Optimal. Leaf size=327 \[ \frac {f^2 \log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{n}+\frac {2 f g x^{3 n} \log \left (c \left (d+e x^n\right )^p\right )}{3 n}+\frac {g^2 x^{6 n} \log \left (c \left (d+e x^n\right )^p\right )}{6 n}-\frac {d^6 g^2 p \log \left (d+e x^n\right )}{6 e^6 n}+\frac {d^5 g^2 p x^n}{6 e^5 n}-\frac {d^4 g^2 p x^{2 n}}{12 e^4 n}+\frac {2 d^3 f g p \log \left (d+e x^n\right )}{3 e^3 n}+\frac {d^3 g^2 p x^{3 n}}{18 e^3 n}-\frac {2 d^2 f g p x^n}{3 e^2 n}-\frac {d^2 g^2 p x^{4 n}}{24 e^2 n}+\frac {f^2 p \text {Li}_2\left (\frac {e x^n}{d}+1\right )}{n}+\frac {d f g p x^{2 n}}{3 e n}+\frac {d g^2 p x^{5 n}}{30 e n}-\frac {2 f g p x^{3 n}}{9 n}-\frac {g^2 p x^{6 n}}{36 n} \]
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Rubi [A] time = 0.33, antiderivative size = 327, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 7, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {2475, 266, 43, 2416, 2394, 2315, 2395} \[ \frac {f^2 p \text {PolyLog}\left (2,\frac {e x^n}{d}+1\right )}{n}+\frac {f^2 \log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{n}+\frac {2 f g x^{3 n} \log \left (c \left (d+e x^n\right )^p\right )}{3 n}+\frac {g^2 x^{6 n} \log \left (c \left (d+e x^n\right )^p\right )}{6 n}-\frac {2 d^2 f g p x^n}{3 e^2 n}+\frac {2 d^3 f g p \log \left (d+e x^n\right )}{3 e^3 n}+\frac {d^5 g^2 p x^n}{6 e^5 n}-\frac {d^4 g^2 p x^{2 n}}{12 e^4 n}+\frac {d^3 g^2 p x^{3 n}}{18 e^3 n}-\frac {d^2 g^2 p x^{4 n}}{24 e^2 n}-\frac {d^6 g^2 p \log \left (d+e x^n\right )}{6 e^6 n}+\frac {d f g p x^{2 n}}{3 e n}+\frac {d g^2 p x^{5 n}}{30 e n}-\frac {2 f g p x^{3 n}}{9 n}-\frac {g^2 p x^{6 n}}{36 n} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rule 2315
Rule 2394
Rule 2395
Rule 2416
Rule 2475
Rubi steps
\begin {align*} \int \frac {\left (f+g x^{3 n}\right )^2 \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 )^2 \log \left (c (d+e x)^p\right )}{x} \, dx,x,x^n\right )}{n}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {f^2 \log \left (c (d+e x)^p\right )}{x}+2 f g x^2 \log \left (c (d+e x)^p\right )+g^2 x^5 \log \left (c (d+e x)^p\right )\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 x^2 \log \left (c (d+e x)^p\right ) \, dx,x,x^n\right )}{n}+\frac {g^2 \operatorname {Subst}\left (\int x^5 \log \left (c (d+e x)^p\right ) \, dx,x,x^n\right )}{n}\\ &=\frac {2 f g x^{3 n} \log \left (c \left (d+e x^n\right )^p\right )}{3 n}+\frac {g^2 x^{6 n} \log \left (c \left (d+e x^n\right )^p\right )}{6 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 {(2 e f g p) \operatorname {Subst}\left (\int \frac {x^3}{d+e x} \, dx,x,x^n\right )}{3 n}-\frac {\left (e g^2 p\right ) \operatorname {Subst}\left (\int \frac {x^6}{d+e x} \, dx,x,x^n\right )}{6 n}\\ &=\frac {2 f g x^{3 n} \log \left (c \left (d+e x^n\right )^p\right )}{3 n}+\frac {g^2 x^{6 n} \log \left (c \left (d+e x^n\right )^p\right )}{6 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 {(2 e f 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 {\left (e g^2 p\right ) \operatorname {Subst}\left (\int \left (-\frac {d^5}{e^6}+\frac {d^4 x}{e^5}-\frac {d^3 x^2}{e^4}+\frac {d^2 x^3}{e^3}-\frac {d x^4}{e^2}+\frac {x^5}{e}+\frac {d^6}{e^6 (d+e x)}\right ) \, dx,x,x^n\right )}{6 n}\\ &=-\frac {2 d^2 f g p x^n}{3 e^2 n}+\frac {d^5 g^2 p x^n}{6 e^5 n}+\frac {d f g p x^{2 n}}{3 e n}-\frac {d^4 g^2 p x^{2 n}}{12 e^4 n}-\frac {2 f g p x^{3 n}}{9 n}+\frac {d^3 g^2 p x^{3 n}}{18 e^3 n}-\frac {d^2 g^2 p x^{4 n}}{24 e^2 n}+\frac {d g^2 p x^{5 n}}{30 e n}-\frac {g^2 p x^{6 n}}{36 n}+\frac {2 d^3 f g p \log \left (d+e x^n\right )}{3 e^3 n}-\frac {d^6 g^2 p \log \left (d+e x^n\right )}{6 e^6 n}+\frac {2 f g x^{3 n} \log \left (c \left (d+e x^n\right )^p\right )}{3 n}+\frac {g^2 x^{6 n} \log \left (c \left (d+e x^n\right )^p\right )}{6 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 = 0.31, size = 209, normalized size = 0.64 \[ \frac {60 e^6 \log \left (c \left (d+e x^n\right )^p\right ) \left (6 f^2 \log \left (-\frac {e x^n}{d}\right )+g x^{3 n} \left (4 f+g x^{3 n}\right )\right )-60 d^3 g p \left (d^3 g-4 e^3 f\right ) \log \left (d+e x^n\right )-e g p x^n \left (-60 d^5 g+30 d^4 e g x^n-20 d^3 e^2 g x^{2 n}+15 d^2 e^3 \left (16 f+g x^{3 n}\right )-12 d e^4 x^n \left (10 f+g x^{3 n}\right )+10 e^5 x^{2 n} \left (8 f+g x^{3 n}\right )\right )+360 e^6 f^2 p \text {Li}_2\left (\frac {e x^n}{d}+1\right )}{360 e^6 n} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 291, normalized size = 0.89 \[ -\frac {360 \, e^{6} f^{2} n p \log \relax (x) \log \left (\frac {e x^{n} + d}{d}\right ) - 360 \, e^{6} f^{2} n \log \relax (c) \log \relax (x) - 12 \, d e^{5} g^{2} p x^{5 \, n} + 15 \, d^{2} e^{4} g^{2} p x^{4 \, n} + 360 \, e^{6} f^{2} p {\rm Li}_2\left (-\frac {e x^{n} + d}{d} + 1\right ) - 30 \, {\left (4 \, d e^{5} f g - d^{4} e^{2} g^{2}\right )} p x^{2 \, n} + 60 \, {\left (4 \, d^{2} e^{4} f g - d^{5} e g^{2}\right )} p x^{n} + 10 \, {\left (e^{6} g^{2} p - 6 \, e^{6} g^{2} \log \relax (c)\right )} x^{6 \, n} - 20 \, {\left (12 \, e^{6} f g \log \relax (c) - {\left (4 \, e^{6} f g - d^{3} e^{3} g^{2}\right )} p\right )} x^{3 \, n} - 60 \, {\left (6 \, e^{6} f^{2} n p \log \relax (x) + e^{6} g^{2} p x^{6 \, n} + 4 \, e^{6} f g p x^{3 \, n} + {\left (4 \, d^{3} e^{3} f g - d^{6} g^{2}\right )} p\right )} \log \left (e x^{n} + d\right )}{360 \, e^{6} n} \]
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 \[ \int \frac {{\left (g x^{3 \, n} + f\right )}^{2} \log \left ({\left (e x^{n} + d\right )}^{p} c\right )}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 4.04, size = 795, normalized size = 2.43 \[ \frac {\left (6 f^{2} n \ln \relax (x )+4 f g \,x^{3 n}+g^{2} x^{6 n}\right ) \ln \left (\left (e \,x^{n}+d \right )^{p}\right )}{6 n}-\frac {g^{2} p \,x^{6 n}}{36 n}+\frac {g^{2} x^{6 n} \ln \relax (c )}{6 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 {d^{5} g^{2} p \,x^{n}}{6 e^{5} n}-\frac {d^{4} g^{2} p \,x^{2 n}}{12 e^{4} n}+\frac {d^{3} g^{2} p \,x^{3 n}}{18 e^{3} n}-f^{2} p \ln \relax (x ) \ln \left (\frac {e \,x^{n}+d}{d}\right )-\frac {2 d^{2} f g p \,x^{n}}{3 e^{2} n}+\frac {2 d^{3} f g p \ln \left (e \,x^{n}+d \right )}{3 e^{3} n}-\frac {2 f g p \,x^{3 n}}{9 n}-\frac {d^{6} g^{2} p \ln \left (e \,x^{n}+d \right )}{6 e^{6} n}+\frac {d f g p \,x^{2 n}}{3 e n}+\frac {2 f g \,x^{3 n} \ln \relax (c )}{3 n}-\frac {i \pi f 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 )}{3 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^{3 n} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (e \,x^{n}+d \right )^{p}\right )^{2}}{3 n}+\frac {i \pi f 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}}{3 n}-\frac {i \pi \,g^{2} x^{6 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 )}{12 n}+\frac {i \pi \,g^{2} x^{6 n} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (e \,x^{n}+d \right )^{p}\right )^{2}}{12 n}-\frac {d^{2} g^{2} p \,x^{4 n}}{24 e^{2} n}+\frac {d \,g^{2} p \,x^{5 n}}{30 e n}-\frac {i \pi \,g^{2} x^{6 n} \mathrm {csgn}\left (i c \left (e \,x^{n}+d \right )^{p}\right )^{3}}{12 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 {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 \,g^{2} x^{6 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}}{12 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^{3 n} \mathrm {csgn}\left (i c \left (e \,x^{n}+d \right )^{p}\right )^{3}}{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 \[ -\frac {180 \, e^{6} f^{2} n^{2} p \log \relax (x)^{2} - 12 \, d e^{5} g^{2} p x^{5 \, n} + 15 \, d^{2} e^{4} g^{2} p x^{4 \, n} + 10 \, {\left (e^{6} g^{2} p - 6 \, e^{6} g^{2} \log \relax (c)\right )} x^{6 \, n} + 20 \, {\left (4 \, e^{6} f g p - d^{3} e^{3} g^{2} p - 12 \, e^{6} f g \log \relax (c)\right )} x^{3 \, n} - 30 \, {\left (4 \, d e^{5} f g p - d^{4} e^{2} g^{2} p\right )} x^{2 \, n} + 60 \, {\left (4 \, d^{2} e^{4} f g p - d^{5} e g^{2} p\right )} x^{n} - 60 \, {\left (6 \, e^{6} f^{2} n \log \relax (x) + e^{6} g^{2} x^{6 \, n} + 4 \, e^{6} f g x^{3 \, n}\right )} \log \left ({\left (e x^{n} + d\right )}^{p}\right ) - 60 \, {\left (4 \, d^{3} e^{3} f g n p - d^{6} g^{2} n p + 6 \, e^{6} f^{2} n \log \relax (c)\right )} \log \relax (x)}{360 \, e^{6} n} + \int \frac {6 \, d e^{6} f^{2} n p \log \relax (x) - 4 \, d^{4} e^{3} f g p + d^{7} g^{2} p}{6 \, {\left (e^{7} x x^{n} + d e^{6} x\right )}}\,{d x} \]
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 \[ \int \frac {\ln \left (c\,{\left (d+e\,x^n\right )}^p\right )\,{\left (f+g\,x^{3\,n}\right )}^2}{x} \,d x \]
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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