Integrand size = 23, antiderivative size = 83 \[ \int \frac {\left (f+g x^n\right ) \log \left (c \left (d+e x^n\right )^p\right )}{x} \, dx=-\frac {g p x^n}{n}+\frac {g \left (d+e x^n\right ) \log \left (c \left (d+e x^n\right )^p\right )}{e n}+\frac {f \log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{n}+\frac {f p \operatorname {PolyLog}\left (2,1+\frac {e x^n}{d}\right )}{n} \]
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Time = 0.08 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {2525, 45, 2463, 2436, 2332, 2441, 2352} \[ \int \frac {\left (f+g x^n\right ) \log \left (c \left (d+e x^n\right )^p\right )}{x} \, dx=\frac {f \log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{n}+\frac {g \left (d+e x^n\right ) \log \left (c \left (d+e x^n\right )^p\right )}{e n}+\frac {f p \operatorname {PolyLog}\left (2,\frac {e x^n}{d}+1\right )}{n}-\frac {g p x^n}{n} \]
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Rule 45
Rule 2332
Rule 2352
Rule 2436
Rule 2441
Rule 2463
Rule 2525
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {(f+g x) \log \left (c (d+e x)^p\right )}{x} \, dx,x,x^n\right )}{n} \\ & = \frac {\text {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 \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 \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 \text {Subst}\left (\int \log \left (c x^p\right ) \, dx,x,d+e x^n\right )}{e 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 {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*}
Time = 0.04 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.82 \[ \int \frac {\left (f+g x^n\right ) \log \left (c \left (d+e x^n\right )^p\right )}{x} \, dx=\frac {-e g p x^n+\left (d g+e g x^n+e f \log \left (-\frac {e x^n}{d}\right )\right ) \log \left (c \left (d+e x^n\right )^p\right )+e f p \operatorname {PolyLog}\left (2,1+\frac {e x^n}{d}\right )}{e n} \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 2.16 (sec) , antiderivative size = 223, normalized size of antiderivative = 2.69
method | result | size |
risch | \(\frac {\left (f \ln \left (x \right ) n +g \,x^{n}\right ) \ln \left (\left (d +e \,x^{n}\right )^{p}\right )}{n}+\left (\frac {i \pi \,\operatorname {csgn}\left (i \left (d +e \,x^{n}\right )^{p}\right ) {\operatorname {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right )}^{2}}{2}-\frac {i \pi \,\operatorname {csgn}\left (i \left (d +e \,x^{n}\right )^{p}\right ) \operatorname {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right ) \operatorname {csgn}\left (i c \right )}{2}-\frac {i \pi {\operatorname {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right )}^{3}}{2}+\frac {i \pi {\operatorname {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right )}^{2} \operatorname {csgn}\left (i c \right )}{2}+\ln \left (c \right )\right ) \left (f \ln \left (x \right )+\frac {g \,x^{n}}{n}\right )-\frac {g p \,x^{n}}{n}+\frac {p g d \ln \left (d +e \,x^{n}\right )}{e n}-\frac {p f \operatorname {dilog}\left (\frac {d +e \,x^{n}}{d}\right )}{n}-p f \ln \left (x \right ) \ln \left (\frac {d +e \,x^{n}}{d}\right )\) | \(223\) |
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Time = 0.39 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.20 \[ \int \frac {\left (f+g x^n\right ) \log \left (c \left (d+e x^n\right )^p\right )}{x} \, dx=-\frac {e f n p \log \left (x\right ) \log \left (\frac {e x^{n} + d}{d}\right ) - e f n \log \left (c\right ) \log \left (x\right ) + e f p {\rm Li}_2\left (-\frac {e x^{n} + d}{d} + 1\right ) + {\left (e g p - e g \log \left (c\right )\right )} x^{n} - {\left (e f n p \log \left (x\right ) + e g p x^{n} + d g p\right )} \log \left (e x^{n} + d\right )}{e n} \]
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\[ \int \frac {\left (f+g x^n\right ) \log \left (c \left (d+e x^n\right )^p\right )}{x} \, dx=\int \frac {\left (f + g x^{n}\right ) \log {\left (c \left (d + e x^{n}\right )^{p} \right )}}{x}\, dx \]
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\[ \int \frac {\left (f+g x^n\right ) \log \left (c \left (d+e x^n\right )^p\right )}{x} \, dx=\int { \frac {{\left (g x^{n} + f\right )} \log \left ({\left (e x^{n} + d\right )}^{p} c\right )}{x} \,d x } \]
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\[ \int \frac {\left (f+g x^n\right ) \log \left (c \left (d+e x^n\right )^p\right )}{x} \, dx=\int { \frac {{\left (g x^{n} + f\right )} \log \left ({\left (e x^{n} + d\right )}^{p} c\right )}{x} \,d x } \]
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Timed out. \[ \int \frac {\left (f+g x^n\right ) \log \left (c \left (d+e x^n\right )^p\right )}{x} \, dx=\int \frac {\ln \left (c\,{\left (d+e\,x^n\right )}^p\right )\,\left (f+g\,x^n\right )}{x} \,d x \]
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