Optimal. Leaf size=113 \[ \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{2 b n}-\frac {1}{2} m \text {Li}_2\left (-\frac {f x^2}{e}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {m \log \left (\frac {f x^2}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 b n}+\frac {1}{4} b m n \text {Li}_3\left (-\frac {f x^2}{e}\right ) \]
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Rubi [A] time = 0.13, antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2375, 2337, 2374, 6589} \[ -\frac {1}{2} m \text {PolyLog}\left (2,-\frac {f x^2}{e}\right ) \left (a+b \log \left (c x^n\right )\right )+\frac {1}{4} b m n \text {PolyLog}\left (3,-\frac {f x^2}{e}\right )+\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{2 b n}-\frac {m \log \left (\frac {f x^2}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 b n} \]
Antiderivative was successfully verified.
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Rule 2337
Rule 2374
Rule 2375
Rule 6589
Rubi steps
\begin {align*} \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )}{x} \, dx &=\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{2 b n}-\frac {(f m) \int \frac {x \left (a+b \log \left (c x^n\right )\right )^2}{e+f x^2} \, dx}{b n}\\ &=\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{2 b n}-\frac {m \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {f x^2}{e}\right )}{2 b n}+m \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {f x^2}{e}\right )}{x} \, dx\\ &=\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{2 b n}-\frac {m \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {f x^2}{e}\right )}{2 b n}-\frac {1}{2} m \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {f x^2}{e}\right )+\frac {1}{2} (b m n) \int \frac {\text {Li}_2\left (-\frac {f x^2}{e}\right )}{x} \, dx\\ &=\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{2 b n}-\frac {m \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {f x^2}{e}\right )}{2 b n}-\frac {1}{2} m \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {f x^2}{e}\right )+\frac {1}{4} b m n \text {Li}_3\left (-\frac {f x^2}{e}\right )\\ \end {align*}
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Mathematica [C] time = 0.09, size = 297, normalized size = 2.63 \[ \frac {1}{2} \left (a \log \left (-\frac {f x^2}{e}\right ) \log \left (d \left (e+f x^2\right )^m\right )+a m \text {Li}_2\left (\frac {f x^2}{e}+1\right )+2 b \log (x) \log \left (c x^n\right ) \log \left (d \left (e+f x^2\right )^m\right )-2 b m \log \left (c x^n\right ) \text {Li}_2\left (-\frac {i \sqrt {f} x}{\sqrt {e}}\right )-2 b m \log \left (c x^n\right ) \text {Li}_2\left (\frac {i \sqrt {f} x}{\sqrt {e}}\right )-2 b m \log (x) \log \left (c x^n\right ) \log \left (1-\frac {i \sqrt {f} x}{\sqrt {e}}\right )-2 b m \log (x) \log \left (c x^n\right ) \log \left (1+\frac {i \sqrt {f} x}{\sqrt {e}}\right )-b n \log ^2(x) \log \left (d \left (e+f x^2\right )^m\right )+2 b m n \text {Li}_3\left (-\frac {i \sqrt {f} x}{\sqrt {e}}\right )+2 b m n \text {Li}_3\left (\frac {i \sqrt {f} x}{\sqrt {e}}\right )+b m n \log ^2(x) \log \left (1-\frac {i \sqrt {f} x}{\sqrt {e}}\right )+b m n \log ^2(x) \log \left (1+\frac {i \sqrt {f} x}{\sqrt {e}}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.67, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b \log \left (c x^{n}\right ) + a\right )} \log \left ({\left (f x^{2} + e\right )}^{m} d\right )}{x}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} \log \left ({\left (f x^{2} + e\right )}^{m} d\right )}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 3.57, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \ln \left (c \,x^{n}\right )+a \right ) \ln \left (d \left (f \,x^{2}+e \right )^{m}\right )}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {1}{2} \, {\left (b m n \log \relax (x)^{2} - 2 \, b m \log \relax (x) \log \left (x^{n}\right ) - 2 \, {\left (b m \log \relax (c) + a m\right )} \log \relax (x)\right )} \log \left (f x^{2} + e\right ) - \int -\frac {b f m n x^{2} \log \relax (x)^{2} + b e \log \relax (c) \log \relax (d) - 2 \, {\left (b f m \log \relax (c) + a f m\right )} x^{2} \log \relax (x) + {\left (b f \log \relax (c) \log \relax (d) + a f \log \relax (d)\right )} x^{2} + a e \log \relax (d) - {\left (2 \, b f m x^{2} \log \relax (x) - b f x^{2} \log \relax (d) - b e \log \relax (d)\right )} \log \left (x^{n}\right )}{f x^{3} + e x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\ln \left (d\,{\left (f\,x^2+e\right )}^m\right )\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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