Optimal. Leaf size=102 \[ \frac {\log ^2\left (f x^p\right ) \left (a+b \log \left (c \left (d+e x^m\right )^n\right )\right )}{2 p}-\frac {b n \log ^2\left (f x^p\right ) \log \left (1+\frac {e x^m}{d}\right )}{2 p}-\frac {b n \log \left (f x^p\right ) \text {Li}_2\left (-\frac {e x^m}{d}\right )}{m}+\frac {b n p \text {Li}_3\left (-\frac {e x^m}{d}\right )}{m^2} \]
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Rubi [A]
time = 0.09, antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2531, 2375,
2421, 6724} \begin {gather*} -\frac {b n \log \left (f x^p\right ) \text {PolyLog}\left (2,-\frac {e x^m}{d}\right )}{m}+\frac {b n p \text {PolyLog}\left (3,-\frac {e x^m}{d}\right )}{m^2}+\frac {\log ^2\left (f x^p\right ) \left (a+b \log \left (c \left (d+e x^m\right )^n\right )\right )}{2 p}-\frac {b n \log ^2\left (f x^p\right ) \log \left (\frac {e x^m}{d}+1\right )}{2 p} \end {gather*}
Antiderivative was successfully verified.
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Rule 2375
Rule 2421
Rule 2531
Rule 6724
Rubi steps
\begin {align*} \int \frac {\log \left (f x^p\right ) \left (a+b \log \left (c \left (d+e x^m\right )^n\right )\right )}{x} \, dx &=\frac {\log ^2\left (f x^p\right ) \left (a+b \log \left (c \left (d+e x^m\right )^n\right )\right )}{2 p}-\frac {(b e m n) \int \frac {x^{-1+m} \log ^2\left (f x^p\right )}{d+e x^m} \, dx}{2 p}\\ &=\frac {\log ^2\left (f x^p\right ) \left (a+b \log \left (c \left (d+e x^m\right )^n\right )\right )}{2 p}-\frac {b n \log ^2\left (f x^p\right ) \log \left (1+\frac {e x^m}{d}\right )}{2 p}+(b n) \int \frac {\log \left (f x^p\right ) \log \left (1+\frac {e x^m}{d}\right )}{x} \, dx\\ &=\frac {\log ^2\left (f x^p\right ) \left (a+b \log \left (c \left (d+e x^m\right )^n\right )\right )}{2 p}-\frac {b n \log ^2\left (f x^p\right ) \log \left (1+\frac {e x^m}{d}\right )}{2 p}-\frac {b n \log \left (f x^p\right ) \text {Li}_2\left (-\frac {e x^m}{d}\right )}{m}+\frac {(b n p) \int \frac {\text {Li}_2\left (-\frac {e x^m}{d}\right )}{x} \, dx}{m}\\ &=\frac {\log ^2\left (f x^p\right ) \left (a+b \log \left (c \left (d+e x^m\right )^n\right )\right )}{2 p}-\frac {b n \log ^2\left (f x^p\right ) \log \left (1+\frac {e x^m}{d}\right )}{2 p}-\frac {b n \log \left (f x^p\right ) \text {Li}_2\left (-\frac {e x^m}{d}\right )}{m}+\frac {b n p \text {Li}_3\left (-\frac {e x^m}{d}\right )}{m^2}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(265\) vs. \(2(102)=204\).
time = 0.13, size = 265, normalized size = 2.60 \begin {gather*} -\frac {1}{6} b m n p \log ^3(x)+\frac {a \log ^2\left (f x^p\right )}{2 p}-\frac {1}{2} b n p \log ^2(x) \log \left (1+\frac {d x^{-m}}{e}\right )+b n p \log ^2(x) \log \left (d+e x^m\right )-\frac {b n p \log (x) \log \left (-\frac {e x^m}{d}\right ) \log \left (d+e x^m\right )}{m}-b n \log (x) \log \left (f x^p\right ) \log \left (d+e x^m\right )+\frac {b n \log \left (-\frac {e x^m}{d}\right ) \log \left (f x^p\right ) \log \left (d+e x^m\right )}{m}-\frac {1}{2} b p \log ^2(x) \log \left (c \left (d+e x^m\right )^n\right )+b \log (x) \log \left (f x^p\right ) \log \left (c \left (d+e x^m\right )^n\right )+\frac {b n p \log (x) \text {Li}_2\left (-\frac {d x^{-m}}{e}\right )}{m}-\frac {b n \left (p \log (x)-\log \left (f x^p\right )\right ) \text {Li}_2\left (1+\frac {e x^m}{d}\right )}{m}+\frac {b n p \text {Li}_3\left (-\frac {d x^{-m}}{e}\right )}{m^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {\ln \left (f \,x^{p}\right ) \left (a +b \ln \left (c \left (d +e \,x^{m}\right )^{n}\right )\right )}{x}\, dx\]
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 [A]
time = 0.43, size = 165, normalized size = 1.62 \begin {gather*} \frac {2 \, b n p {\rm polylog}\left (3, -\frac {x^{m} e}{d}\right ) + 2 \, {\left (b m^{2} \log \left (c\right ) + a m^{2}\right )} \log \left (f\right ) \log \left (x\right ) + {\left (b m^{2} p \log \left (c\right ) + a m^{2} p\right )} \log \left (x\right )^{2} - 2 \, {\left (b m n p \log \left (x\right ) + b m n \log \left (f\right )\right )} {\rm Li}_2\left (-\frac {x^{m} e + d}{d} + 1\right ) + {\left (b m^{2} n p \log \left (x\right )^{2} + 2 \, b m^{2} n \log \left (f\right ) \log \left (x\right )\right )} \log \left (x^{m} e + d\right ) - {\left (b m^{2} n p \log \left (x\right )^{2} + 2 \, b m^{2} n \log \left (f\right ) \log \left (x\right )\right )} \log \left (\frac {x^{m} e + d}{d}\right )}{2 \, m^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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 = -0.01 \begin {gather*} \int \frac {\ln \left (f\,x^p\right )\,\left (a+b\,\ln \left (c\,{\left (d+e\,x^m\right )}^n\right )\right )}{x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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