Optimal. Leaf size=176 \[ \frac{f^2 p \text{PolyLog}\left (2,\frac{e x^n}{d}+1\right )}{n}+\frac{f^2 \log \left (-\frac{e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{n}+\frac{2 f g \left (d+e x^n\right ) \log \left (c \left (d+e x^n\right )^p\right )}{e n}+\frac{g^2 x^{2 n} \log \left (c \left (d+e x^n\right )^p\right )}{2 n}-\frac{d^2 g^2 p \log \left (d+e x^n\right )}{2 e^2 n}+\frac{d g^2 p x^n}{2 e n}-\frac{2 f g p x^n}{n}-\frac{g^2 p x^{2 n}}{4 n} \]
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Rubi [A] time = 0.196758, antiderivative size = 176, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.32, 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 2475
Rule 43
Rule 2416
Rule 2389
Rule 2295
Rule 2394
Rule 2315
Rule 2395
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*}
Mathematica [A] time = 0.189758, size = 124, normalized size = 0.7 \[ \frac{4 e^2 f^2 p \text{PolyLog}\left (2,\frac{e x^n}{d}+1\right )+2 e \log \left (c \left (d+e x^n\right )^p\right ) \left (2 e f^2 \log \left (-\frac{e x^n}{d}\right )+4 d f g+e g x^n \left (4 f+g x^n\right )\right )-2 d^2 g^2 p \log \left (d+e x^n\right )-e g p x^n \left (-2 d g+8 e f+e g x^n\right )}{4 e^2 n} \]
Antiderivative was successfully verified.
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Maple [C] time = 5.079, size = 665, normalized size = 3.8 \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{2 \, e^{2} f^{2} n^{2} p \log \left (x\right )^{2} +{\left (e^{2} g^{2} p - 2 \, e^{2} g^{2} \log \left (c\right )\right )} x^{2 \, n} + 2 \,{\left (4 \, e^{2} f g p - d e g^{2} p - 4 \, e^{2} f g \log \left (c\right )\right )} x^{n} - 2 \,{\left (2 \, e^{2} f^{2} n \log \left (x\right ) + 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 \left (c\right )\right )} \log \left (x\right )}{4 \, e^{2} n} + \int \frac{2 \, d e^{2} f^{2} n p \log \left (x\right ) - 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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.15976, size = 437, normalized size = 2.48 \begin{align*} -\frac{4 \, e^{2} f^{2} n p \log \left (x\right ) \log \left (\frac{e x^{n} + d}{d}\right ) - 4 \, e^{2} f^{2} n \log \left (c\right ) \log \left (x\right ) + 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 \left (c\right )\right )} x^{2 \, n} - 2 \,{\left (4 \, e^{2} f g \log \left (c\right ) -{\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 \left (x\right ) + 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} \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^{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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