Optimal. Leaf size=144 \[ \frac {f \log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{n}+\frac {g x^{3 n} \log \left (c \left (d+e x^n\right )^p\right )}{3 n}+\frac {d^3 g p \log \left (d+e x^n\right )}{3 e^3 n}-\frac {d^2 g p x^n}{3 e^2 n}+\frac {f p \text {Li}_2\left (\frac {e x^n}{d}+1\right )}{n}+\frac {d g p x^{2 n}}{6 e n}-\frac {g p x^{3 n}}{9 n} \]
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Rubi [A] time = 0.17, antiderivative size = 144, 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} \[ \frac {f p \text {PolyLog}\left (2,\frac {e x^n}{d}+1\right )}{n}+\frac {f \log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{n}+\frac {g x^{3 n} \log \left (c \left (d+e x^n\right )^p\right )}{3 n}-\frac {d^2 g p x^n}{3 e^2 n}+\frac {d^3 g p \log \left (d+e x^n\right )}{3 e^3 n}+\frac {d g p x^{2 n}}{6 e n}-\frac {g p x^{3 n}}{9 n} \]
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^{3 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^3\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^2 \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^2 \log \left (c (d+e x)^p\right ) \, dx,x,x^n\right )}{n}\\ &=\frac {g x^{3 n} \log \left (c \left (d+e x^n\right )^p\right )}{3 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^3}{d+e x} \, dx,x,x^n\right )}{3 n}\\ &=\frac {g x^{3 n} \log \left (c \left (d+e x^n\right )^p\right )}{3 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^2}{e^3}-\frac {d x}{e^2}+\frac {x^2}{e}-\frac {d^3}{e^3 (d+e x)}\right ) \, dx,x,x^n\right )}{3 n}\\ &=-\frac {d^2 g p x^n}{3 e^2 n}+\frac {d g p x^{2 n}}{6 e n}-\frac {g p x^{3 n}}{9 n}+\frac {d^3 g p \log \left (d+e x^n\right )}{3 e^3 n}+\frac {g x^{3 n} \log \left (c \left (d+e x^n\right )^p\right )}{3 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 = 0.17, size = 118, normalized size = 0.82 \[ \frac {18 f \left (\log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )+p \text {Li}_2\left (\frac {e x^n}{d}+1\right )\right )+6 g x^{3 n} \log \left (c \left (d+e x^n\right )^p\right )-\frac {g p \left (e x^n \left (6 d^2-3 d e x^n+2 e^2 x^{2 n}\right )-6 d^3 \log \left (d+e x^n\right )\right )}{e^3}}{18 n} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.81, size = 148, normalized size = 1.03 \[ -\frac {18 \, e^{3} f n p \log \relax (x) \log \left (\frac {e x^{n} + d}{d}\right ) - 18 \, e^{3} f n \log \relax (c) \log \relax (x) - 3 \, d e^{2} g p x^{2 \, n} + 6 \, d^{2} e g p x^{n} + 18 \, e^{3} f p {\rm Li}_2\left (-\frac {e x^{n} + d}{d} + 1\right ) + 2 \, {\left (e^{3} g p - 3 \, e^{3} g \log \relax (c)\right )} x^{3 \, n} - 6 \, {\left (3 \, e^{3} f n p \log \relax (x) + e^{3} g p x^{3 \, n} + d^{3} g p\right )} \log \left (e x^{n} + d\right )}{18 \, e^{3} n} \]
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 \[ \int \frac {{\left (g x^{3 \, n} + f\right )} \log \left ({\left (e x^{n} + d\right )}^{p} c\right )}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 3.01, size = 428, normalized size = 2.97 \[ -\frac {i \pi f \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i \left (e \,x^{n}+d \right )^{p}\right ) \mathrm {csgn}\left (i c \left (e \,x^{n}+d \right )^{p}\right ) \ln \relax (x )}{2}+\frac {i \pi f \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (e \,x^{n}+d \right )^{p}\right )^{2} \ln \relax (x )}{2}+\frac {i \pi f \,\mathrm {csgn}\left (i \left (e \,x^{n}+d \right )^{p}\right ) \mathrm {csgn}\left (i c \left (e \,x^{n}+d \right )^{p}\right )^{2} \ln \relax (x )}{2}-\frac {i \pi f \mathrm {csgn}\left (i c \left (e \,x^{n}+d \right )^{p}\right )^{3} \ln \relax (x )}{2}-f p \ln \relax (x ) \ln \left (\frac {e \,x^{n}+d}{d}\right )-\frac {i \pi g \,x^{3 n} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i \left (e \,x^{n}+d \right )^{p}\right ) \mathrm {csgn}\left (i c \left (e \,x^{n}+d \right )^{p}\right )}{6 n}+\frac {i \pi g \,x^{3 n} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (e \,x^{n}+d \right )^{p}\right )^{2}}{6 n}+\frac {i \pi g \,x^{3 n} \mathrm {csgn}\left (i \left (e \,x^{n}+d \right )^{p}\right ) \mathrm {csgn}\left (i c \left (e \,x^{n}+d \right )^{p}\right )^{2}}{6 n}-\frac {i \pi g \,x^{3 n} \mathrm {csgn}\left (i c \left (e \,x^{n}+d \right )^{p}\right )^{3}}{6 n}+f \ln \relax (c ) \ln \relax (x )+\frac {d^{3} g p \ln \left (e \,x^{n}+d \right )}{3 e^{3} n}-\frac {d^{2} g p \,x^{n}}{3 e^{2} n}+\frac {d g p \,x^{2 n}}{6 e n}-\frac {f p \dilog \left (\frac {e \,x^{n}+d}{d}\right )}{n}-\frac {g p \,x^{3 n}}{9 n}+\frac {g \,x^{3 n} \ln \relax (c )}{3 n}+\frac {\left (3 f n \ln \relax (x )+g \,x^{3 n}\right ) \ln \left (\left (e \,x^{n}+d \right )^{p}\right )}{3 n} \]
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 \[ -\frac {9 \, e^{3} f n^{2} p \log \relax (x)^{2} - 3 \, d e^{2} g p x^{2 \, n} + 6 \, d^{2} e g p x^{n} + 2 \, {\left (e^{3} g p - 3 \, e^{3} g \log \relax (c)\right )} x^{3 \, n} - 6 \, {\left (3 \, e^{3} f n \log \relax (x) + e^{3} g x^{3 \, n}\right )} \log \left ({\left (e x^{n} + d\right )}^{p}\right ) - 6 \, {\left (d^{3} g n p + 3 \, e^{3} f n \log \relax (c)\right )} \log \relax (x)}{18 \, e^{3} n} + \int \frac {3 \, d e^{3} f n p \log \relax (x) - d^{4} g p}{3 \, {\left (e^{4} x x^{n} + d e^{3} x\right )}}\,{d x} \]
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 \[ \int \frac {\ln \left (c\,{\left (d+e\,x^n\right )}^p\right )\,\left (f+g\,x^{3\,n}\right )}{x} \,d x \]
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 \[ \int \frac {\left (f + g x^{3 n}\right ) \log {\left (c \left (d + e x^{n}\right )^{p} \right )}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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