Integrand size = 23, antiderivative size = 23 \[ \int \frac {(f x)^m \left (a+b \log \left (c x^n\right )\right )}{d+e x} \, dx=\text {Int}\left (\frac {(f x)^m \left (a+b \log \left (c x^n\right )\right )}{d+e x},x\right ) \]
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Not integrable
Time = 0.04 (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 = {} \[ \int \frac {(f x)^m \left (a+b \log \left (c x^n\right )\right )}{d+e x} \, dx=\int \frac {(f x)^m \left (a+b \log \left (c x^n\right )\right )}{d+e x} \, dx \]
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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} \, dx \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(72\) vs. \(2(26)=52\).
Time = 0.13 (sec) , antiderivative size = 72, normalized size of antiderivative = 3.13 \[ \int \frac {(f x)^m \left (a+b \log \left (c x^n\right )\right )}{d+e x} \, dx=\frac {x (f x)^m \left (-b n \, _3F_2\left (1,1+m,1+m;2+m,2+m;-\frac {e x}{d}\right )+(1+m) \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,-\frac {e x}{d}\right ) \left (a+b \log \left (c x^n\right )\right )\right )}{d (1+m)^2} \]
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Not integrable
Time = 0.04 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00
\[\int \frac {\left (f x \right )^{m} \left (a +b \ln \left (c \,x^{n}\right )\right )}{e x +d}d x\]
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Not integrable
Time = 0.32 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.35 \[ \int \frac {(f x)^m \left (a+b \log \left (c x^n\right )\right )}{d+e x} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} \left (f x\right )^{m}}{e x + d} \,d x } \]
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Not integrable
Time = 2.08 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.87 \[ \int \frac {(f x)^m \left (a+b \log \left (c x^n\right )\right )}{d+e x} \, dx=\int \frac {\left (f x\right )^{m} \left (a + b \log {\left (c x^{n} \right )}\right )}{d + e x}\, dx \]
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Not integrable
Time = 0.25 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {(f x)^m \left (a+b \log \left (c x^n\right )\right )}{d+e x} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} \left (f x\right )^{m}}{e x + d} \,d x } \]
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Not integrable
Time = 0.34 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {(f x)^m \left (a+b \log \left (c x^n\right )\right )}{d+e x} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} \left (f x\right )^{m}}{e x + d} \,d x } \]
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Not integrable
Time = 0.47 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {(f x)^m \left (a+b \log \left (c x^n\right )\right )}{d+e x} \, dx=\int \frac {{\left (f\,x\right )}^m\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{d+e\,x} \,d x \]
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