Optimal. Leaf size=75 \[ \frac {2 p \log \left (f x^p\right ) \text {Li}_2\left (-\frac {e x^m}{d}\right )}{e m^2}+\frac {\log ^2\left (f x^p\right ) \log \left (\frac {e x^m}{d}+1\right )}{e m}-\frac {2 p^2 \text {Li}_3\left (-\frac {e x^m}{d}\right )}{e m^3} \]
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Rubi [A] time = 0.12, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {2337, 2374, 6589} \[ \frac {2 p \log \left (f x^p\right ) \text {PolyLog}\left (2,-\frac {e x^m}{d}\right )}{e m^2}-\frac {2 p^2 \text {PolyLog}\left (3,-\frac {e x^m}{d}\right )}{e m^3}+\frac {\log ^2\left (f x^p\right ) \log \left (\frac {e x^m}{d}+1\right )}{e m} \]
Antiderivative was successfully verified.
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Rule 2337
Rule 2374
Rule 6589
Rubi steps
\begin {align*} \int \frac {x^{-1+m} \log ^2\left (f x^p\right )}{d+e x^m} \, dx &=\frac {\log ^2\left (f x^p\right ) \log \left (1+\frac {e x^m}{d}\right )}{e m}-\frac {(2 p) \int \frac {\log \left (f x^p\right ) \log \left (1+\frac {e x^m}{d}\right )}{x} \, dx}{e m}\\ &=\frac {\log ^2\left (f x^p\right ) \log \left (1+\frac {e x^m}{d}\right )}{e m}+\frac {2 p \log \left (f x^p\right ) \text {Li}_2\left (-\frac {e x^m}{d}\right )}{e m^2}-\frac {\left (2 p^2\right ) \int \frac {\text {Li}_2\left (-\frac {e x^m}{d}\right )}{x} \, dx}{e m^2}\\ &=\frac {\log ^2\left (f x^p\right ) \log \left (1+\frac {e x^m}{d}\right )}{e m}+\frac {2 p \log \left (f x^p\right ) \text {Li}_2\left (-\frac {e x^m}{d}\right )}{e m^2}-\frac {2 p^2 \text {Li}_3\left (-\frac {e x^m}{d}\right )}{e m^3}\\ \end {align*}
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Mathematica [B] time = 0.14, size = 210, normalized size = 2.80 \[ \frac {3 m^2 \log ^2\left (f x^p\right ) \log \left (d+e x^m\right )+6 m p \left (p \log (x)-\log \left (f x^p\right )\right ) \text {Li}_2\left (\frac {e x^m}{d}+1\right )-6 m p \log \left (f x^p\right ) \log \left (-\frac {e x^m}{d}\right ) \log \left (d+e x^m\right )+3 m^2 p^2 \log ^2(x) \log \left (\frac {d x^{-m}}{e}+1\right )-3 m^2 p^2 \log ^2(x) \log \left (d+e x^m\right )-6 p^2 \text {Li}_3\left (-\frac {d x^{-m}}{e}\right )-6 m p^2 \log (x) \text {Li}_2\left (-\frac {d x^{-m}}{e}\right )+6 m p^2 \log (x) \log \left (-\frac {e x^m}{d}\right ) \log \left (d+e x^m\right )+m^3 p^2 \log ^3(x)}{3 e m^3} \]
Antiderivative was successfully verified.
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fricas [C] time = 0.82, size = 105, normalized size = 1.40 \[ \frac {m^{2} \log \left (e x^{m} + d\right ) \log \relax (f)^{2} - 2 \, p^{2} {\rm polylog}\left (3, -\frac {e x^{m}}{d}\right ) + 2 \, {\left (m p^{2} \log \relax (x) + m p \log \relax (f)\right )} {\rm Li}_2\left (-\frac {e x^{m} + d}{d} + 1\right ) + {\left (m^{2} p^{2} \log \relax (x)^{2} + 2 \, m^{2} p \log \relax (f) \log \relax (x)\right )} \log \left (\frac {e x^{m} + d}{d}\right )}{e m^{3}} \]
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 \[ \int \frac {x^{m - 1} \log \left (f x^{p}\right )^{2}}{e x^{m} + d}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.48, size = 1373, normalized size = 18.31 \[ \text {result too large to display} \]
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 \[ \int \frac {x^{m - 1} \log \left (f x^{p}\right )^{2}}{e x^{m} + d}\,{d x} \]
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 \[ \int \frac {x^{m-1}\,{\ln \left (f\,x^p\right )}^2}{d+e\,x^m} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{m - 1} \log {\left (f x^{p} \right )}^{2}}{d + e x^{m}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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