Integrand size = 29, antiderivative size = 29 \[ \int \frac {\left (f+g x^{2 n}\right )^2 \log ^q\left (c \left (d+e x^n\right )^p\right )}{x} \, dx=\frac {4^{-1-q} g^2 \left (d+e x^n\right )^4 \left (c \left (d+e x^n\right )^p\right )^{-4/p} \Gamma \left (1+q,-\frac {4 \log \left (c \left (d+e x^n\right )^p\right )}{p}\right ) \log ^q\left (c \left (d+e x^n\right )^p\right ) \left (-\frac {\log \left (c \left (d+e x^n\right )^p\right )}{p}\right )^{-q}}{e^4 n}-\frac {3^{-q} d g^2 \left (d+e x^n\right )^3 \left (c \left (d+e x^n\right )^p\right )^{-3/p} \Gamma \left (1+q,-\frac {3 \log \left (c \left (d+e x^n\right )^p\right )}{p}\right ) \log ^q\left (c \left (d+e x^n\right )^p\right ) \left (-\frac {\log \left (c \left (d+e x^n\right )^p\right )}{p}\right )^{-q}}{e^4 n}+\frac {2^{-q} f g \left (d+e x^n\right )^2 \left (c \left (d+e x^n\right )^p\right )^{-2/p} \Gamma \left (1+q,-\frac {2 \log \left (c \left (d+e x^n\right )^p\right )}{p}\right ) \log ^q\left (c \left (d+e x^n\right )^p\right ) \left (-\frac {\log \left (c \left (d+e x^n\right )^p\right )}{p}\right )^{-q}}{e^2 n}+\frac {3\ 2^{-1-q} d^2 g^2 \left (d+e x^n\right )^2 \left (c \left (d+e x^n\right )^p\right )^{-2/p} \Gamma \left (1+q,-\frac {2 \log \left (c \left (d+e x^n\right )^p\right )}{p}\right ) \log ^q\left (c \left (d+e x^n\right )^p\right ) \left (-\frac {\log \left (c \left (d+e x^n\right )^p\right )}{p}\right )^{-q}}{e^4 n}-\frac {2 d f g \left (d+e x^n\right ) \left (c \left (d+e x^n\right )^p\right )^{-1/p} \Gamma \left (1+q,-\frac {\log \left (c \left (d+e x^n\right )^p\right )}{p}\right ) \log ^q\left (c \left (d+e x^n\right )^p\right ) \left (-\frac {\log \left (c \left (d+e x^n\right )^p\right )}{p}\right )^{-q}}{e^2 n}-\frac {d^3 g^2 \left (d+e x^n\right ) \left (c \left (d+e x^n\right )^p\right )^{-1/p} \Gamma \left (1+q,-\frac {\log \left (c \left (d+e x^n\right )^p\right )}{p}\right ) \log ^q\left (c \left (d+e x^n\right )^p\right ) \left (-\frac {\log \left (c \left (d+e x^n\right )^p\right )}{p}\right )^{-q}}{e^4 n}+f^2 \text {Int}\left (\frac {\log ^q\left (c \left (d+e x^n\right )^p\right )}{x},x\right ) \]
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Not integrable
Time = 0.05 (sec) , antiderivative size = 29, 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 = {} \[ \int \frac {\left (f+g x^{2 n}\right )^2 \log ^q\left (c \left (d+e x^n\right )^p\right )}{x} \, dx=\int \frac {\left (f+g x^{2 n}\right )^2 \log ^q\left (c \left (d+e x^n\right )^p\right )}{x} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {\left (f+g x^{2 n}\right )^2 \log ^q\left (c \left (d+e x^n\right )^p\right )}{x} \, dx \\ \end{align*}
Not integrable
Time = 0.25 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.07 \[ \int \frac {\left (f+g x^{2 n}\right )^2 \log ^q\left (c \left (d+e x^n\right )^p\right )}{x} \, dx=\int \frac {\left (f+g x^{2 n}\right )^2 \log ^q\left (c \left (d+e x^n\right )^p\right )}{x} \, dx \]
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Not integrable
Time = 0.28 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00
\[\int \frac {\left (f +g \,x^{2 n}\right )^{2} {\ln \left (c \left (d +e \,x^{n}\right )^{p}\right )}^{q}}{x}d x\]
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Not integrable
Time = 0.31 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.45 \[ \int \frac {\left (f+g x^{2 n}\right )^2 \log ^q\left (c \left (d+e x^n\right )^p\right )}{x} \, dx=\int { \frac {{\left (g x^{2 \, n} + f\right )}^{2} \log \left ({\left (e x^{n} + d\right )}^{p} c\right )^{q}}{x} \,d x } \]
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Timed out. \[ \int \frac {\left (f+g x^{2 n}\right )^2 \log ^q\left (c \left (d+e x^n\right )^p\right )}{x} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {\left (f+g x^{2 n}\right )^2 \log ^q\left (c \left (d+e x^n\right )^p\right )}{x} \, dx=\text {Exception raised: RuntimeError} \]
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Not integrable
Time = 3.77 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.07 \[ \int \frac {\left (f+g x^{2 n}\right )^2 \log ^q\left (c \left (d+e x^n\right )^p\right )}{x} \, dx=\int { \frac {{\left (g x^{2 \, n} + f\right )}^{2} \log \left ({\left (e x^{n} + d\right )}^{p} c\right )^{q}}{x} \,d x } \]
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Not integrable
Time = 1.51 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.07 \[ \int \frac {\left (f+g x^{2 n}\right )^2 \log ^q\left (c \left (d+e x^n\right )^p\right )}{x} \, dx=\int \frac {{\ln \left (c\,{\left (d+e\,x^n\right )}^p\right )}^q\,{\left (f+g\,x^{2\,n}\right )}^2}{x} \,d x \]
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