Optimal. Leaf size=124 \[ \frac {f \log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{n}+\frac {g x^{2 n} \log \left (c \left (d+e x^n\right )^p\right )}{2 n}-\frac {d^2 g p \log \left (d+e x^n\right )}{2 e^2 n}+\frac {f p \text {Li}_2\left (\frac {e x^n}{d}+1\right )}{n}+\frac {d g p x^n}{2 e n}-\frac {g p x^{2 n}}{4 n} \]
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Rubi [A] time = 0.14, antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {2475, 14, 2416, 2394, 2315, 2395, 43} \[ \frac {f p \text {PolyLog}\left (2,\frac {e x^n}{d}+1\right )}{n}+\frac {f \log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{n}+\frac {g x^{2 n} \log \left (c \left (d+e x^n\right )^p\right )}{2 n}-\frac {d^2 g p \log \left (d+e x^n\right )}{2 e^2 n}+\frac {d g p x^n}{2 e n}-\frac {g p x^{2 n}}{4 n} \]
Antiderivative was successfully verified.
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Rule 14
Rule 43
Rule 2315
Rule 2394
Rule 2395
Rule 2416
Rule 2475
Rubi steps
\begin {align*} \int \frac {\left (f+g x^{2 n}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{x} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (f+g x^2\right ) \log \left (c (d+e x)^p\right )}{x} \, dx,x,x^n\right )}{n}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {f \log \left (c (d+e x)^p\right )}{x}+g x \log \left (c (d+e x)^p\right )\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 x \log \left (c (d+e x)^p\right ) \, dx,x,x^n\right )}{n}\\ &=\frac {g x^{2 n} \log \left (c \left (d+e x^n\right )^p\right )}{2 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 {x^2}{d+e x} \, dx,x,x^n\right )}{2 n}\\ &=\frac {g x^{2 n} \log \left (c \left (d+e x^n\right )^p\right )}{2 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 \left (-\frac {d}{e^2}+\frac {x}{e}+\frac {d^2}{e^2 (d+e x)}\right ) \, dx,x,x^n\right )}{2 n}\\ &=\frac {d g p x^n}{2 e n}-\frac {g p x^{2 n}}{4 n}-\frac {d^2 g p \log \left (d+e x^n\right )}{2 e^2 n}+\frac {g x^{2 n} \log \left (c \left (d+e x^n\right )^p\right )}{2 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 = 0.12, size = 100, normalized size = 0.81 \[ \frac {2 e^2 \log \left (c \left (d+e x^n\right )^p\right ) \left (2 f \log \left (-\frac {e x^n}{d}\right )+g x^{2 n}\right )-2 d^2 g p \log \left (d+e x^n\right )+4 e^2 f p \text {Li}_2\left (\frac {e x^n}{d}+1\right )-e g p x^n \left (e x^n-2 d\right )}{4 e^2 n} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.63, size = 133, normalized size = 1.07 \[ -\frac {4 \, e^{2} f n p \log \relax (x) \log \left (\frac {e x^{n} + d}{d}\right ) - 4 \, e^{2} f n \log \relax (c) \log \relax (x) - 2 \, d e g p x^{n} + 4 \, e^{2} f p {\rm Li}_2\left (-\frac {e x^{n} + d}{d} + 1\right ) + {\left (e^{2} g p - 2 \, e^{2} g \log \relax (c)\right )} x^{2 \, n} - 2 \, {\left (2 \, e^{2} f n p \log \relax (x) + e^{2} g p x^{2 \, n} - d^{2} g p\right )} \log \left (e x^{n} + d\right )}{4 \, e^{2} n} \]
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 \[ \int \frac {{\left (g x^{2 \, n} + f\right )} \log \left ({\left (e x^{n} + d\right )}^{p} c\right )}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 2.91, size = 410, normalized size = 3.31 \[ -\frac {i \pi f \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i \left (e \,x^{n}+d \right )^{p}\right ) \mathrm {csgn}\left (i c \left (e \,x^{n}+d \right )^{p}\right ) \ln \relax (x )}{2}+\frac {i \pi f \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (e \,x^{n}+d \right )^{p}\right )^{2} \ln \relax (x )}{2}+\frac {i \pi f \,\mathrm {csgn}\left (i \left (e \,x^{n}+d \right )^{p}\right ) \mathrm {csgn}\left (i c \left (e \,x^{n}+d \right )^{p}\right )^{2} \ln \relax (x )}{2}-\frac {i \pi f \mathrm {csgn}\left (i c \left (e \,x^{n}+d \right )^{p}\right )^{3} \ln \relax (x )}{2}-f p \ln \relax (x ) \ln \left (\frac {e \,x^{n}+d}{d}\right )-\frac {i \pi g \,x^{2 n} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i \left (e \,x^{n}+d \right )^{p}\right ) \mathrm {csgn}\left (i c \left (e \,x^{n}+d \right )^{p}\right )}{4 n}+\frac {i \pi g \,x^{2 n} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (e \,x^{n}+d \right )^{p}\right )^{2}}{4 n}+\frac {i \pi g \,x^{2 n} \mathrm {csgn}\left (i \left (e \,x^{n}+d \right )^{p}\right ) \mathrm {csgn}\left (i c \left (e \,x^{n}+d \right )^{p}\right )^{2}}{4 n}-\frac {i \pi g \,x^{2 n} \mathrm {csgn}\left (i c \left (e \,x^{n}+d \right )^{p}\right )^{3}}{4 n}+f \ln \relax (c ) \ln \relax (x )-\frac {d^{2} g p \ln \left (e \,x^{n}+d \right )}{2 e^{2} n}+\frac {d g p \,x^{n}}{2 e n}-\frac {f p \dilog \left (\frac {e \,x^{n}+d}{d}\right )}{n}-\frac {g p \,x^{2 n}}{4 n}+\frac {g \,x^{2 n} \ln \relax (c )}{2 n}+\frac {\left (2 f n \ln \relax (x )+g \,x^{2 n}\right ) \ln \left (\left (e \,x^{n}+d \right )^{p}\right )}{2 n} \]
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 \[ -\frac {2 \, e^{2} f n^{2} p \log \relax (x)^{2} - 2 \, d e g p x^{n} + {\left (e^{2} g p - 2 \, e^{2} g \log \relax (c)\right )} x^{2 \, n} - 2 \, {\left (2 \, e^{2} f n \log \relax (x) + e^{2} g x^{2 \, n}\right )} \log \left ({\left (e x^{n} + d\right )}^{p}\right ) + 2 \, {\left (d^{2} g n p - 2 \, e^{2} f n \log \relax (c)\right )} \log \relax (x)}{4 \, e^{2} n} + \int \frac {2 \, d e^{2} f n p \log \relax (x) + d^{3} g p}{2 \, {\left (e^{3} x x^{n} + d e^{2} x\right )}}\,{d x} \]
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 \[ \int \frac {\ln \left (c\,{\left (d+e\,x^n\right )}^p\right )\,\left (f+g\,x^{2\,n}\right )}{x} \,d x \]
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 \[ \int \frac {\left (f + g x^{2 n}\right ) \log {\left (c \left (d + e x^{n}\right )^{p} \right )}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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