Optimal. Leaf size=179 \[ -\frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )}{x}+\frac {2 \sqrt {f} m \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {e}}-\frac {b n \log \left (d \left (e+f x^2\right )^m\right )}{x}-\frac {i b \sqrt {f} m n \text {Li}_2\left (-\frac {i \sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {e}}+\frac {i b \sqrt {f} m n \text {Li}_2\left (\frac {i \sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {e}}+\frac {2 b \sqrt {f} m n \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {e}} \]
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Rubi [A] time = 0.13, antiderivative size = 179, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {2455, 205, 2376, 4848, 2391} \[ -\frac {i b \sqrt {f} m n \text {PolyLog}\left (2,-\frac {i \sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {e}}+\frac {i b \sqrt {f} m n \text {PolyLog}\left (2,\frac {i \sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {e}}-\frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )}{x}+\frac {2 \sqrt {f} m \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {e}}-\frac {b n \log \left (d \left (e+f x^2\right )^m\right )}{x}+\frac {2 b \sqrt {f} m n \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {e}} \]
Antiderivative was successfully verified.
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Rule 205
Rule 2376
Rule 2391
Rule 2455
Rule 4848
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^2} \, dx &=\frac {2 \sqrt {f} m \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {e}}-\frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )}{x}-(b n) \int \left (\frac {2 \sqrt {f} m \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {e} x}-\frac {\log \left (d \left (e+f x^2\right )^m\right )}{x^2}\right ) \, dx\\ &=\frac {2 \sqrt {f} m \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {e}}-\frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )}{x}+(b n) \int \frac {\log \left (d \left (e+f x^2\right )^m\right )}{x^2} \, dx-\frac {\left (2 b \sqrt {f} m n\right ) \int \frac {\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{x} \, dx}{\sqrt {e}}\\ &=\frac {2 \sqrt {f} m \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {e}}-\frac {b n \log \left (d \left (e+f x^2\right )^m\right )}{x}-\frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )}{x}-\frac {\left (i b \sqrt {f} m n\right ) \int \frac {\log \left (1-\frac {i \sqrt {f} x}{\sqrt {e}}\right )}{x} \, dx}{\sqrt {e}}+\frac {\left (i b \sqrt {f} m n\right ) \int \frac {\log \left (1+\frac {i \sqrt {f} x}{\sqrt {e}}\right )}{x} \, dx}{\sqrt {e}}+(2 b f m n) \int \frac {1}{e+f x^2} \, dx\\ &=\frac {2 b \sqrt {f} m n \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {e}}+\frac {2 \sqrt {f} m \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {e}}-\frac {b n \log \left (d \left (e+f x^2\right )^m\right )}{x}-\frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )}{x}-\frac {i b \sqrt {f} m n \text {Li}_2\left (-\frac {i \sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {e}}+\frac {i b \sqrt {f} m n \text {Li}_2\left (\frac {i \sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {e}}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 305, normalized size = 1.70 \[ \frac {-a \sqrt {e} \log \left (d \left (e+f x^2\right )^m\right )+2 a \sqrt {f} m x \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )-b \sqrt {e} \log \left (c x^n\right ) \log \left (d \left (e+f x^2\right )^m\right )+2 b \sqrt {f} m x \log \left (c x^n\right ) \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )-b \sqrt {e} n \log \left (d \left (e+f x^2\right )^m\right )-i b \sqrt {f} m n x \text {Li}_2\left (-\frac {i \sqrt {f} x}{\sqrt {e}}\right )+i b \sqrt {f} m n x \text {Li}_2\left (\frac {i \sqrt {f} x}{\sqrt {e}}\right )+i b \sqrt {f} m n x \log (x) \log \left (1-\frac {i \sqrt {f} x}{\sqrt {e}}\right )-i b \sqrt {f} m n x \log (x) \log \left (1+\frac {i \sqrt {f} x}{\sqrt {e}}\right )+2 b \sqrt {f} m n x \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )-2 b \sqrt {f} m n x \log (x) \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {e} x} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.76, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b \log \left (c x^{n}\right ) + a\right )} \log \left ({\left (f x^{2} + e\right )}^{m} d\right )}{x^{2}}, x\right ) \]
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 \[ \int \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} \log \left ({\left (f x^{2} + e\right )}^{m} d\right )}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.55, size = 1972, normalized size = 11.02 \[ \text {Expression too large to display} \]
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 \[ -\frac {{\left (b m \log \left (x^{n}\right ) + {\left (m n + m \log \relax (c)\right )} b + a m\right )} \log \left (f x^{2} + e\right )}{x} + \int \frac {b e \log \relax (c) \log \relax (d) + {\left ({\left (2 \, f m + f \log \relax (d)\right )} a + {\left (2 \, f m n + {\left (2 \, f m + f \log \relax (d)\right )} \log \relax (c)\right )} b\right )} x^{2} + a e \log \relax (d) + {\left ({\left (2 \, f m + f \log \relax (d)\right )} b x^{2} + b e \log \relax (d)\right )} \log \left (x^{n}\right )}{f x^{4} + e x^{2}}\,{d x} \]
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 \[ \int \frac {\ln \left (d\,{\left (f\,x^2+e\right )}^m\right )\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{x^2} \,d x \]
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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