac {g^2 i (a+b x)^3 (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{4 b}+\frac {\left ((b c-a d)^4 g^2 i\right ) \text {Subst}\left (\int \frac {x^2 \left (A+B \log \left (e x^n\right )\right )^2}{(b-d x)^4} \, dx,x,\frac {a+b x}{c+d x}\right )}{4 b}-\frac {\left (B (b c-a d)^4 g^2 i n\right ) \text {Subst}\left (\int \frac {x^2 \left (A+B \log \left (e x^n\right )\right )}{(b-d x)^4} \, dx,x,\frac {a+b x}{c+d x}\right )}{2 b} \\ & = -\frac {B (b c-a d) g^2 i n (a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{6 b^2}+\frac {(b c-a d) g^2 i (a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{12 b^2}+\frac {g^2 i (a+b x)^3 (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{4 b}-\frac {\left (B (b c-a d)^4 g^2 i n\right ) \text {Subst}\left (\int \frac {x^2 \left (A+B \log \left (e x^n\right )\right )}{(b-d x)^3} \, dx,x,\frac {a+b x}{c+d x}\right )}{6 b^2}+\frac {\left (B^2 (b c-a d)^4 g^2 i n^2\right ) \text {Subst}\left (\int \frac {x^2}{(b-d x)^3} \, dx,x,\frac {a+b x}{c+d x}\right )}{6 b^2} \\ & = -\frac {B (b c-a d)^2 g^2 i n (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{12 b^2 d}-\frac {B (b c-a d) g^2 i n (a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{6 b^2}+\frac {(b c-a d) g^2 i (a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{12 b^2}+\frac {g^2 i (a+b x)^3 (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{4 b}+\frac {\left (B (b c-a d)^4 g^2 i n\right ) \text {Subst}\left (\int \frac {x \left (2 A+B n+2 B \log \left (e x^n\right )\right )}{(b-d x)^2} \, dx,x,\frac {a+b x}{c+d x}\right )}{12 b^2 d}+\frac {\left (B^2 (b c-a d)^4 g^2 i n^2\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 )}{6 b^2} \\ & = -\frac {B^2 (b c-a d)^3 g^2 i n^2 x}{3 b d^2}+\frac {B^2 (b c-a d)^2 g^2 i n^2 (c+d x)^2}{12 d^3}-\frac {B (b c-a d)^2 g^2 i n (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{12 b^2 d}-\frac {B (b c-a d) g^2 i n (a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{6 b^2}+\frac {(b c-a d) g^2 i (a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{12 b^2}+\frac {g^2 i (a+b x)^3 (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{4 b}+\frac {B (b c-a d)^3 g^2 i n (a+b x) \left (2 A+B n+2 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{12 b^2 d^2}+\frac {B^2 (b c-a d)^4 g^2 i n^2 \log (c+d x)}{6 b^2 d^3}-\frac {\left (B (b c-a d)^4 g^2 i n\right ) \text {Subst}\left (\int \frac {2 A+3 B n+2 B \log \left (e x^n\right )}{b-d x} \, dx,x,\frac {a+b x}{c+d x}\right )}{12 b^2 d^2} \\ & = -\frac {B^2 (b c-a d)^3 g^2 i n^2 x}{3 b d^2}+\frac {B^2 (b c-a d)^2 g^2 i n^2 (c+d x)^2}{12 d^3}-\frac {B (b c-a d)^2 g^2 i n (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{12 b^2 d}-\frac {B (b c-a d) g^2 i n (a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{6 b^2}+\frac {(b c-a d) g^2 i (a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{12 b^2}+\frac {g^2 i (a+b x)^3 (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{4 b}+\frac {B (b c-a d)^3 g^2 i n (a+b x) \left (2 A+B n+2 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{12 b^2 d^2}+\frac {B (b c-a d)^4 g^2 i n \left (2 A+3 B n+2 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log \left (\frac {b c-a d}{b (c+d x)}\right )}{12 b^2 d^3}+\frac {B^2 (b c-a d)^4 g^2 i n^2 \log (c+d x)}{6 b^2 d^3}-\frac {\left (B^2 (b c-a d)^4 g^2 i n^2\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 )}{6 b^2 d^3} \\ & = -\frac {B^2 (b c-a d)^3 g^2 i n^2 x}{3 b d^2}+\frac {B^2 (b c-a d)^2 g^2 i n^2 (c+d x)^2}{12 d^3}-\frac {B (b c-a d)^2 g^2 i n (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{12 b^2 d}-\frac {B (b c-a d) g^2 i n (a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{6 b^2}+\frac {(b c-a d) g^2 i (a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{12 b^2}+\frac {g^2 i (a+b x)^3 (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{4 b}+\frac {B (b c-a d)^3 g^2 i n (a+b x) \left (2 A+B n+2 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{12 b^2 d^2}+\frac {B (b c-a d)^4 g^2 i n \left (2 A+3 B n+2 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log \left (\frac {b c-a d}{b (c+d x)}\right )}{12 b^2 d^3}+\frac {B^2 (b c-a d)^4 g^2 i n^2 \log (c+d x)}{6 b^2 d^3}+\frac {B^2 (b c-a d)^4 g^2 i n^2 \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right )}{6 b^2 d^3} \\ \end{align*}
Time = 0.34 (sec) , antiderivative size = 716, normalized size of antiderivative = 1.47 \[ \int (a g+b g x)^2 (c i+d i x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx=\frac {g^2 i \left (4 (b c-a d) (a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2+3 d (a+b x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2+\frac {4 B (b c-a d)^2 n \left (2 A b d (b c-a d) x+2 B d (b c-a d) (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-d^2 (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )-2 B (b c-a d)^2 n \log (c+d x)-2 (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)+B (b c-a d) n (b d x+(-b c+a d) \log (c+d x))+B (b c-a d)^2 n \left (\left (2 \log \left (\frac {d (a+b x)}{-b c+a d}\right )-\log (c+d x)\right ) \log (c+d x)+2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )\right )\right )}{d^3}-\frac {B (b c-a d) n \left (6 A b d (b c-a d)^2 x+6 B d (b c-a d)^2 (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+3 d^2 (-b c+a d) (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )+2 d^3 (a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )-6 B (b c-a d)^3 n \log (c+d x)-6 (b c-a d)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)+B (b c-a d) n \left (2 b d (b c-a d) x-d^2 (a+b x)^2-2 (b c-a d)^2 \log (c+d x)\right )+3 B (b c-a d)^2 n (b d x+(-b c+a d) \log (c+d x))+3 B (b c-a d)^3 n \left (\left (2 \log \left (\frac {d (a+b x)}{-b c+a d}\right )-\log (c+d x)\right ) \log (c+d x)+2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )\right )\right )}{d^3}\right )}{12 b^2} \]
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\[\int \left (b g x +a g \right )^{2} \left (d i x +c i \right ) {\left (A +B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )\right )}^{2}d x\]
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\[ \int (a g+b g x)^2 (c i+d i x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx=\int { {\left (b g x + a g\right )}^{2} {\left (d i x + c i\right )} {\left (B \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right ) + A\right )}^{2} \,d x } \]
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Timed out. \[ \int (a g+b g x)^2 (c i+d i x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 2691 vs. \(2 (466) = 932\).
Time = 0.73 (sec) , antiderivative size = 2691, normalized size of antiderivative = 5.53 \[ \int (a g+b g x)^2 (c i+d i x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx=\text {Too large to display} \]
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\[ \int (a g+b g x)^2 (c i+d i x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx=\int { {\left (b g x + a g\right )}^{2} {\left (d i x + c i\right )} {\left (B \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right ) + A\right )}^{2} \,d x } \]
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Timed out. \[ \int (a g+b g x)^2 (c i+d i x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx=\int {\left (a\,g+b\,g\,x\right )}^2\,\left (c\,i+d\,i\,x\right )\,{\left (A+B\,\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\right )}^2 \,d x \]
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