Optimal. Leaf size=97 \[ \frac {e g p \log (x)}{d}-\frac {e g p \log \left (d+e x^n\right )}{d n}-\frac {g x^{-n} \log \left (c \left (d+e x^n\right )^p\right )}{n}+\frac {f \log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{n}+\frac {f p \text {Li}_2\left (1+\frac {e x^n}{d}\right )}{n} \]
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Rubi [A]
time = 0.09, antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 9, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {2525, 14, 2463,
2442, 36, 29, 31, 2441, 2352} \begin {gather*} \frac {f p \text {PolyLog}\left (2,\frac {e x^n}{d}+1\right )}{n}+\frac {f \log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{n}-\frac {g x^{-n} \log \left (c \left (d+e x^n\right )^p\right )}{n}-\frac {e g p \log \left (d+e x^n\right )}{d n}+\frac {e g p \log (x)}{d} \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 29
Rule 31
Rule 36
Rule 2352
Rule 2441
Rule 2442
Rule 2463
Rule 2525
Rubi steps
\begin {align*} \int \frac {\left (f+g x^{-n}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{x} \, dx &=\frac {\text {Subst}\left (\int \frac {\left (f+\frac {g}{x}\right ) \log \left (c (d+e x)^p\right )}{x} \, dx,x,x^n\right )}{n}\\ &=\frac {\text {Subst}\left (\int \left (\frac {g \log \left (c (d+e x)^p\right )}{x^2}+\frac {f \log \left (c (d+e x)^p\right )}{x}\right ) \, dx,x,x^n\right )}{n}\\ &=\frac {f \text {Subst}\left (\int \frac {\log \left (c (d+e x)^p\right )}{x} \, dx,x,x^n\right )}{n}+\frac {g \text {Subst}\left (\int \frac {\log \left (c (d+e x)^p\right )}{x^2} \, dx,x,x^n\right )}{n}\\ &=-\frac {g x^{-n} \log \left (c \left (d+e x^n\right )^p\right )}{n}+\frac {f \log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{n}-\frac {(e f p) \text {Subst}\left (\int \frac {\log \left (-\frac {e x}{d}\right )}{d+e x} \, dx,x,x^n\right )}{n}+\frac {(e g p) \text {Subst}\left (\int \frac {1}{x (d+e x)} \, dx,x,x^n\right )}{n}\\ &=-\frac {g x^{-n} \log \left (c \left (d+e x^n\right )^p\right )}{n}+\frac {f \log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{n}+\frac {f p \text {Li}_2\left (1+\frac {e x^n}{d}\right )}{n}+\frac {(e g p) \text {Subst}\left (\int \frac {1}{x} \, dx,x,x^n\right )}{d n}-\frac {\left (e^2 g p\right ) \text {Subst}\left (\int \frac {1}{d+e x} \, dx,x,x^n\right )}{d n}\\ &=\frac {e g p \log (x)}{d}-\frac {e g p \log \left (d+e x^n\right )}{d n}-\frac {g x^{-n} \log \left (c \left (d+e x^n\right )^p\right )}{n}+\frac {f \log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{n}+\frac {f p \text {Li}_2\left (1+\frac {e x^n}{d}\right )}{n}\\ \end {align*}
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Mathematica [A]
time = 0.21, size = 100, normalized size = 1.03 \begin {gather*} -\frac {e g p \log \left (d n \left (d+e x^n\right )\right )+d g x^{-n} \log \left (c \left (d+e x^n\right )^p\right )-n \log (x) \left (e g p+d f \log \left (c \left (d+e x^n\right )^p\right )-d f p \log \left (1+\frac {e x^n}{d}\right )\right )+d f p \text {Li}_2\left (-\frac {e x^n}{d}\right )}{d n} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.10, size = 423, normalized size = 4.36
method | result | size |
risch | \(\frac {\left (f \ln \left (x \right ) n \,x^{n}-g \right ) x^{-n} \ln \left (\left (d +e \,x^{n}\right )^{p}\right )}{n}+\frac {i \pi \,\mathrm {csgn}\left (i \left (d +e \,x^{n}\right )^{p}\right ) \mathrm {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right )^{2} f \ln \left (x^{n}\right )}{2 n}-\frac {i \pi \,\mathrm {csgn}\left (i \left (d +e \,x^{n}\right )^{p}\right ) \mathrm {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right )^{2} g \,x^{-n}}{2 n}-\frac {i \pi \,\mathrm {csgn}\left (i \left (d +e \,x^{n}\right )^{p}\right ) \mathrm {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right ) \mathrm {csgn}\left (i c \right ) f \ln \left (x^{n}\right )}{2 n}+\frac {i \pi \,\mathrm {csgn}\left (i \left (d +e \,x^{n}\right )^{p}\right ) \mathrm {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right ) \mathrm {csgn}\left (i c \right ) g \,x^{-n}}{2 n}-\frac {i \pi \mathrm {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right )^{3} f \ln \left (x^{n}\right )}{2 n}+\frac {i \pi \mathrm {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right )^{3} g \,x^{-n}}{2 n}+\frac {i \pi \mathrm {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right )^{2} \mathrm {csgn}\left (i c \right ) f \ln \left (x^{n}\right )}{2 n}-\frac {i \pi \mathrm {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right )^{2} \mathrm {csgn}\left (i c \right ) g \,x^{-n}}{2 n}+\frac {\ln \left (c \right ) f \ln \left (x^{n}\right )}{n}-\frac {\ln \left (c \right ) g \,x^{-n}}{n}-\frac {p f \dilog \left (\frac {d +e \,x^{n}}{d}\right )}{n}-p f \ln \left (x \right ) \ln \left (\frac {d +e \,x^{n}}{d}\right )-\frac {e g p \ln \left (d +e \,x^{n}\right )}{d n}+\frac {p e g \ln \left (x^{n}\right )}{n d}\) | \(423\) |
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 = 0.36, size = 119, normalized size = 1.23 \begin {gather*} -\frac {d f n p x^{n} \log \left (x\right ) \log \left (\frac {x^{n} e + d}{d}\right ) + d f p x^{n} {\rm Li}_2\left (-\frac {x^{n} e + d}{d} + 1\right ) + d g \log \left (c\right ) - {\left (g n p e + d f n \log \left (c\right )\right )} x^{n} \log \left (x\right ) + {\left (d g p - {\left (d f n p \log \left (x\right ) - g p e\right )} x^{n}\right )} \log \left (x^{n} e + d\right )}{d n x^{n}} \end {gather*}
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 \begin {gather*} \int \frac {x^{- n} \left (f x^{n} + g\right ) \log {\left (c \left (d + e x^{n}\right )^{p} \right )}}{x}\, dx \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 (c\,{\left (d+e\,x^n\right )}^p\right )\,\left (f+\frac {g}{x^n}\right )}{x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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