Optimal. Leaf size=97 \[ \frac {f \log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{n}-\frac {g x^{-n} \log \left (c \left (d+e x^n\right )^p\right )}{n}+\frac {f p \text {Li}_2\left (\frac {e x^n}{d}+1\right )}{n}-\frac {e g p \log \left (d+e x^n\right )}{d n}+\frac {e g p \log (x)}{d} \]
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Rubi [A] time = 0.14, 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 = {2475, 14, 2416, 2395, 36, 29, 31, 2394, 2315} \[ \frac {f p \text {PolyLog}\left (2,\frac {e x^n}{d}+1\right )}{n}+\frac {f \log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{n}-\frac {g x^{-n} \log \left (c \left (d+e x^n\right )^p\right )}{n}-\frac {e g p \log \left (d+e x^n\right )}{d n}+\frac {e g p \log (x)}{d} \]
Antiderivative was successfully verified.
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Rule 14
Rule 29
Rule 31
Rule 36
Rule 2315
Rule 2394
Rule 2395
Rule 2416
Rule 2475
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 {\operatorname {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 {\operatorname {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 \operatorname {Subst}\left (\int \frac {\log \left (c (d+e x)^p\right )}{x} \, dx,x,x^n\right )}{n}+\frac {g \operatorname {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) \operatorname {Subst}\left (\int \frac {\log \left (-\frac {e x}{d}\right )}{d+e x} \, dx,x,x^n\right )}{n}+\frac {(e g p) \operatorname {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) \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,x^n\right )}{d n}-\frac {\left (e^2 g p\right ) \operatorname {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 = 0.10, size = 87, normalized size = 0.90 \[ \frac {f \left (\log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )+p \text {Li}_2\left (\frac {e x^n}{d}+1\right )\right )-g x^{-n} \log \left (c \left (d+e x^n\right )^p\right )+\frac {e g p \left (n \log (x)-\log \left (d+e x^n\right )\right )}{d}}{n} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 114, normalized size = 1.18 \[ -\frac {d f n p x^{n} \log \relax (x) \log \left (\frac {e x^{n} + d}{d}\right ) + d f p x^{n} {\rm Li}_2\left (-\frac {e x^{n} + d}{d} + 1\right ) + d g \log \relax (c) - {\left (e g n p + d f n \log \relax (c)\right )} x^{n} \log \relax (x) + {\left (d g p - {\left (d f n p \log \relax (x) - e g p\right )} x^{n}\right )} \log \left (e x^{n} + d\right )}{d n x^{n}} \]
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 \[ \int \frac {{\left (f + \frac {g}{x^{n}}\right )} \log \left ({\left (e x^{n} + d\right )}^{p} c\right )}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 3.41, size = 423, normalized size = 4.36 \[ -\frac {i \pi f \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i \left (e \,x^{n}+d \right )^{p}\right ) \mathrm {csgn}\left (i c \left (e \,x^{n}+d \right )^{p}\right ) \ln \left (x^{n}\right )}{2 n}+\frac {i \pi f \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (e \,x^{n}+d \right )^{p}\right )^{2} \ln \left (x^{n}\right )}{2 n}+\frac {i \pi f \,\mathrm {csgn}\left (i \left (e \,x^{n}+d \right )^{p}\right ) \mathrm {csgn}\left (i c \left (e \,x^{n}+d \right )^{p}\right )^{2} \ln \left (x^{n}\right )}{2 n}-\frac {i \pi f \mathrm {csgn}\left (i c \left (e \,x^{n}+d \right )^{p}\right )^{3} \ln \left (x^{n}\right )}{2 n}-f p \ln \relax (x ) \ln \left (\frac {e \,x^{n}+d}{d}\right )+\frac {i \pi g \,x^{-n} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i \left (e \,x^{n}+d \right )^{p}\right ) \mathrm {csgn}\left (i c \left (e \,x^{n}+d \right )^{p}\right )}{2 n}-\frac {i \pi g \,x^{-n} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (e \,x^{n}+d \right )^{p}\right )^{2}}{2 n}-\frac {i \pi g \,x^{-n} \mathrm {csgn}\left (i \left (e \,x^{n}+d \right )^{p}\right ) \mathrm {csgn}\left (i c \left (e \,x^{n}+d \right )^{p}\right )^{2}}{2 n}+\frac {i \pi g \,x^{-n} \mathrm {csgn}\left (i c \left (e \,x^{n}+d \right )^{p}\right )^{3}}{2 n}+\frac {e g p \ln \left (x^{n}\right )}{d n}-\frac {e g p \ln \left (e \,x^{n}+d \right )}{d n}-\frac {f p \dilog \left (\frac {e \,x^{n}+d}{d}\right )}{n}+\frac {f \ln \relax (c ) \ln \left (x^{n}\right )}{n}-\frac {g \,x^{-n} \ln \relax (c )}{n}+\frac {\left (f n \,x^{n} \ln \relax (x )-g \right ) x^{-n} \ln \left (\left (e \,x^{n}+d \right )^{p}\right )}{n} \]
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 \[ -\frac {{\left (f n^{2} p \log \relax (x)^{2} - 2 \, f n \log \relax (c) \log \relax (x)\right )} x^{n} - 2 \, {\left (f n x^{n} \log \relax (x) - g\right )} \log \left ({\left (e x^{n} + d\right )}^{p}\right ) + 2 \, g \log \relax (c)}{2 \, n x^{n}} + \int \frac {d f n p \log \relax (x) + e g p}{e x x^{n} + d x}\,{d x} \]
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 \[ \int \frac {\ln \left (c\,{\left (d+e\,x^n\right )}^p\right )\,\left (f+\frac {g}{x^n}\right )}{x} \,d x \]
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 \[ \int \frac {x^{- n} \left (f x^{n} + g\right ) \log {\left (c \left (d + e x^{n}\right )^{p} \right )}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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