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 = 0.03, antiderivative size = 70, normalized size of antiderivative = 1.00, 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 364
Rule 2455
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*}
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Mathematica [A] time = 0.03, 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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fricas [F] time = 0.51, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\log \left ({\left (e x^{n} + d\right )}^{p} c\right )}{x^{4}}, x\right ) \]
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 \[ \int \frac {\log \left ({\left (e x^{n} + d\right )}^{p} c\right )}{x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.01, size = 0, normalized size = 0.00 \[ \int \frac {\ln \left (c \left (e \,x^{n}+d \right )^{p}\right )}{x^{4}}\, dx \]
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 \[ -d n p \int \frac {1}{3 \, {\left (e x^{4} x^{n} + d x^{4}\right )}}\,{d x} - \frac {n p + 3 \, \log \left ({\left (e x^{n} + d\right )}^{p}\right ) + 3 \, \log \relax (c)}{9 \, x^{3}} \]
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 \[ \int \frac {\ln \left (c\,{\left (d+e\,x^n\right )}^p\right )}{x^4} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 32.22, size = 51, normalized size = 0.73 \[ - \frac {\log {\left (c \left (d + e x^{n}\right )^{p} \right )}}{3 x^{3}} + \frac {p \Phi \left (\frac {d x^{- n} e^{i \pi }}{e}, 1, \frac {3}{n}\right ) \Gamma \left (- \frac {3}{n}\right )}{n x^{3} \Gamma \left (1 - \frac {3}{n}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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