Integrand size = 21, antiderivative size = 21 \[ \int \frac {x \left (a+b \log \left (c x^n\right )\right )}{d+e x^r} \, dx=\text {Int}\left (\frac {x \left (a+b \log \left (c x^n\right )\right )}{d+e x^r},x\right ) \]
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Not integrable
Time = 0.03 (sec) , antiderivative size = 21, 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 = {} \[ \int \frac {x \left (a+b \log \left (c x^n\right )\right )}{d+e x^r} \, dx=\int \frac {x \left (a+b \log \left (c x^n\right )\right )}{d+e x^r} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {x \left (a+b \log \left (c x^n\right )\right )}{d+e x^r} \, dx \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(87\) vs. \(2(24)=48\).
Time = 0.08 (sec) , antiderivative size = 87, normalized size of antiderivative = 4.14 \[ \int \frac {x \left (a+b \log \left (c x^n\right )\right )}{d+e x^r} \, dx=\frac {x^2 \left (-b n \, _3F_2\left (1,\frac {2}{r},\frac {2}{r};1+\frac {2}{r},1+\frac {2}{r};-\frac {e x^r}{d}\right )+2 \operatorname {Hypergeometric2F1}\left (1,\frac {2}{r},\frac {2+r}{r},-\frac {e x^r}{d}\right ) \left (a+b \log \left (c x^n\right )\right )\right )}{4 d} \]
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Not integrable
Time = 0.05 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00
\[\int \frac {x \left (a +b \ln \left (c \,x^{n}\right )\right )}{d +e \,x^{r}}d x\]
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Not integrable
Time = 0.27 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.19 \[ \int \frac {x \left (a+b \log \left (c x^n\right )\right )}{d+e x^r} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} x}{e x^{r} + d} \,d x } \]
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Not integrable
Time = 2.49 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.90 \[ \int \frac {x \left (a+b \log \left (c x^n\right )\right )}{d+e x^r} \, dx=\int \frac {x \left (a + b \log {\left (c x^{n} \right )}\right )}{d + e x^{r}}\, dx \]
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Not integrable
Time = 0.23 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int \frac {x \left (a+b \log \left (c x^n\right )\right )}{d+e x^r} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} x}{e x^{r} + d} \,d x } \]
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Not integrable
Time = 0.30 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int \frac {x \left (a+b \log \left (c x^n\right )\right )}{d+e x^r} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} x}{e x^{r} + d} \,d x } \]
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Not integrable
Time = 0.43 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int \frac {x \left (a+b \log \left (c x^n\right )\right )}{d+e x^r} \, dx=\int \frac {x\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{d+e\,x^r} \,d x \]
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