Optimal. Leaf size=80 \[ \frac {e p x^n (f x)^{-n} \log (x)}{d f}-\frac {e p x^n (f x)^{-n} \log \left (d+e x^n\right )}{d f n}-\frac {(f x)^{-n} \log \left (c \left (d+e x^n\right )^p\right )}{f n} \]
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Rubi [A]
time = 0.02, antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {2505, 19, 272,
36, 29, 31} \begin {gather*} -\frac {(f x)^{-n} \log \left (c \left (d+e x^n\right )^p\right )}{f n}+\frac {e p x^n \log (x) (f x)^{-n}}{d f}-\frac {e p x^n (f x)^{-n} \log \left (d+e x^n\right )}{d f n} \end {gather*}
Antiderivative was successfully verified.
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Rule 19
Rule 29
Rule 31
Rule 36
Rule 272
Rule 2505
Rubi steps
\begin {align*} \int (f x)^{-1-n} \log \left (c \left (d+e x^n\right )^p\right ) \, dx &=-\frac {(f x)^{-n} \log \left (c \left (d+e x^n\right )^p\right )}{f n}+\frac {(e p) \int \frac {x^{-1+n} (f x)^{-n}}{d+e x^n} \, dx}{f}\\ &=-\frac {(f x)^{-n} \log \left (c \left (d+e x^n\right )^p\right )}{f n}+\frac {\left (e p x^n (f x)^{-n}\right ) \int \frac {1}{x \left (d+e x^n\right )} \, dx}{f}\\ &=-\frac {(f x)^{-n} \log \left (c \left (d+e x^n\right )^p\right )}{f n}+\frac {\left (e p x^n (f x)^{-n}\right ) \text {Subst}\left (\int \frac {1}{x (d+e x)} \, dx,x,x^n\right )}{f n}\\ &=-\frac {(f x)^{-n} \log \left (c \left (d+e x^n\right )^p\right )}{f n}+\frac {\left (e p x^n (f x)^{-n}\right ) \text {Subst}\left (\int \frac {1}{x} \, dx,x,x^n\right )}{d f n}-\frac {\left (e^2 p x^n (f x)^{-n}\right ) \text {Subst}\left (\int \frac {1}{d+e x} \, dx,x,x^n\right )}{d f n}\\ &=\frac {e p x^n (f x)^{-n} \log (x)}{d f}-\frac {e p x^n (f x)^{-n} \log \left (d+e x^n\right )}{d f n}-\frac {(f x)^{-n} \log \left (c \left (d+e x^n\right )^p\right )}{f n}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 57, normalized size = 0.71 \begin {gather*} -\frac {(f x)^{-n} \left (-e n p x^n \log (x)+e p x^n \log \left (d+e x^n\right )+d \log \left (c \left (d+e x^n\right )^p\right )\right )}{d f n} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.06, size = 0, normalized size = 0.00 \[\int \left (f x \right )^{-1-n} \ln \left (c \left (d +e \,x^{n}\right )^{p}\right )\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 71, normalized size = 0.89 \begin {gather*} \frac {e p {\left (\frac {\log \left (x\right )}{d f^{n}} - \frac {\log \left (\frac {e x^{n} + d}{e}\right )}{d f^{n} n}\right )}}{f} - \frac {\log \left ({\left (e x^{n} + d\right )}^{p} c\right )}{\left (f x\right )^{n} f n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 78, normalized size = 0.98 \begin {gather*} \frac {f^{-n - 1} n p x^{n} e \log \left (x\right ) - d f^{-n - 1} \log \left (c\right ) - {\left (f^{-n - 1} p x^{n} e + d f^{-n - 1} p\right )} \log \left (x^{n} e + d\right )}{d n x^{n}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (f x\right )^{- n - 1} \log {\left (c \left (d + e x^{n}\right )^{p} \right )}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {\ln \left (c\,{\left (d+e\,x^n\right )}^p\right )}{{\left (f\,x\right )}^{n+1}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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