Integrand size = 32, antiderivative size = 565 \[ \int (f+g x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx=\frac {B^2 (b c-a d)^2 g^2 n^2 x}{3 b^2 d^2}-\frac {2 B (b c-a d) g (3 b d f-2 b c g-a d g) n (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 b^3 d^2}-\frac {B (b c-a d) g^2 n (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 b d^3}-\frac {(b f-a g)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{3 b^3 g}+\frac {(f+g x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{3 g}+\frac {2 B (b c-a d) \left (a^2 d^2 g^2-a b d g (3 d f-c g)+b^2 \left (3 d^2 f^2-3 c d f g+c^2 g^2\right )\right ) n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log \left (\frac {b c-a d}{b (c+d x)}\right )}{3 b^3 d^3}+\frac {B^2 (b c-a d)^3 g^2 n^2 \log \left (\frac {a+b x}{c+d x}\right )}{3 b^3 d^3}+\frac {B^2 (b c-a d)^3 g^2 n^2 \log (c+d x)}{3 b^3 d^3}+\frac {2 B^2 (b c-a d)^2 g (3 b d f-2 b c g-a d g) n^2 \log (c+d x)}{3 b^3 d^3}+\frac {2 B^2 (b c-a d) \left (a^2 d^2 g^2-a b d g (3 d f-c g)+b^2 \left (3 d^2 f^2-3 c d f g+c^2 g^2\right )\right ) n^2 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{3 b^3 d^3} \]
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Time = 0.74 (sec) , antiderivative size = 565, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {2553, 2398, 2404, 2338, 2356, 46, 2351, 31, 2354, 2438} \[ \int (f+g x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx=\frac {2 B n (b c-a d) \left (a^2 d^2 g^2-a b d g (3 d f-c g)+b^2 \left (c^2 g^2-3 c d f g+3 d^2 f^2\right )\right ) \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{3 b^3 d^3}+\frac {2 B^2 n^2 (b c-a d) \left (a^2 d^2 g^2-a b d g (3 d f-c g)+b^2 \left (c^2 g^2-3 c d f g+3 d^2 f^2\right )\right ) \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{3 b^3 d^3}-\frac {2 B g n (a+b x) (b c-a d) (-a d g-2 b c g+3 b d f) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{3 b^3 d^2}-\frac {(b f-a g)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{3 b^3 g}-\frac {B g^2 n (c+d x)^2 (b c-a d) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{3 b d^3}+\frac {(f+g x)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{3 g}+\frac {2 B^2 g n^2 (b c-a d)^2 \log (c+d x) (-a d g-2 b c g+3 b d f)}{3 b^3 d^3}+\frac {B^2 g^2 n^2 (b c-a d)^3 \log \left (\frac {a+b x}{c+d x}\right )}{3 b^3 d^3}+\frac {B^2 g^2 n^2 (b c-a d)^3 \log (c+d x)}{3 b^3 d^3}+\frac {B^2 g^2 n^2 x (b c-a d)^2}{3 b^2 d^2} \]
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Rule 31
Rule 46
Rule 2338
Rule 2351
Rule 2354
Rule 2356
Rule 2398
Rule 2404
Rule 2438
Rule 2553
Rubi steps \begin{align*} \text {integral}& = (b c-a d) \text {Subst}\left (\int \frac {(b f-a g-(d f-c g) 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 ) \\ & = \frac {(f+g x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{3 g}-\frac {(2 B n) \text {Subst}\left (\int \frac {(b f-a g+(-d f+c g) x)^3 \left (A+B \log \left (e x^n\right )\right )}{x (b-d x)^3} \, dx,x,\frac {a+b x}{c+d x}\right )}{3 g} \\ & = \frac {(f+g x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{3 g}-\frac {(2 B n) \text {Subst}\left (\int \left (\frac {(b f-a g)^3 \left (A+B \log \left (e x^n\right )\right )}{b^3 x}+\frac {(b c-a d)^3 g^3 \left (A+B \log \left (e x^n\right )\right )}{b d^2 (b-d x)^3}+\frac {(b c-a d)^2 g^2 (3 b d f-2 b c g-a d g) \left (A+B \log \left (e x^n\right )\right )}{b^2 d^2 (b-d x)^2}+\frac {(b c-a d) g \left (a^2 d^2 g^2-a b d g (3 d f-c g)+b^2 \left (3 d^2 f^2-3 c d f g+c^2 g^2\right )\right ) \left (A+B \log \left (e x^n\right )\right )}{b^3 d^2 (b-d x)}\right ) \, dx,x,\frac {a+b x}{c+d x}\right )}{3 g} \\ & = \frac {(f+g x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{3 g}-\frac {\left (2 B (b c-a d)^3 g^2 n\right ) \text {Subst}\left (\int \frac {A+B \log \left (e x^n\right )}{(b-d x)^3} \, dx,x,\frac {a+b x}{c+d x}\right )}{3 b d^2}-\frac {\left (2 B (b f-a g)^3 n\right ) \text {Subst}\left (\int \frac {A+B \log \left (e x^n\right )}{x} \, dx,x,\frac {a+b x}{c+d x}\right )}{3 b^3 g}-\frac {\left (2 B (b c-a d)^2 g (3 b d f-2 b c g-a d g) n\right ) \text {Subst}\left (\int \frac {A+B \log \left (e x^n\right )}{(b-d x)^2} \, dx,x,\frac {a+b x}{c+d x}\right )}{3 b^2 d^2}-\frac {\left (2 B (b c-a d) \left (a^2 d^2 g^2-a b d g (3 d f-c g)+b^2 \left (3 d^2 f^2-3 c d f g+c^2 g^2\right )\right ) n\right ) \text {Subst}\left (\int \frac {A+B \log \left (e x^n\right )}{b-d x} \, dx,x,\frac {a+b x}{c+d x}\right )}{3 b^3 d^2} \\ & = -\frac {2 B (b c-a d) g (3 b d f-2 b c g-a d g) n (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 b^3 d^2}-\frac {B (b c-a d) g^2 n (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 b d^3}-\frac {(b f-a g)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{3 b^3 g}+\frac {(f+g x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{3 g}+\frac {2 B (b c-a d) \left (a^2 d^2 g^2-a b d g (3 d f-c g)+b^2 \left (3 d^2 f^2-3 c d f g+c^2 g^2\right )\right ) n \left (A+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 )}{3 b^3 d^3}+\frac {\left (B^2 (b c-a d)^3 g^2 n^2\right ) \text {Subst}\left (\int \frac {1}{x (b-d x)^2} \, dx,x,\frac {a+b x}{c+d x}\right )}{3 b d^3}+\frac {\left (2 B^2 (b c-a d)^2 g (3 b d f-2 b c g-a d g) n^2\right ) \text {Subst}\left (\int \frac {1}{b-d x} \, dx,x,\frac {a+b x}{c+d x}\right )}{3 b^3 d^2}-\frac {\left (2 B^2 (b c-a d) \left (a^2 d^2 g^2-a b d g (3 d f-c g)+b^2 \left (3 d^2 f^2-3 c d f g+c^2 g^2\right )\right ) 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 )}{3 b^3 d^3} \\ & = -\frac {2 B (b c-a d) g (3 b d f-2 b c g-a d g) n (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 b^3 d^2}-\frac {B (b c-a d) g^2 n (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 b d^3}-\frac {(b f-a g)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{3 b^3 g}+\frac {(f+g x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{3 g}+\frac {2 B (b c-a d) \left (a^2 d^2 g^2-a b d g (3 d f-c g)+b^2 \left (3 d^2 f^2-3 c d f g+c^2 g^2\right )\right ) n \left (A+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 )}{3 b^3 d^3}+\frac {2 B^2 (b c-a d)^2 g (3 b d f-2 b c g-a d g) n^2 \log (c+d x)}{3 b^3 d^3}+\frac {2 B^2 (b c-a d) \left (a^2 d^2 g^2-a b d g (3 d f-c g)+b^2 \left (3 d^2 f^2-3 c d f g+c^2 g^2\right )\right ) n^2 \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right )}{3 b^3 d^3}+\frac {\left (B^2 (b c-a d)^3 g^2 n^2\right ) \text {Subst}\left (\int \left (\frac {1}{b^2 x}+\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 )}{3 b d^3} \\ & = \frac {B^2 (b c-a d)^2 g^2 n^2 x}{3 b^2 d^2}-\frac {2 B (b c-a d) g (3 b d f-2 b c g-a d g) n (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 b^3 d^2}-\frac {B (b c-a d) g^2 n (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 b d^3}-\frac {(b f-a g)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{3 b^3 g}+\frac {(f+g x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{3 g}+\frac {2 B (b c-a d) \left (a^2 d^2 g^2-a b d g (3 d f-c g)+b^2 \left (3 d^2 f^2-3 c d f g+c^2 g^2\right )\right ) n \left (A+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 )}{3 b^3 d^3}+\frac {B^2 (b c-a d)^3 g^2 n^2 \log \left (\frac {a+b x}{c+d x}\right )}{3 b^3 d^3}+\frac {B^2 (b c-a d)^3 g^2 n^2 \log (c+d x)}{3 b^3 d^3}+\frac {2 B^2 (b c-a d)^2 g (3 b d f-2 b c g-a d g) n^2 \log (c+d x)}{3 b^3 d^3}+\frac {2 B^2 (b c-a d) \left (a^2 d^2 g^2-a b d g (3 d f-c g)+b^2 \left (3 d^2 f^2-3 c d f g+c^2 g^2\right )\right ) n^2 \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right )}{3 b^3 d^3} \\ \end{align*}
Time = 0.36 (sec) , antiderivative size = 506, normalized size of antiderivative = 0.90 \[ \int (f+g x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx=\frac {(f+g x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2-\frac {B n \left (2 A b d (b c-a d) g^2 (3 b d f-b c g-a d g) x+2 B d (b c-a d) g^2 (3 b d f-b c g-a d g) (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+b^2 d^2 (b c-a d) g^3 x^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )+2 d^3 (b f-a g)^3 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )+2 B (b c-a d)^2 g^2 (-3 b d f+b c g+a d g) n \log (c+d x)-2 b^3 (d f-c g)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)-B (b c-a d) g^3 n \left (a^2 d^2 \log (a+b x)-b \left (d (-b c+a d) x+b c^2 \log (c+d x)\right )\right )-B d^3 (b f-a g)^3 n \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{-b c+a d}\right )\right )+b^3 B (d f-c g)^3 n \left (\left (2 \log \left (\frac {d (a+b x)}{-b c+a d}\right )-\log (c+d x)\right ) \log (c+d x)+2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )\right )\right )}{b^3 d^3}}{3 g} \]
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\[\int \left (g x +f \right )^{2} {\left (A +B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )\right )}^{2}d x\]
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\[ \int (f+g x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx=\int { {\left (g x + f\right )}^{2} {\left (B \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right ) + A\right )}^{2} \,d x } \]
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Timed out. \[ \int (f+g x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1659 vs. \(2 (544) = 1088\).
Time = 0.69 (sec) , antiderivative size = 1659, normalized size of antiderivative = 2.94 \[ \int (f+g x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx=\text {Too large to display} \]
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\[ \int (f+g x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx=\int { {\left (g x + f\right )}^{2} {\left (B \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right ) + A\right )}^{2} \,d x } \]
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Timed out. \[ \int (f+g x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx=\int {\left (f+g\,x\right )}^2\,{\left (A+B\,\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\right )}^2 \,d x \]
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