Integrand size = 23, antiderivative size = 23 \[ \int \frac {(f x)^m \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^2} \, dx=\text {Int}\left (\frac {(f x)^m \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^2},x\right ) \]
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Not integrable
Time = 0.03 (sec) , antiderivative size = 23, 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 = {} \[ \int \frac {(f x)^m \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^2} \, dx=\int \frac {(f x)^m \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^2} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {(f x)^m \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^2} \, dx \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(72\) vs. \(2(26)=52\).
Time = 0.12 (sec) , antiderivative size = 72, normalized size of antiderivative = 3.13 \[ \int \frac {(f x)^m \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^2} \, dx=\frac {x (f x)^m \left (-b n \, _3F_2\left (2,1+m,1+m;2+m,2+m;-\frac {e x}{d}\right )+(1+m) \operatorname {Hypergeometric2F1}\left (2,1+m,2+m,-\frac {e x}{d}\right ) \left (a+b \log \left (c x^n\right )\right )\right )}{d^2 (1+m)^2} \]
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Not integrable
Time = 0.04 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00
\[\int \frac {\left (f x \right )^{m} \left (a +b \ln \left (c \,x^{n}\right )\right )}{\left (e x +d \right )^{2}}d x\]
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Not integrable
Time = 0.34 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.83 \[ \int \frac {(f x)^m \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^2} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} \left (f x\right )^{m}}{{\left (e x + d\right )}^{2}} \,d x } \]
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Not integrable
Time = 4.03 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96 \[ \int \frac {(f x)^m \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^2} \, dx=\int \frac {\left (f x\right )^{m} \left (a + b \log {\left (c x^{n} \right )}\right )}{\left (d + e x\right )^{2}}\, dx \]
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Not integrable
Time = 0.27 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {(f x)^m \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^2} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} \left (f x\right )^{m}}{{\left (e x + d\right )}^{2}} \,d x } \]
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Not integrable
Time = 0.34 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {(f x)^m \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^2} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} \left (f x\right )^{m}}{{\left (e x + d\right )}^{2}} \,d x } \]
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Not integrable
Time = 0.51 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {(f x)^m \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^2} \, dx=\int \frac {{\left (f\,x\right )}^m\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{{\left (d+e\,x\right )}^2} \,d x \]
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