Integrand size = 23, antiderivative size = 23 \[ \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^r\right )^2} \, dx=\text {Int}\left (\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^r\right )^2},x\right ) \]
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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 = {} \[ \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^r\right )^2} \, dx=\int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^r\right )^2} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^r\right )^2} \, dx \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(140\) vs. \(2(26)=52\).
Time = 0.16 (sec) , antiderivative size = 140, normalized size of antiderivative = 6.09 \[ \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^r\right )^2} \, dx=\frac {x^3 \left (-b n (-3+r) \left (d+e x^r\right ) \, _3F_2\left (1,\frac {3}{r},\frac {3}{r};1+\frac {3}{r},1+\frac {3}{r};-\frac {e x^r}{d}\right )+9 d \left (a+b \log \left (c x^n\right )\right )+3 \left (d+e x^r\right ) \operatorname {Hypergeometric2F1}\left (1,\frac {3}{r},\frac {3+r}{r},-\frac {e x^r}{d}\right ) \left (-b n+a (-3+r)+b (-3+r) \log \left (c x^n\right )\right )\right )}{9 d^2 r \left (d+e x^r\right )} \]
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Not integrable
Time = 0.07 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00
\[\int \frac {x^{2} \left (a +b \ln \left (c \,x^{n}\right )\right )}{\left (d +e \,x^{r}\right )^{2}}d x\]
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Not integrable
Time = 0.26 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.83 \[ \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^r\right )^2} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} x^{2}}{{\left (e x^{r} + d\right )}^{2}} \,d x } \]
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Not integrable
Time = 23.10 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96 \[ \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^r\right )^2} \, dx=\int \frac {x^{2} \left (a + b \log {\left (c x^{n} \right )}\right )}{\left (d + e x^{r}\right )^{2}}\, dx \]
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Not integrable
Time = 0.23 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^r\right )^2} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} x^{2}}{{\left (e x^{r} + d\right )}^{2}} \,d x } \]
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Not integrable
Time = 0.30 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^r\right )^2} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} x^{2}}{{\left (e x^{r} + d\right )}^{2}} \,d x } \]
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Not integrable
Time = 0.53 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^r\right )^2} \, dx=\int \frac {x^2\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{{\left (d+e\,x^r\right )}^2} \,d x \]
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