Optimal. Leaf size=607 \[ -\frac {2 b n \left (e x \log \left (-\frac {e x}{d}\right )-(d+e x) \log (d+e x)\right ) \left (m \log (x)-\log \left (f x^m\right )\right ) \left (a+b \log \left (c (d+e x)^n\right )-b n \log (d+e x)\right )}{d x}-\frac {\left (\log \left (f x^m\right )+m (-\log (x))+m\right ) \left (a+b \log \left (c (d+e x)^n\right )-b n \log (d+e x)\right )^2}{x}+\frac {b m n \left (e x \left (\log ^2(x)-2 \left (\text {Li}_2\left (-\frac {e x}{d}\right )+\log (x) \log \left (\frac {e x}{d}+1\right )\right )\right )+2 e x \log \left (-\frac {e x}{d}\right )-2 (d+e x) \log (d+e x)-2 d \log (x) \log (d+e x)\right ) \left (a+b \log \left (c (d+e x)^n\right )-b n \log (d+e x)\right )}{d x}-\frac {m \log (x) \left (a+b \log \left (c (d+e x)^n\right )-b n \log (d+e x)\right )^2}{x}-\frac {2 b^2 e n^2 \text {Li}_2\left (-\frac {e x}{d}\right ) \log \left (f x^m\right )}{d}-\frac {b^2 e n^2 \log ^2(d+e x) \log \left (f x^m\right )}{d}-\frac {b^2 n^2 \log ^2(d+e x) \log \left (f x^m\right )}{x}+\frac {2 b^2 e n^2 \log (x) \log (d+e x) \log \left (f x^m\right )}{d}-\frac {2 b^2 e n^2 \log (x) \log \left (\frac {e x}{d}+1\right ) \log \left (f x^m\right )}{d}+\frac {2 b^2 e m n^2 \text {Li}_3\left (-\frac {e x}{d}\right )}{d}-\frac {2 b^2 e m n^2 \text {Li}_3\left (\frac {e x}{d}+1\right )}{d}+\frac {2 b^2 e m n^2 \text {Li}_2\left (\frac {e x}{d}+1\right ) (\log (d+e x)+1)}{d}-\frac {b^2 e m n^2 \log ^2(d+e x)}{d}+\frac {b^2 e m n^2 \log \left (-\frac {e x}{d}\right ) \log ^2(d+e x)}{d}-\frac {b^2 m n^2 \log ^2(d+e x)}{x}-\frac {b^2 e m n^2 \log ^2(x) \log (d+e x)}{d}+\frac {b^2 e m n^2 \log ^2(x) \log \left (\frac {e x}{d}+1\right )}{d}+\frac {2 b^2 e m n^2 \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{d} \]
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Rubi [F] time = 0.03, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x^2} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x^2} \, dx &=\int \frac {\log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x^2} \, dx\\ \end {align*}
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Mathematica [A] time = 0.71, size = 513, normalized size = 0.85 \[ \frac {2 b n \left ((d+e x) \log (d+e x)-e x \log \left (-\frac {e x}{d}\right )\right ) \left (m \log (x)-\log \left (f x^m\right )\right ) \left (a+b \log \left (c (d+e x)^n\right )-b n \log (d+e x)\right )+d \left (-\log \left (f x^m\right )+m \log (x)-m\right ) \left (a+b \log \left (c (d+e x)^n\right )-b n \log (d+e x)\right )^2-b m n \left (-e x \left (\log ^2(x)-2 \left (\text {Li}_2\left (-\frac {e x}{d}\right )+\log (x) \log \left (\frac {e x}{d}+1\right )\right )\right )-2 e x \log \left (-\frac {e x}{d}\right )+2 (d+e x) \log (d+e x)+2 d \log (x) \log (d+e x)\right ) \left (a+b \log \left (c (d+e x)^n\right )-b n \log (d+e x)\right )-d m \log (x) \left (a+b \log \left (c (d+e x)^n\right )-b n \log (d+e x)\right )^2+b^2 n^2 \left (2 e x \text {Li}_2\left (\frac {e x}{d}+1\right ) \left (m \log (d+e x)+\log \left (f x^m\right )+m (-\log (x))+m\right )-d \log ^2(d+e x) \log \left (f x^m\right )-e x \log ^2(d+e x) \log \left (f x^m\right )+2 e x \log \left (-\frac {e x}{d}\right ) \log (d+e x) \log \left (f x^m\right )+2 e m x \text {Li}_3\left (-\frac {e x}{d}\right )-2 e m x \text {Li}_3\left (\frac {e x}{d}+1\right )-2 e m x \log (x) \text {Li}_2\left (-\frac {e x}{d}\right )+e m x \log ^2(x) \log (d+e x)-e m x \log ^2(x) \log \left (\frac {e x}{d}+1\right )-d m \log ^2(d+e x)-e m x \log ^2(d+e x)+e m x \log \left (-\frac {e x}{d}\right ) \log ^2(d+e x)-2 e m x \log (x) \log \left (-\frac {e x}{d}\right ) \log (d+e x)+2 e m x \log \left (-\frac {e x}{d}\right ) \log (d+e x)\right )}{d x} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{2} \log \left ({\left (e x + d\right )}^{n} c\right )^{2} \log \left (f x^{m}\right ) + 2 \, a b \log \left ({\left (e x + d\right )}^{n} c\right ) \log \left (f x^{m}\right ) + a^{2} \log \left (f x^{m}\right )}{x^{2}}, x\right ) \]
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 \[ \int \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2} \log \left (f x^{m}\right )}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.73, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \ln \left (c \left (e x +d \right )^{n}\right )+a \right )^{2} \ln \left (f \,x^{m}\right )}{x^{2}}\, dx \]
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 \[ -\frac {{\left (b^{2} {\left (m + \log \relax (f)\right )} + b^{2} \log \left (x^{m}\right )\right )} \log \left ({\left (e x + d\right )}^{n}\right )^{2}}{x} + \int \frac {b^{2} d \log \relax (c)^{2} \log \relax (f) + 2 \, a b d \log \relax (c) \log \relax (f) + a^{2} d \log \relax (f) + {\left (b^{2} e \log \relax (c)^{2} \log \relax (f) + 2 \, a b e \log \relax (c) \log \relax (f) + a^{2} e \log \relax (f)\right )} x + 2 \, {\left (b^{2} d \log \relax (c) \log \relax (f) + a b d \log \relax (f) + {\left (a b e \log \relax (f) + {\left (e \log \relax (c) \log \relax (f) + {\left (m n + n \log \relax (f)\right )} e\right )} b^{2}\right )} x + {\left (b^{2} d \log \relax (c) + a b d + {\left ({\left (e n + e \log \relax (c)\right )} b^{2} + a b e\right )} x\right )} \log \left (x^{m}\right )\right )} \log \left ({\left (e x + d\right )}^{n}\right ) + {\left (b^{2} d \log \relax (c)^{2} + 2 \, a b d \log \relax (c) + a^{2} d + {\left (b^{2} e \log \relax (c)^{2} + 2 \, a b e \log \relax (c) + a^{2} e\right )} x\right )} \log \left (x^{m}\right )}{e x^{3} + d x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\ln \left (f\,x^m\right )\,{\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}^2}{x^2} \,d x \]
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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