Optimal. Leaf size=234 \[ \frac{(f+g x)^4 \log \left (c \left (d+e x^n\right )^p\right )}{4 g}-\frac{3 e f^2 g n p x^{n+2} \, _2F_1\left (1,\frac{n+2}{n};2 \left (1+\frac{1}{n}\right );-\frac{e x^n}{d}\right )}{2 d (n+2)}-\frac{f^4 p \log \left (d+e x^n\right )}{4 g}-\frac{e f^3 n p x^{n+1} \, _2F_1\left (1,1+\frac{1}{n};2+\frac{1}{n};-\frac{e x^n}{d}\right )}{d (n+1)}-\frac{e f g^2 n p x^{n+3} \, _2F_1\left (1,\frac{n+3}{n};2+\frac{3}{n};-\frac{e x^n}{d}\right )}{d (n+3)}-\frac{e g^3 n p x^{n+4} \, _2F_1\left (1,\frac{n+4}{n};2 \left (1+\frac{2}{n}\right );-\frac{e x^n}{d}\right )}{4 d (n+4)} \]
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Rubi [A] time = 0.232796, antiderivative size = 234, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {2463, 1844, 260, 364} \[ \frac{(f+g x)^4 \log \left (c \left (d+e x^n\right )^p\right )}{4 g}-\frac{3 e f^2 g n p x^{n+2} \, _2F_1\left (1,\frac{n+2}{n};2 \left (1+\frac{1}{n}\right );-\frac{e x^n}{d}\right )}{2 d (n+2)}-\frac{f^4 p \log \left (d+e x^n\right )}{4 g}-\frac{e f^3 n p x^{n+1} \, _2F_1\left (1,1+\frac{1}{n};2+\frac{1}{n};-\frac{e x^n}{d}\right )}{d (n+1)}-\frac{e f g^2 n p x^{n+3} \, _2F_1\left (1,\frac{n+3}{n};2+\frac{3}{n};-\frac{e x^n}{d}\right )}{d (n+3)}-\frac{e g^3 n p x^{n+4} \, _2F_1\left (1,\frac{n+4}{n};2 \left (1+\frac{2}{n}\right );-\frac{e x^n}{d}\right )}{4 d (n+4)} \]
Antiderivative was successfully verified.
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Rule 2463
Rule 1844
Rule 260
Rule 364
Rubi steps
\begin{align*} \int (f+g x)^3 \log \left (c \left (d+e x^n\right )^p\right ) \, dx &=\frac{(f+g x)^4 \log \left (c \left (d+e x^n\right )^p\right )}{4 g}-\frac{(e n p) \int \frac{x^{-1+n} (f+g x)^4}{d+e x^n} \, dx}{4 g}\\ &=\frac{(f+g x)^4 \log \left (c \left (d+e x^n\right )^p\right )}{4 g}-\frac{(e n p) \int \left (\frac{f^4 x^{-1+n}}{d+e x^n}+\frac{4 f^3 g x^n}{d+e x^n}+\frac{6 f^2 g^2 x^{1+n}}{d+e x^n}+\frac{4 f g^3 x^{2+n}}{d+e x^n}+\frac{g^4 x^{3+n}}{d+e x^n}\right ) \, dx}{4 g}\\ &=\frac{(f+g x)^4 \log \left (c \left (d+e x^n\right )^p\right )}{4 g}-\left (e f^3 n p\right ) \int \frac{x^n}{d+e x^n} \, dx-\frac{\left (e f^4 n p\right ) \int \frac{x^{-1+n}}{d+e x^n} \, dx}{4 g}-\frac{1}{2} \left (3 e f^2 g n p\right ) \int \frac{x^{1+n}}{d+e x^n} \, dx-\left (e f g^2 n p\right ) \int \frac{x^{2+n}}{d+e x^n} \, dx-\frac{1}{4} \left (e g^3 n p\right ) \int \frac{x^{3+n}}{d+e x^n} \, dx\\ &=-\frac{e f^3 n p x^{1+n} \, _2F_1\left (1,1+\frac{1}{n};2+\frac{1}{n};-\frac{e x^n}{d}\right )}{d (1+n)}-\frac{3 e f^2 g n p x^{2+n} \, _2F_1\left (1,\frac{2+n}{n};2 \left (1+\frac{1}{n}\right );-\frac{e x^n}{d}\right )}{2 d (2+n)}-\frac{e f g^2 n p x^{3+n} \, _2F_1\left (1,\frac{3+n}{n};2+\frac{3}{n};-\frac{e x^n}{d}\right )}{d (3+n)}-\frac{e g^3 n p x^{4+n} \, _2F_1\left (1,\frac{4+n}{n};2 \left (1+\frac{2}{n}\right );-\frac{e x^n}{d}\right )}{4 d (4+n)}-\frac{f^4 p \log \left (d+e x^n\right )}{4 g}+\frac{(f+g x)^4 \log \left (c \left (d+e x^n\right )^p\right )}{4 g}\\ \end{align*}
Mathematica [A] time = 0.515633, size = 224, normalized size = 0.96 \[ \frac{(f+g x)^4 \log \left (c \left (d+e x^n\right )^p\right )-e n p \left (\frac{6 f^2 g^2 x^{n+2} \, _2F_1\left (1,\frac{n+2}{n};2 \left (1+\frac{1}{n}\right );-\frac{e x^n}{d}\right )}{d (n+2)}+\frac{4 f^3 g x^{n+1} \, _2F_1\left (1,1+\frac{1}{n};2+\frac{1}{n};-\frac{e x^n}{d}\right )}{d (n+1)}+\frac{f^4 \log \left (d+e x^n\right )}{e n}+\frac{4 f g^3 x^{n+3} \, _2F_1\left (1,\frac{n+3}{n};2+\frac{3}{n};-\frac{e x^n}{d}\right )}{d (n+3)}+\frac{g^4 x^{n+4} \, _2F_1\left (1,\frac{n+4}{n};2+\frac{4}{n};-\frac{e x^n}{d}\right )}{d (n+4)}\right )}{4 g} \]
Antiderivative was successfully verified.
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Maple [F] time = 2.714, size = 0, normalized size = 0. \begin{align*} \int \left ( gx+f \right ) ^{3}\ln \left ( c \left ( d+e{x}^{n} \right ) ^{p} \right ) \, dx \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{1}{16} \,{\left (g^{3} n p - 4 \, g^{3} \log \left (c\right )\right )} x^{4} - \frac{1}{3} \,{\left (f g^{2} n p - 3 \, f g^{2} \log \left (c\right )\right )} x^{3} - \frac{3}{4} \,{\left (f^{2} g n p - 2 \, f^{2} g \log \left (c\right )\right )} x^{2} -{\left (f^{3} n p - f^{3} \log \left (c\right )\right )} x + \frac{1}{4} \,{\left (g^{3} x^{4} + 4 \, f g^{2} x^{3} + 6 \, f^{2} g x^{2} + 4 \, f^{3} x\right )} \log \left ({\left (e x^{n} + d\right )}^{p}\right ) + \int \frac{d g^{3} n p x^{3} + 4 \, d f g^{2} n p x^{2} + 6 \, d f^{2} g n p x + 4 \, d f^{3} n p}{4 \,{\left (e x^{n} + d\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (g^{3} x^{3} + 3 \, f g^{2} x^{2} + 3 \, f^{2} g x + f^{3}\right )} \log \left ({\left (e x^{n} + d\right )}^{p} c\right ), x\right ) \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{\left (g x + f\right )}^{3} \log \left ({\left (e x^{n} + d\right )}^{p} c\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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