Integrand size = 23, antiderivative size = 23 \[ \int \frac {a+b \log \left (c x^n\right )}{x^3 \left (d+e x^r\right )^2} \, dx=\text {Int}\left (\frac {a+b \log \left (c x^n\right )}{x^3 \left (d+e x^r\right )^2},x\right ) \]
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Not integrable
Time = 0.05 (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 = {} \[ \int \frac {a+b \log \left (c x^n\right )}{x^3 \left (d+e x^r\right )^2} \, dx=\int \frac {a+b \log \left (c x^n\right )}{x^3 \left (d+e x^r\right )^2} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {a+b \log \left (c x^n\right )}{x^3 \left (d+e x^r\right )^2} \, dx \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(139\) vs. \(2(26)=52\).
Time = 0.16 (sec) , antiderivative size = 139, normalized size of antiderivative = 6.04 \[ \int \frac {a+b \log \left (c x^n\right )}{x^3 \left (d+e x^r\right )^2} \, dx=-\frac {b n (2+r) \left (d+e x^r\right ) \, _3F_2\left (1,-\frac {2}{r},-\frac {2}{r};1-\frac {2}{r},1-\frac {2}{r};-\frac {e x^r}{d}\right )-4 d \left (a+b \log \left (c x^n\right )\right )+2 \left (d+e x^r\right ) \operatorname {Hypergeometric2F1}\left (1,-\frac {2}{r},\frac {-2+r}{r},-\frac {e x^r}{d}\right ) \left (-b n+a (2+r)+b (2+r) \log \left (c x^n\right )\right )}{4 d^2 r x^2 \left (d+e x^r\right )} \]
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Not integrable
Time = 0.06 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00
\[\int \frac {a +b \ln \left (c \,x^{n}\right )}{x^{3} \left (d +e \,x^{r}\right )^{2}}d x\]
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Not integrable
Time = 0.27 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.96 \[ \int \frac {a+b \log \left (c x^n\right )}{x^3 \left (d+e x^r\right )^2} \, dx=\int { \frac {b \log \left (c x^{n}\right ) + a}{{\left (e x^{r} + d\right )}^{2} x^{3}} \,d x } \]
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Not integrable
Time = 117.22 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96 \[ \int \frac {a+b \log \left (c x^n\right )}{x^3 \left (d+e x^r\right )^2} \, dx=\int \frac {a + b \log {\left (c x^{n} \right )}}{x^{3} \left (d + e x^{r}\right )^{2}}\, dx \]
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Not integrable
Time = 0.23 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {a+b \log \left (c x^n\right )}{x^3 \left (d+e x^r\right )^2} \, dx=\int { \frac {b \log \left (c x^{n}\right ) + a}{{\left (e x^{r} + d\right )}^{2} x^{3}} \,d x } \]
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Not integrable
Time = 0.32 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {a+b \log \left (c x^n\right )}{x^3 \left (d+e x^r\right )^2} \, dx=\int { \frac {b \log \left (c x^{n}\right ) + a}{{\left (e x^{r} + d\right )}^{2} x^{3}} \,d x } \]
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Not integrable
Time = 0.46 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {a+b \log \left (c x^n\right )}{x^3 \left (d+e x^r\right )^2} \, dx=\int \frac {a+b\,\ln \left (c\,x^n\right )}{x^3\,{\left (d+e\,x^r\right )}^2} \,d x \]
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