Integrand size = 26, antiderivative size = 26 \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2}}{(f+g x)^3} \, dx=-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2}}{2 g (f+g x)^2}+\frac {5 b e n \text {Int}\left (\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}{(d+e x) (f+g x)^2},x\right )}{4 g} \]
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Not integrable
Time = 0.16 (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 = {} \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2}}{(f+g x)^3} \, dx=\int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2}}{(f+g x)^3} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = -\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2}}{2 g (f+g x)^2}+\frac {(5 b e n) \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}{(d+e x) (f+g x)^2} \, dx}{4 g} \\ \end{align*}
Not integrable
Time = 5.38 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.08 \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2}}{(f+g x)^3} \, dx=\int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2}}{(f+g x)^3} \, dx \]
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Not integrable
Time = 0.12 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92
\[\int \frac {{\left (a +b \ln \left (c \left (e x +d \right )^{n}\right )\right )}^{\frac {5}{2}}}{\left (g x +f \right )^{3}}d x\]
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Exception generated. \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2}}{(f+g x)^3} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2}}{(f+g x)^3} \, dx=\text {Timed out} \]
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Not integrable
Time = 0.67 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2}}{(f+g x)^3} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{\frac {5}{2}}}{{\left (g x + f\right )}^{3}} \,d x } \]
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Not integrable
Time = 0.89 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2}}{(f+g x)^3} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{\frac {5}{2}}}{{\left (g x + f\right )}^{3}} \,d x } \]
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Not integrable
Time = 1.49 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2}}{(f+g x)^3} \, dx=\int \frac {{\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}^{5/2}}{{\left (f+g\,x\right )}^3} \,d x \]
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