Integrand size = 25, antiderivative size = 25 \[ \int \frac {(f x)^m \left (a+b \log \left (c x^n\right )\right )}{d+e x^2} \, dx=\text {Int}\left (\frac {(f x)^m \left (a+b \log \left (c x^n\right )\right )}{d+e x^2},x\right ) \]
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Not integrable
Time = 0.04 (sec) , antiderivative size = 25, 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. \(108\) vs. \(2(28)=56\).
Time = 0.70 (sec) , antiderivative size = 108, normalized size of antiderivative = 4.32 \[ \int \frac {(f x)^m \left (a+b \log \left (c x^n\right )\right )}{d+e x^2} \, dx=\frac {x (f x)^m \left (-b n \, _3F_2\left (1,\frac {1}{2}+\frac {m}{2},\frac {1}{2}+\frac {m}{2};\frac {3}{2}+\frac {m}{2},\frac {3}{2}+\frac {m}{2};-\frac {e x^2}{d}\right )+(1+m) \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{2},\frac {3+m}{2},-\frac {e x^2}{d}\right ) \left (a+b \log \left (c x^n\right )\right )\right )}{d (1+m)^2} \]
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Not integrable
Time = 0.03 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00
\[\int \frac {\left (f x \right )^{m} \left (a +b \ln \left (c \,x^{n}\right )\right )}{e \,x^{2}+d}d x\]
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Not integrable
Time = 0.28 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.32 \[ \int \frac {(f x)^m \left (a+b \log \left (c x^n\right )\right )}{d+e x^2} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} \left (f x\right )^{m}}{e x^{2} + d} \,d x } \]
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Not integrable
Time = 7.11 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.88 \[ \int \frac {(f x)^m \left (a+b \log \left (c x^n\right )\right )}{d+e x^2} \, dx=\int \frac {\left (f x\right )^{m} \left (a + b \log {\left (c x^{n} \right )}\right )}{d + e x^{2}}\, dx \]
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Not integrable
Time = 0.22 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \frac {(f x)^m \left (a+b \log \left (c x^n\right )\right )}{d+e x^2} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} \left (f x\right )^{m}}{e x^{2} + d} \,d x } \]
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Not integrable
Time = 0.38 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \frac {(f x)^m \left (a+b \log \left (c x^n\right )\right )}{d+e x^2} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} \left (f x\right )^{m}}{e x^{2} + d} \,d x } \]
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Not integrable
Time = 0.41 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \frac {(f x)^m \left (a+b \log \left (c x^n\right )\right )}{d+e x^2} \, dx=\int \frac {{\left (f\,x\right )}^m\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{e\,x^2+d} \,d x \]
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