Optimal. Leaf size=194 \[ x \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )+\frac {2 \sqrt {e} m \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {f}}-2 m x \left (a+b \log \left (c x^n\right )\right )-b n x \log \left (d \left (e+f x^2\right )^m\right )-\frac {i b \sqrt {e} m n \text {Li}_2\left (-\frac {i \sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {f}}+\frac {i b \sqrt {e} m n \text {Li}_2\left (\frac {i \sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {f}}-\frac {2 b \sqrt {e} m n \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {f}}+4 b m n x \]
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Rubi [A] time = 0.12, antiderivative size = 194, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {2448, 321, 205, 2370, 4848, 2391} \[ -\frac {i b \sqrt {e} m n \text {PolyLog}\left (2,-\frac {i \sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {f}}+\frac {i b \sqrt {e} m n \text {PolyLog}\left (2,\frac {i \sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {f}}+x \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )+\frac {2 \sqrt {e} m \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {f}}-2 m x \left (a+b \log \left (c x^n\right )\right )-b n x \log \left (d \left (e+f x^2\right )^m\right )-\frac {2 b \sqrt {e} m n \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {f}}+4 b m n x \]
Antiderivative was successfully verified.
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Rule 205
Rule 321
Rule 2370
Rule 2391
Rule 2448
Rule 4848
Rubi steps
\begin {align*} \int \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right ) \, dx &=-2 m x \left (a+b \log \left (c x^n\right )\right )+\frac {2 \sqrt {e} m \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {f}}+x \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )-(b n) \int \left (-2 m+\frac {2 \sqrt {e} m \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {f} x}+\log \left (d \left (e+f x^2\right )^m\right )\right ) \, dx\\ &=2 b m n x-2 m x \left (a+b \log \left (c x^n\right )\right )+\frac {2 \sqrt {e} m \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {f}}+x \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )-(b n) \int \log \left (d \left (e+f x^2\right )^m\right ) \, dx-\frac {\left (2 b \sqrt {e} m n\right ) \int \frac {\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{x} \, dx}{\sqrt {f}}\\ &=2 b m n x-2 m x \left (a+b \log \left (c x^n\right )\right )+\frac {2 \sqrt {e} m \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {f}}-b n x \log \left (d \left (e+f x^2\right )^m\right )+x \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )-\frac {\left (i b \sqrt {e} m n\right ) \int \frac {\log \left (1-\frac {i \sqrt {f} x}{\sqrt {e}}\right )}{x} \, dx}{\sqrt {f}}+\frac {\left (i b \sqrt {e} m n\right ) \int \frac {\log \left (1+\frac {i \sqrt {f} x}{\sqrt {e}}\right )}{x} \, dx}{\sqrt {f}}+(2 b f m n) \int \frac {x^2}{e+f x^2} \, dx\\ &=4 b m n x-2 m x \left (a+b \log \left (c x^n\right )\right )+\frac {2 \sqrt {e} m \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {f}}-b n x \log \left (d \left (e+f x^2\right )^m\right )+x \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )-\frac {i b \sqrt {e} m n \text {Li}_2\left (-\frac {i \sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {f}}+\frac {i b \sqrt {e} m n \text {Li}_2\left (\frac {i \sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {f}}-(2 b e m n) \int \frac {1}{e+f x^2} \, dx\\ &=4 b m n x-\frac {2 b \sqrt {e} m n \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {f}}-2 m x \left (a+b \log \left (c x^n\right )\right )+\frac {2 \sqrt {e} m \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {f}}-b n x \log \left (d \left (e+f x^2\right )^m\right )+x \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )-\frac {i b \sqrt {e} m n \text {Li}_2\left (-\frac {i \sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {f}}+\frac {i b \sqrt {e} m n \text {Li}_2\left (\frac {i \sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {f}}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 332, normalized size = 1.71 \[ \frac {a \sqrt {f} x \log \left (d \left (e+f x^2\right )^m\right )+2 a \sqrt {e} m \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )-2 a \sqrt {f} m x+b \sqrt {f} x \log \left (c x^n\right ) \log \left (d \left (e+f x^2\right )^m\right )+2 b \sqrt {e} m \log \left (c x^n\right ) \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )-2 b \sqrt {f} m x \log \left (c x^n\right )-b \sqrt {f} n x \log \left (d \left (e+f x^2\right )^m\right )-i b \sqrt {e} m n \text {Li}_2\left (-\frac {i \sqrt {f} x}{\sqrt {e}}\right )+i b \sqrt {e} m n \text {Li}_2\left (\frac {i \sqrt {f} x}{\sqrt {e}}\right )+i b \sqrt {e} m n \log (x) \log \left (1-\frac {i \sqrt {f} x}{\sqrt {e}}\right )-i b \sqrt {e} m n \log (x) \log \left (1+\frac {i \sqrt {f} x}{\sqrt {e}}\right )-2 b \sqrt {e} m n \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )-2 b \sqrt {e} m n \log (x) \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )+4 b \sqrt {f} m n x}{\sqrt {f}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.78, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} \log \left ({\left (f x^{2} + e\right )}^{m} d\right ), x\right ) \]
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 \[ \int {\left (b \log \left (c x^{n}\right ) + a\right )} \log \left ({\left (f x^{2} + e\right )}^{m} d\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.50, size = 2001, normalized size = 10.31 \[ \text {Expression too large to display} \]
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 \[ {\left (b m x \log \left (x^{n}\right ) - {\left ({\left (m n - m \log \relax (c)\right )} b - a m\right )} x\right )} \log \left (f x^{2} + e\right ) + \int \frac {b e \log \relax (c) \log \relax (d) - {\left ({\left (2 \, f m - f \log \relax (d)\right )} a - {\left (2 \, f m n - {\left (2 \, f m - f \log \relax (d)\right )} \log \relax (c)\right )} b\right )} x^{2} + a e \log \relax (d) - {\left ({\left (2 \, f m - f \log \relax (d)\right )} b x^{2} - b e \log \relax (d)\right )} \log \left (x^{n}\right )}{f x^{2} + e}\,{d x} \]
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 \[ \int \ln \left (d\,{\left (f\,x^2+e\right )}^m\right )\,\left (a+b\,\ln \left (c\,x^n\right )\right ) \,d x \]
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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