Integrand size = 45, antiderivative size = 45 \[ \int \frac {\log \left (h (f+g x)^m\right )}{(a+b x) (c+d x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )} \, dx=\frac {b \text {Int}\left (\frac {\log \left (h (f+g x)^m\right )}{(a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )},x\right )}{b c-a d}-\frac {d \text {Int}\left (\frac {\log \left (h (f+g x)^m\right )}{(c+d x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )},x\right )}{b c-a d} \]
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Not integrable
Time = 0.38 (sec) , antiderivative size = 45, 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 = {} \[ \int \frac {\log \left (h (f+g x)^m\right )}{(a+b x) (c+d x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )} \, dx=\int \frac {\log \left (h (f+g x)^m\right )}{(a+b x) (c+d x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {b \log \left (h (f+g x)^m\right )}{(b c-a d) (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}-\frac {d \log \left (h (f+g x)^m\right )}{(b c-a d) (c+d x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}\right ) \, dx \\ & = \frac {b \int \frac {\log \left (h (f+g x)^m\right )}{(a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )} \, dx}{b c-a d}-\frac {d \int \frac {\log \left (h (f+g x)^m\right )}{(c+d x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )} \, dx}{b c-a d} \\ \end{align*}
Not integrable
Time = 1.24 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.04 \[ \int \frac {\log \left (h (f+g x)^m\right )}{(a+b x) (c+d x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )} \, dx=\int \frac {\log \left (h (f+g x)^m\right )}{(a+b x) (c+d x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )} \, dx \]
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Not integrable
Time = 0.25 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.00
\[\int \frac {\ln \left (h \left (g x +f \right )^{m}\right )}{\left (b x +a \right ) \left (d x +c \right ) \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )}d x\]
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Not integrable
Time = 0.30 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.20 \[ \int \frac {\log \left (h (f+g x)^m\right )}{(a+b x) (c+d x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )} \, dx=\int { \frac {\log \left ({\left (g x + f\right )}^{m} h\right )}{{\left (b x + a\right )} {\left (d x + c\right )} \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right )} \,d x } \]
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Timed out. \[ \int \frac {\log \left (h (f+g x)^m\right )}{(a+b x) (c+d x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )} \, dx=\text {Timed out} \]
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Not integrable
Time = 1.72 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.04 \[ \int \frac {\log \left (h (f+g x)^m\right )}{(a+b x) (c+d x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )} \, dx=\int { \frac {\log \left ({\left (g x + f\right )}^{m} h\right )}{{\left (b x + a\right )} {\left (d x + c\right )} \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right )} \,d x } \]
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Not integrable
Time = 0.41 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.04 \[ \int \frac {\log \left (h (f+g x)^m\right )}{(a+b x) (c+d x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )} \, dx=\int { \frac {\log \left ({\left (g x + f\right )}^{m} h\right )}{{\left (b x + a\right )} {\left (d x + c\right )} \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right )} \,d x } \]
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Not integrable
Time = 1.35 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.04 \[ \int \frac {\log \left (h (f+g x)^m\right )}{(a+b x) (c+d x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )} \, dx=\int \frac {\ln \left (h\,{\left (f+g\,x\right )}^m\right )}{\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\,\left (a+b\,x\right )\,\left (c+d\,x\right )} \,d x \]
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