Optimal. Leaf size=83 \[ \frac {f \log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{n}+\frac {g \left (d+e x^n\right ) \log \left (c \left (d+e x^n\right )^p\right )}{e n}+\frac {f p \text {Li}_2\left (\frac {e x^n}{d}+1\right )}{n}-\frac {g p x^n}{n} \]
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Rubi [A] time = 0.11, antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {2475, 43, 2416, 2389, 2295, 2394, 2315} \[ \frac {f p \text {PolyLog}\left (2,\frac {e x^n}{d}+1\right )}{n}+\frac {f \log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{n}+\frac {g \left (d+e x^n\right ) \log \left (c \left (d+e x^n\right )^p\right )}{e n}-\frac {g p x^n}{n} \]
Antiderivative was successfully verified.
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Rule 43
Rule 2295
Rule 2315
Rule 2389
Rule 2394
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 {(f+g x) \log \left (c (d+e x)^p\right )}{x} \, dx,x,x^n\right )}{n}\\ &=\frac {\operatorname {Subst}\left (\int \left (g \log \left (c (d+e x)^p\right )+\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 \log \left (c (d+e x)^p\right ) \, dx,x,x^n\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 \operatorname {Subst}\left (\int \log \left (c x^p\right ) \, dx,x,d+e x^n\right )}{e 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 {g p x^n}{n}+\frac {g \left (d+e x^n\right ) \log \left (c \left (d+e x^n\right )^p\right )}{e 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 = 0.06, size = 68, normalized size = 0.82 \[ \frac {\log \left (c \left (d+e x^n\right )^p\right ) \left (e f \log \left (-\frac {e x^n}{d}\right )+d g+e g x^n\right )+e f p \text {Li}_2\left (\frac {e x^n}{d}+1\right )-e g p x^n}{e n} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 100, normalized size = 1.20 \[ -\frac {e f n p \log \relax (x) \log \left (\frac {e x^{n} + d}{d}\right ) - e f n \log \relax (c) \log \relax (x) + e f p {\rm Li}_2\left (-\frac {e x^{n} + d}{d} + 1\right ) + {\left (e g p - e g \log \relax (c)\right )} x^{n} - {\left (e f n p \log \relax (x) + e g p x^{n} + d g p\right )} \log \left (e x^{n} + d\right )}{e n} \]
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 \[ \int \frac {{\left (g x^{n} + f\right )} \log \left ({\left (e x^{n} + d\right )}^{p} c\right )}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 3.50, size = 376, normalized size = 4.53 \[ -\frac {i \pi f \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i \left (e \,x^{n}+d \right )^{p}\right ) \mathrm {csgn}\left (i c \left (e \,x^{n}+d \right )^{p}\right ) \ln \relax (x )}{2}+\frac {i \pi f \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (e \,x^{n}+d \right )^{p}\right )^{2} \ln \relax (x )}{2}+\frac {i \pi f \,\mathrm {csgn}\left (i \left (e \,x^{n}+d \right )^{p}\right ) \mathrm {csgn}\left (i c \left (e \,x^{n}+d \right )^{p}\right )^{2} \ln \relax (x )}{2}-\frac {i \pi f \mathrm {csgn}\left (i c \left (e \,x^{n}+d \right )^{p}\right )^{3} \ln \relax (x )}{2}-f p \ln \relax (x ) \ln \left (\frac {e \,x^{n}+d}{d}\right )-\frac {i \pi g \,x^{n} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i \left (e \,x^{n}+d \right )^{p}\right ) \mathrm {csgn}\left (i c \left (e \,x^{n}+d \right )^{p}\right )}{2 n}+\frac {i \pi g \,x^{n} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (e \,x^{n}+d \right )^{p}\right )^{2}}{2 n}+\frac {i \pi g \,x^{n} \mathrm {csgn}\left (i \left (e \,x^{n}+d \right )^{p}\right ) \mathrm {csgn}\left (i c \left (e \,x^{n}+d \right )^{p}\right )^{2}}{2 n}-\frac {i \pi g \,x^{n} \mathrm {csgn}\left (i c \left (e \,x^{n}+d \right )^{p}\right )^{3}}{2 n}+f \ln \relax (c ) \ln \relax (x )+\frac {d g p \ln \left (e \,x^{n}+d \right )}{e n}-\frac {f p \dilog \left (\frac {e \,x^{n}+d}{d}\right )}{n}-\frac {g p \,x^{n}}{n}+\frac {g \,x^{n} \ln \relax (c )}{n}+\frac {\left (f n \ln \relax (x )+g \,x^{n}\right ) \ln \left (\left (e \,x^{n}+d \right )^{p}\right )}{n} \]
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 \[ -\frac {e f n^{2} p \log \relax (x)^{2} + 2 \, {\left (e g p - e g \log \relax (c)\right )} x^{n} - 2 \, {\left (e f n \log \relax (x) + e g x^{n}\right )} \log \left ({\left (e x^{n} + d\right )}^{p}\right ) - 2 \, {\left (d g n p + e f n \log \relax (c)\right )} \log \relax (x)}{2 \, e n} + \int \frac {d e f n p \log \relax (x) - d^{2} g p}{e^{2} x x^{n} + d e x}\,{d x} \]
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 \[ \int \frac {\ln \left (c\,{\left (d+e\,x^n\right )}^p\right )\,\left (f+g\,x^n\right )}{x} \,d x \]
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 \[ \int \frac {\left (f + g x^{n}\right ) \log {\left (c \left (d + e x^{n}\right )^{p} \right )}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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