Optimal. Leaf size=327 \[ \frac{f^2 p \text{PolyLog}\left (2,\frac{e x^n}{d}+1\right )}{n}+\frac{f^2 \log \left (-\frac{e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{n}+\frac{2 f g x^{3 n} \log \left (c \left (d+e x^n\right )^p\right )}{3 n}+\frac{g^2 x^{6 n} \log \left (c \left (d+e x^n\right )^p\right )}{6 n}-\frac{2 d^2 f g p x^n}{3 e^2 n}+\frac{2 d^3 f g p \log \left (d+e x^n\right )}{3 e^3 n}+\frac{d^5 g^2 p x^n}{6 e^5 n}-\frac{d^4 g^2 p x^{2 n}}{12 e^4 n}+\frac{d^3 g^2 p x^{3 n}}{18 e^3 n}-\frac{d^2 g^2 p x^{4 n}}{24 e^2 n}-\frac{d^6 g^2 p \log \left (d+e x^n\right )}{6 e^6 n}+\frac{d f g p x^{2 n}}{3 e n}+\frac{d g^2 p x^{5 n}}{30 e n}-\frac{2 f g p x^{3 n}}{9 n}-\frac{g^2 p x^{6 n}}{36 n} \]
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Rubi [A] time = 0.325745, antiderivative size = 327, normalized size of antiderivative = 1., 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 2475
Rule 266
Rule 43
Rule 2416
Rule 2394
Rule 2315
Rule 2395
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*}
Mathematica [A] time = 0.374937, size = 209, normalized size = 0.64 \[ \frac{360 e^6 f^2 p \text{PolyLog}\left (2,\frac{e x^n}{d}+1\right )+60 e^6 \log \left (c \left (d+e x^n\right )^p\right ) \left (6 f^2 \log \left (-\frac{e x^n}{d}\right )+g x^{3 n} \left (4 f+g x^{3 n}\right )\right )-e g p x^n \left (15 d^2 e^3 \left (16 f+g x^{3 n}\right )-20 d^3 e^2 g x^{2 n}+30 d^4 e g x^n-60 d^5 g-12 d e^4 x^n \left (10 f+g x^{3 n}\right )+10 e^5 x^{2 n} \left (8 f+g x^{3 n}\right )\right )-60 d^3 g p \left (d^3 g-4 e^3 f\right ) \log \left (d+e x^n\right )}{360 e^6 n} \]
Antiderivative was successfully verified.
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Maple [C] time = 5.224, size = 795, normalized size = 2.4 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{180 \, e^{6} f^{2} n^{2} p \log \left (x\right )^{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 \left (c\right )\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 \left (c\right )\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 \left (x\right ) + 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 \left (c\right )\right )} \log \left (x\right )}{360 \, e^{6} n} + \int \frac{6 \, d e^{6} f^{2} n p \log \left (x\right ) - 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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.2259, size = 659, normalized size = 2.02 \begin{align*} -\frac{360 \, e^{6} f^{2} n p \log \left (x\right ) \log \left (\frac{e x^{n} + d}{d}\right ) - 360 \, e^{6} f^{2} n \log \left (c\right ) \log \left (x\right ) - 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 \left (c\right )\right )} x^{6 \, n} - 20 \,{\left (12 \, e^{6} f g \log \left (c\right ) -{\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 \left (x\right ) + 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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (g x^{3 \, n} + f\right )}^{2} \log \left ({\left (e x^{n} + d\right )}^{p} c\right )}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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