Optimal. Leaf size=70 \[ -\frac{\log \left (c \left (d+e x^n\right )^p\right )}{3 x^3}-\frac{e n p x^{n-3} \, _2F_1\left (1,-\frac{3-n}{n};2-\frac{3}{n};-\frac{e x^n}{d}\right )}{3 d (3-n)} \]
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Rubi [A] time = 0.0293658, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {2455, 364} \[ -\frac{\log \left (c \left (d+e x^n\right )^p\right )}{3 x^3}-\frac{e n p x^{n-3} \, _2F_1\left (1,-\frac{3-n}{n};2-\frac{3}{n};-\frac{e x^n}{d}\right )}{3 d (3-n)} \]
Antiderivative was successfully verified.
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Rule 2455
Rule 364
Rubi steps
\begin{align*} \int \frac{\log \left (c \left (d+e x^n\right )^p\right )}{x^4} \, dx &=-\frac{\log \left (c \left (d+e x^n\right )^p\right )}{3 x^3}+\frac{1}{3} (e n p) \int \frac{x^{-4+n}}{d+e x^n} \, dx\\ &=-\frac{e n p x^{-3+n} \, _2F_1\left (1,-\frac{3-n}{n};2-\frac{3}{n};-\frac{e x^n}{d}\right )}{3 d (3-n)}-\frac{\log \left (c \left (d+e x^n\right )^p\right )}{3 x^3}\\ \end{align*}
Mathematica [A] time = 0.0263766, size = 62, normalized size = 0.89 \[ \frac{\frac{e n p x^n \, _2F_1\left (1,\frac{n-3}{n};2-\frac{3}{n};-\frac{e x^n}{d}\right )}{d (n-3)}-\log \left (c \left (d+e x^n\right )^p\right )}{3 x^3} \]
Antiderivative was successfully verified.
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Maple [F] time = 1.529, size = 0, normalized size = 0. \begin{align*} \int{\frac{\ln \left ( c \left ( d+e{x}^{n} \right ) ^{p} \right ) }{{x}^{4}}}\, 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*} -d n p \int \frac{1}{3 \,{\left (e x^{4} x^{n} + d x^{4}\right )}}\,{d x} - \frac{n p + 3 \, \log \left ({\left (e x^{n} + d\right )}^{p}\right ) + 3 \, \log \left (c\right )}{9 \, x^{3}} \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 (\frac{\log \left ({\left (e x^{n} + d\right )}^{p} c\right )}{x^{4}}, 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 \frac{\log \left ({\left (e x^{n} + d\right )}^{p} c\right )}{x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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