Integrand size = 43, antiderivative size = 43 \[ \int \frac {1}{(f+g x) (a h+b h x) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2} \, dx=\text {Subst}\left (\text {Int}\left (\frac {1}{(f+g x) (a h+b h x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2},x\right ),e \left (\frac {a+b x}{c+d x}\right )^n,e (a+b x)^n (c+d x)^{-n}\right ) \]
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Not integrable
Time = 0.16 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {1}{(f+g x) (a h+b h x) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2} \, dx=\int \frac {1}{(f+g x) (a h+b h x) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {1}{(f+g x) (a h+b h x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2} \, dx,e \left (\frac {a+b x}{c+d x}\right )^n,e (a+b x)^n (c+d x)^{-n}\right ) \\ \end{align*}
Not integrable
Time = 0.33 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.05 \[ \int \frac {1}{(f+g x) (a h+b h x) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2} \, dx=\int \frac {1}{(f+g x) (a h+b h x) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2} \, dx \]
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Not integrable
Time = 0.03 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.00
\[\int \frac {1}{\left (g x +f \right ) \left (b h x +a h \right ) {\left (A +B \ln \left (e \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right )\right )}^{2}}d x\]
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Not integrable
Time = 0.31 (sec) , antiderivative size = 145, normalized size of antiderivative = 3.37 \[ \int \frac {1}{(f+g x) (a h+b h x) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2} \, dx=\int { \frac {1}{{\left (b h x + a h\right )} {\left (g x + f\right )} {\left (B \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) + A\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {1}{(f+g x) (a h+b h x) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2} \, dx=\text {Timed out} \]
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Not integrable
Time = 0.38 (sec) , antiderivative size = 506, normalized size of antiderivative = 11.77 \[ \int \frac {1}{(f+g x) (a h+b h x) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2} \, dx=\int { \frac {1}{{\left (b h x + a h\right )} {\left (g x + f\right )} {\left (B \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) + A\right )}^{2}} \,d x } \]
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Not integrable
Time = 0.34 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.05 \[ \int \frac {1}{(f+g x) (a h+b h x) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2} \, dx=\int { \frac {1}{{\left (b h x + a h\right )} {\left (g x + f\right )} {\left (B \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) + A\right )}^{2}} \,d x } \]
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Not integrable
Time = 1.29 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.05 \[ \int \frac {1}{(f+g x) (a h+b h x) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2} \, dx=\int \frac {1}{\left (f+g\,x\right )\,\left (a\,h+b\,h\,x\right )\,{\left (A+B\,\ln \left (\frac {e\,{\left (a+b\,x\right )}^n}{{\left (c+d\,x\right )}^n}\right )\right )}^2} \,d x \]
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