Optimal. Leaf size=176 \[ \frac {f^2 \log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{n}+\frac {2 f g \left (d+e x^n\right ) \log \left (c \left (d+e x^n\right )^p\right )}{e n}+\frac {g^2 x^{2 n} \log \left (c \left (d+e x^n\right )^p\right )}{2 n}-\frac {d^2 g^2 p \log \left (d+e x^n\right )}{2 e^2 n}+\frac {f^2 p \text {Li}_2\left (\frac {e x^n}{d}+1\right )}{n}+\frac {d g^2 p x^n}{2 e n}-\frac {2 f g p x^n}{n}-\frac {g^2 p x^{2 n}}{4 n} \]
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Rubi [A] time = 0.20, antiderivative size = 176, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {2475, 43, 2416, 2389, 2295, 2394, 2315, 2395} \[ \frac {f^2 p \text {PolyLog}\left (2,\frac {e x^n}{d}+1\right )}{n}+\frac {f^2 \log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{n}+\frac {2 f g \left (d+e x^n\right ) \log \left (c \left (d+e x^n\right )^p\right )}{e n}+\frac {g^2 x^{2 n} \log \left (c \left (d+e x^n\right )^p\right )}{2 n}-\frac {d^2 g^2 p \log \left (d+e x^n\right )}{2 e^2 n}+\frac {d g^2 p x^n}{2 e n}-\frac {2 f g p x^n}{n}-\frac {g^2 p x^{2 n}}{4 n} \]
Antiderivative was successfully verified.
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Rule 43
Rule 2295
Rule 2315
Rule 2389
Rule 2394
Rule 2395
Rule 2416
Rule 2475
Rubi steps
\begin {align*} \int \frac {\left (f+g x^n\right )^2 \log \left (c \left (d+e x^n\right )^p\right )}{x} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {(f+g x)^2 \log \left (c (d+e x)^p\right )}{x} \, dx,x,x^n\right )}{n}\\ &=\frac {\operatorname {Subst}\left (\int \left (2 f g \log \left (c (d+e x)^p\right )+\frac {f^2 \log \left (c (d+e x)^p\right )}{x}+g^2 x \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 \log \left (c (d+e x)^p\right ) \, dx,x,x^n\right )}{n}+\frac {g^2 \operatorname {Subst}\left (\int x \log \left (c (d+e x)^p\right ) \, dx,x,x^n\right )}{n}\\ &=\frac {g^2 x^{2 n} \log \left (c \left (d+e x^n\right )^p\right )}{2 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) \operatorname {Subst}\left (\int \log \left (c x^p\right ) \, dx,x,d+e x^n\right )}{e 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 {\left (e g^2 p\right ) \operatorname {Subst}\left (\int \frac {x^2}{d+e x} \, dx,x,x^n\right )}{2 n}\\ &=-\frac {2 f g p x^n}{n}+\frac {g^2 x^{2 n} \log \left (c \left (d+e x^n\right )^p\right )}{2 n}+\frac {2 f g \left (d+e x^n\right ) \log \left (c \left (d+e x^n\right )^p\right )}{e 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 {\left (e g^2 p\right ) \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 {2 f g p x^n}{n}+\frac {d g^2 p x^n}{2 e n}-\frac {g^2 p x^{2 n}}{4 n}-\frac {d^2 g^2 p \log \left (d+e x^n\right )}{2 e^2 n}+\frac {g^2 x^{2 n} \log \left (c \left (d+e x^n\right )^p\right )}{2 n}+\frac {2 f g \left (d+e x^n\right ) \log \left (c \left (d+e x^n\right )^p\right )}{e 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 = 0.19, size = 124, normalized size = 0.70 \[ \frac {2 e \log \left (c \left (d+e x^n\right )^p\right ) \left (2 e f^2 \log \left (-\frac {e x^n}{d}\right )+4 d f g+e g x^n \left (4 f+g x^n\right )\right )-2 d^2 g^2 p \log \left (d+e x^n\right )+4 e^2 f^2 p \text {Li}_2\left (\frac {e x^n}{d}+1\right )-e g p x^n \left (-2 d g+8 e f+e g x^n\right )}{4 e^2 n} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 192, normalized size = 1.09 \[ -\frac {4 \, e^{2} f^{2} n p \log \relax (x) \log \left (\frac {e x^{n} + d}{d}\right ) - 4 \, e^{2} f^{2} n \log \relax (c) \log \relax (x) + 4 \, e^{2} f^{2} p {\rm Li}_2\left (-\frac {e x^{n} + d}{d} + 1\right ) + {\left (e^{2} g^{2} p - 2 \, e^{2} g^{2} \log \relax (c)\right )} x^{2 \, n} - 2 \, {\left (4 \, e^{2} f g \log \relax (c) - {\left (4 \, e^{2} f g - d e g^{2}\right )} p\right )} x^{n} - 2 \, {\left (2 \, e^{2} f^{2} n p \log \relax (x) + e^{2} g^{2} p x^{2 \, n} + 4 \, e^{2} f g p x^{n} + {\left (4 \, d e f g - d^{2} g^{2}\right )} p\right )} \log \left (e x^{n} + d\right )}{4 \, e^{2} n} \]
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 \[ \int \frac {{\left (g x^{n} + f\right )}^{2} \log \left ({\left (e x^{n} + d\right )}^{p} c\right )}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 3.80, size = 665, normalized size = 3.78 \[ -\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^{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^{2} \mathrm {csgn}\left (i c \left (e \,x^{n}+d \right )^{p}\right )^{3} \ln \left (x^{n}\right )}{2 n}-f^{2} p \ln \relax (x ) \ln \left (\frac {e \,x^{n}+d}{d}\right )-\frac {i \pi f g \,x^{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 )}{n}+\frac {i \pi f g \,x^{n} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (e \,x^{n}+d \right )^{p}\right )^{2}}{n}+\frac {i \pi f g \,x^{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}}{n}-\frac {i \pi f g \,x^{n} \mathrm {csgn}\left (i c \left (e \,x^{n}+d \right )^{p}\right )^{3}}{n}-\frac {i \pi \,g^{2} 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^{2} 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^{2} 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^{2} x^{2 n} \mathrm {csgn}\left (i c \left (e \,x^{n}+d \right )^{p}\right )^{3}}{4 n}-\frac {d^{2} g^{2} p \ln \left (e \,x^{n}+d \right )}{2 e^{2} n}+\frac {2 d f g p \ln \left (e \,x^{n}+d \right )}{e n}+\frac {d \,g^{2} p \,x^{n}}{2 e 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 {2 f g p \,x^{n}}{n}+\frac {2 f g \,x^{n} \ln \relax (c )}{n}-\frac {g^{2} p \,x^{2 n}}{4 n}+\frac {g^{2} x^{2 n} \ln \relax (c )}{2 n}+\frac {\left (2 f^{2} n \ln \relax (x )+4 f g \,x^{n}+g^{2} 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 \[ -\frac {2 \, e^{2} f^{2} n^{2} p \log \relax (x)^{2} + {\left (e^{2} g^{2} p - 2 \, e^{2} g^{2} \log \relax (c)\right )} x^{2 \, n} + 2 \, {\left (4 \, e^{2} f g p - d e g^{2} p - 4 \, e^{2} f g \log \relax (c)\right )} x^{n} - 2 \, {\left (2 \, e^{2} f^{2} n \log \relax (x) + e^{2} g^{2} x^{2 \, n} + 4 \, e^{2} f g x^{n}\right )} \log \left ({\left (e x^{n} + d\right )}^{p}\right ) - 2 \, {\left (4 \, d e f g n p - d^{2} g^{2} n p + 2 \, e^{2} f^{2} n \log \relax (c)\right )} \log \relax (x)}{4 \, e^{2} n} + \int \frac {2 \, d e^{2} f^{2} n p \log \relax (x) - 4 \, d^{2} e f g p + d^{3} g^{2} p}{2 \, {\left (e^{3} x x^{n} + d e^{2} x\right )}}\,{d x} \]
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 \[ \int \frac {\ln \left (c\,{\left (d+e\,x^n\right )}^p\right )\,{\left (f+g\,x^n\right )}^2}{x} \,d x \]
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 \[ \int \frac {\left (f + g x^{n}\right )^{2} \log {\left (c \left (d + e x^{n}\right )^{p} \right )}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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