Optimal. Leaf size=257 \[ \frac {f^2 \log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{n}-\frac {f g x^{-2 n} \log \left (c \left (d+e x^n\right )^p\right )}{n}-\frac {g^2 x^{-4 n} \log \left (c \left (d+e x^n\right )^p\right )}{4 n}+\frac {e^4 g^2 p \log \left (d+e x^n\right )}{4 d^4 n}-\frac {e^4 g^2 p \log (x)}{4 d^4}-\frac {e^3 g^2 p x^{-n}}{4 d^3 n}+\frac {e^2 f g p \log \left (d+e x^n\right )}{d^2 n}-\frac {e^2 f g p \log (x)}{d^2}+\frac {e^2 g^2 p x^{-2 n}}{8 d^2 n}+\frac {f^2 p \text {Li}_2\left (\frac {e x^n}{d}+1\right )}{n}-\frac {e f g p x^{-n}}{d n}-\frac {e g^2 p x^{-3 n}}{12 d n} \]
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Rubi [A] time = 0.32, antiderivative size = 257, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 9, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2475, 263, 266, 43, 2416, 2395, 44, 2394, 2315} \[ \frac {f^2 p \text {PolyLog}\left (2,\frac {e x^n}{d}+1\right )}{n}+\frac {f^2 \log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{n}-\frac {f g x^{-2 n} \log \left (c \left (d+e x^n\right )^p\right )}{n}-\frac {g^2 x^{-4 n} \log \left (c \left (d+e x^n\right )^p\right )}{4 n}+\frac {e^2 f g p \log \left (d+e x^n\right )}{d^2 n}-\frac {e^2 f g p \log (x)}{d^2}+\frac {e^2 g^2 p x^{-2 n}}{8 d^2 n}-\frac {e^3 g^2 p x^{-n}}{4 d^3 n}+\frac {e^4 g^2 p \log \left (d+e x^n\right )}{4 d^4 n}-\frac {e^4 g^2 p \log (x)}{4 d^4}-\frac {e f g p x^{-n}}{d n}-\frac {e g^2 p x^{-3 n}}{12 d n} \]
Antiderivative was successfully verified.
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Rule 43
Rule 44
Rule 263
Rule 266
Rule 2315
Rule 2394
Rule 2395
Rule 2416
Rule 2475
Rubi steps
\begin {align*} \int \frac {\left (f+g x^{-2 n}\right )^2 \log \left (c \left (d+e x^n\right )^p\right )}{x} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (f+\frac {g}{x^2}\right )^2 \log \left (c (d+e x)^p\right )}{x} \, dx,x,x^n\right )}{n}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {g^2 \log \left (c (d+e x)^p\right )}{x^5}+\frac {2 f g \log \left (c (d+e x)^p\right )}{x^3}+\frac {f^2 \log \left (c (d+e x)^p\right )}{x}\right ) \, dx,x,x^n\right )}{n}\\ &=\frac {f^2 \operatorname {Subst}\left (\int \frac {\log \left (c (d+e x)^p\right )}{x} \, dx,x,x^n\right )}{n}+\frac {(2 f g) \operatorname {Subst}\left (\int \frac {\log \left (c (d+e x)^p\right )}{x^3} \, dx,x,x^n\right )}{n}+\frac {g^2 \operatorname {Subst}\left (\int \frac {\log \left (c (d+e x)^p\right )}{x^5} \, dx,x,x^n\right )}{n}\\ &=-\frac {g^2 x^{-4 n} \log \left (c \left (d+e x^n\right )^p\right )}{4 n}-\frac {f g x^{-2 n} \log \left (c \left (d+e x^n\right )^p\right )}{n}+\frac {f^2 \log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{n}-\frac {\left (e f^2 p\right ) \operatorname {Subst}\left (\int \frac {\log \left (-\frac {e x}{d}\right )}{d+e x} \, dx,x,x^n\right )}{n}+\frac {(e f g p) \operatorname {Subst}\left (\int \frac {1}{x^2 (d+e x)} \, dx,x,x^n\right )}{n}+\frac {\left (e g^2 p\right ) \operatorname {Subst}\left (\int \frac {1}{x^4 (d+e x)} \, dx,x,x^n\right )}{4 n}\\ &=-\frac {g^2 x^{-4 n} \log \left (c \left (d+e x^n\right )^p\right )}{4 n}-\frac {f g x^{-2 n} \log \left (c \left (d+e x^n\right )^p\right )}{n}+\frac {f^2 \log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{n}+\frac {f^2 p \text {Li}_2\left (1+\frac {e x^n}{d}\right )}{n}+\frac {(e f g p) \operatorname {Subst}\left (\int \left (\frac {1}{d x^2}-\frac {e}{d^2 x}+\frac {e^2}{d^2 (d+e x)}\right ) \, dx,x,x^n\right )}{n}+\frac {\left (e g^2 p\right ) \operatorname {Subst}\left (\int \left (\frac {1}{d x^4}-\frac {e}{d^2 x^3}+\frac {e^2}{d^3 x^2}-\frac {e^3}{d^4 x}+\frac {e^4}{d^4 (d+e x)}\right ) \, dx,x,x^n\right )}{4 n}\\ &=-\frac {e g^2 p x^{-3 n}}{12 d n}+\frac {e^2 g^2 p x^{-2 n}}{8 d^2 n}-\frac {e f g p x^{-n}}{d n}-\frac {e^3 g^2 p x^{-n}}{4 d^3 n}-\frac {e^2 f g p \log (x)}{d^2}-\frac {e^4 g^2 p \log (x)}{4 d^4}+\frac {e^2 f g p \log \left (d+e x^n\right )}{d^2 n}+\frac {e^4 g^2 p \log \left (d+e x^n\right )}{4 d^4 n}-\frac {g^2 x^{-4 n} \log \left (c \left (d+e x^n\right )^p\right )}{4 n}-\frac {f g x^{-2 n} \log \left (c \left (d+e x^n\right )^p\right )}{n}+\frac {f^2 \log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{n}+\frac {f^2 p \text {Li}_2\left (1+\frac {e x^n}{d}\right )}{n}\\ \end {align*}
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Mathematica [A] time = 0.58, size = 188, normalized size = 0.73 \[ -\frac {-24 f^2 \left (\log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )+p \text {Li}_2\left (\frac {e x^n}{d}+1\right )\right )+24 f g x^{-2 n} \log \left (c \left (d+e x^n\right )^p\right )+6 g^2 x^{-4 n} \log \left (c \left (d+e x^n\right )^p\right )+\frac {24 e f g p \left (-e \log \left (d+e x^n\right )+d x^{-n}+e n \log (x)\right )}{d^2}+\frac {e g^2 p \left (d x^{-3 n} \left (2 d^2-3 d e x^n+6 e^2 x^{2 n}\right )-6 e^3 \log \left (d+e x^n\right )+6 e^3 n \log (x)\right )}{d^4}}{24 n} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 265, normalized size = 1.03 \[ -\frac {24 \, d^{4} f^{2} n p x^{4 \, n} \log \relax (x) \log \left (\frac {e x^{n} + d}{d}\right ) + 24 \, d^{4} f^{2} p x^{4 \, n} {\rm Li}_2\left (-\frac {e x^{n} + d}{d} + 1\right ) + 2 \, d^{3} e g^{2} p x^{n} + 6 \, d^{4} g^{2} \log \relax (c) + 6 \, {\left (4 \, d^{3} e f g + d e^{3} g^{2}\right )} p x^{3 \, n} - 6 \, {\left (4 \, d^{4} f^{2} n \log \relax (c) - {\left (4 \, d^{2} e^{2} f g + e^{4} g^{2}\right )} n p\right )} x^{4 \, n} \log \relax (x) - 3 \, {\left (d^{2} e^{2} g^{2} p - 8 \, d^{4} f g \log \relax (c)\right )} x^{2 \, n} + 6 \, {\left (4 \, d^{4} f g p x^{2 \, n} + d^{4} g^{2} p - {\left (4 \, d^{4} f^{2} n p \log \relax (x) + {\left (4 \, d^{2} e^{2} f g + e^{4} g^{2}\right )} p\right )} x^{4 \, n}\right )} \log \left (e x^{n} + d\right )}{24 \, d^{4} n x^{4 \, n}} \]
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 \[ \int \frac {{\left (f + \frac {g}{x^{2 \, n}}\right )}^{2} \log \left ({\left (e x^{n} + d\right )}^{p} c\right )}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 3.64, size = 755, normalized size = 2.94 \[ -\frac {e \,g^{2} p \,x^{-3 n}}{12 d n}-\frac {f^{2} p \dilog \left (\frac {e \,x^{n}+d}{d}\right )}{n}+\frac {f^{2} \ln \relax (c ) \ln \left (x^{n}\right )}{n}+\frac {\left (4 f^{2} n \,x^{4 n} \ln \relax (x )-4 f g \,x^{2 n}-g^{2}\right ) x^{-4 n} \ln \left (\left (e \,x^{n}+d \right )^{p}\right )}{4 n}-\frac {g^{2} x^{-4 n} \ln \relax (c )}{4 n}-\frac {e^{4} g^{2} p \ln \left (x^{n}\right )}{4 d^{4} n}-\frac {i \pi \,g^{2} x^{-4 n} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (e \,x^{n}+d \right )^{p}\right )^{2}}{8 n}-\frac {i \pi \,g^{2} x^{-4 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}}{8 n}-f^{2} p \ln \relax (x ) \ln \left (\frac {e \,x^{n}+d}{d}\right )+\frac {e^{2} g^{2} p \,x^{-2 n}}{8 d^{2} n}-\frac {e^{3} g^{2} p \,x^{-n}}{4 d^{3} n}-\frac {e^{2} f g p \ln \left (x^{n}\right )}{d^{2} n}+\frac {e^{4} g^{2} p \ln \left (e \,x^{n}+d \right )}{4 d^{4} n}+\frac {i \pi f g \,x^{-2 n} \mathrm {csgn}\left (i c \left (e \,x^{n}+d \right )^{p}\right )^{3}}{2 n}-\frac {i \pi \,f^{2} \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 g \,x^{-2 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 f 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}}{2 n}+\frac {i \pi \,g^{2} x^{-4 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 )}{8 n}-\frac {f g \,x^{-2 n} \ln \relax (c )}{n}-\frac {e f g p \,x^{-n}}{d n}+\frac {i \pi \,g^{2} x^{-4 n} \mathrm {csgn}\left (i c \left (e \,x^{n}+d \right )^{p}\right )^{3}}{8 n}-\frac {i \pi \,f^{2} \mathrm {csgn}\left (i c \left (e \,x^{n}+d \right )^{p}\right )^{3} \ln \left (x^{n}\right )}{2 n}+\frac {e^{2} f g p \ln \left (e \,x^{n}+d \right )}{d^{2} n}+\frac {i \pi \,f^{2} \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^{2} \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 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 )}{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 \[ -\frac {2 \, d^{2} e g^{2} p x^{n} + 6 \, d^{3} g^{2} \log \relax (c) + 12 \, {\left (d^{3} f^{2} n^{2} p \log \relax (x)^{2} - 2 \, d^{3} f^{2} n \log \relax (c) \log \relax (x)\right )} x^{4 \, n} + 6 \, {\left (4 \, d^{2} e f g p + e^{3} g^{2} p\right )} x^{3 \, n} - 3 \, {\left (d e^{2} g^{2} p - 8 \, d^{3} f g \log \relax (c)\right )} x^{2 \, n} - 6 \, {\left (4 \, d^{3} f^{2} n x^{4 \, n} \log \relax (x) - 4 \, d^{3} f g x^{2 \, n} - d^{3} g^{2}\right )} \log \left ({\left (e x^{n} + d\right )}^{p}\right )}{24 \, d^{3} n x^{4 \, n}} + \int \frac {4 \, d^{4} f^{2} n p \log \relax (x) - 4 \, d^{2} e^{2} f g p - e^{4} g^{2} p}{4 \, {\left (d^{3} e x x^{n} + d^{4} x\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\ln \left (c\,{\left (d+e\,x^n\right )}^p\right )\,{\left (f+\frac {g}{x^{2\,n}}\right )}^2}{x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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