Optimal. Leaf size=607 \[ -\frac {b^2 e m n^2 \log ^2(x) \log (d+e x)}{d}+\frac {2 b^2 e m n^2 \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{d}+\frac {2 b^2 e n^2 \log (x) \log \left (f x^m\right ) \log (d+e x)}{d}-\frac {b^2 e m n^2 \log ^2(d+e x)}{d}-\frac {b^2 m n^2 \log ^2(d+e x)}{x}+\frac {b^2 e m n^2 \log \left (-\frac {e x}{d}\right ) \log ^2(d+e x)}{d}-\frac {b^2 e n^2 \log \left (f x^m\right ) \log ^2(d+e x)}{d}-\frac {b^2 n^2 \log \left (f x^m\right ) \log ^2(d+e x)}{x}-\frac {2 b n \left (m \log (x)-\log \left (f x^m\right )\right ) \left (e x \log \left (-\frac {e x}{d}\right )-(d+e x) \log (d+e x)\right ) \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )}{d x}-\frac {m \log (x) \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2}{x}-\frac {\left (m-m \log (x)+\log \left (f x^m\right )\right ) \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2}{x}+\frac {b^2 e m n^2 \log ^2(x) \log \left (1+\frac {e x}{d}\right )}{d}-\frac {2 b^2 e n^2 \log (x) \log \left (f x^m\right ) \log \left (1+\frac {e x}{d}\right )}{d}-\frac {2 b^2 e n^2 \log \left (f x^m\right ) \text {Li}_2\left (-\frac {e x}{d}\right )}{d}+\frac {b m n \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right ) \left (2 e x \log \left (-\frac {e x}{d}\right )-2 (d+e x) \log (d+e x)-2 d \log (x) \log (d+e x)+e x \left (\log ^2(x)-2 \left (\log (x) \log \left (1+\frac {e x}{d}\right )+\text {Li}_2\left (-\frac {e x}{d}\right )\right )\right )\right )}{d x}+\frac {2 b^2 e m n^2 (1+\log (d+e x)) \text {Li}_2\left (1+\frac {e x}{d}\right )}{d}+\frac {2 b^2 e m n^2 \text {Li}_3\left (-\frac {e x}{d}\right )}{d}-\frac {2 b^2 e m n^2 \text {Li}_3\left (1+\frac {e x}{d}\right )}{d} \]
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Rubi [F]
time = 0.02, 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 = {}
\begin {gather*} \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 {gather*}
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 = 0.45, size = 513, normalized size = 0.85 \begin {gather*} \frac {2 b n \left (m \log (x)-\log \left (f x^m\right )\right ) \left (-e x \log \left (-\frac {e x}{d}\right )+(d+e x) \log (d+e x)\right ) \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )-d m \log (x) \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2+d \left (-m+m \log (x)-\log \left (f x^m\right )\right ) \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2-b m n \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right ) \left (-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)-e x \left (\log ^2(x)-2 \left (\log (x) \log \left (1+\frac {e x}{d}\right )+\text {Li}_2\left (-\frac {e x}{d}\right )\right )\right )\right )+b^2 n^2 \left (e m x \log ^2(x) \log (d+e x)+2 e m x \log \left (-\frac {e x}{d}\right ) \log (d+e x)-2 e m x \log (x) \log \left (-\frac {e x}{d}\right ) \log (d+e x)+2 e x \log \left (-\frac {e x}{d}\right ) \log \left (f x^m\right ) \log (d+e x)-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)-d \log \left (f x^m\right ) \log ^2(d+e x)-e x \log \left (f x^m\right ) \log ^2(d+e x)-e m x \log ^2(x) \log \left (1+\frac {e x}{d}\right )-2 e m x \log (x) \text {Li}_2\left (-\frac {e x}{d}\right )+2 e x \left (m-m \log (x)+\log \left (f x^m\right )+m \log (d+e x)\right ) \text {Li}_2\left (1+\frac {e x}{d}\right )+2 e m x \text {Li}_3\left (-\frac {e x}{d}\right )-2 e m x \text {Li}_3\left (1+\frac {e x}{d}\right )\right )}{d x} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.05, size = 0, normalized size = 0.00 \[\int \frac {\ln \left (f \,x^{m}\right ) \left (a +b \ln \left (c \left (e x +d \right )^{n}\right )\right )^{2}}{x^{2}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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