Integrand size = 26, antiderivative size = 26 \[ \int \frac {(f+g x)^m}{\left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}} \, dx=\text {Int}\left (\frac {(f+g x)^m}{\left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}},x\right ) \]
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Not integrable
Time = 0.04 (sec) , antiderivative size = 26, 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 = {} \[ \int \frac {(f+g x)^m}{\left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}} \, dx=\int \frac {(f+g x)^m}{\left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {(f+g x)^m}{\left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}} \, dx \\ \end{align*}
Not integrable
Time = 8.52 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.08 \[ \int \frac {(f+g x)^m}{\left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}} \, dx=\int \frac {(f+g x)^m}{\left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}} \, dx \]
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Not integrable
Time = 0.13 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92
\[\int \frac {\left (g x +f \right )^{m}}{{\left (a +b \ln \left (c \left (e x +d \right )^{n}\right )\right )}^{\frac {3}{2}}}d x\]
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Not integrable
Time = 0.36 (sec) , antiderivative size = 62, normalized size of antiderivative = 2.38 \[ \int \frac {(f+g x)^m}{\left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}} \, dx=\int { \frac {{\left (g x + f\right )}^{m}}{{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
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Exception generated. \[ \int \frac {(f+g x)^m}{\left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}} \, dx=\text {Exception raised: HeuristicGCDFailed} \]
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Not integrable
Time = 0.66 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {(f+g x)^m}{\left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}} \, dx=\int { \frac {{\left (g x + f\right )}^{m}}{{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
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Not integrable
Time = 0.45 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {(f+g x)^m}{\left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}} \, dx=\int { \frac {{\left (g x + f\right )}^{m}}{{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
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Not integrable
Time = 1.63 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {(f+g x)^m}{\left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}} \, dx=\int \frac {{\left (f+g\,x\right )}^m}{{\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}^{3/2}} \,d x \]
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