Optimal. Leaf size=69 \[ -\frac {p (f x)^n}{f n}+\frac {d p x^{-n} (f x)^n \log \left (d+e x^n\right )}{e f n}+\frac {(f x)^n \log \left (c \left (d+e x^n\right )^p\right )}{f n} \]
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Rubi [A]
time = 0.03, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2505, 20, 272,
45} \begin {gather*} \frac {(f x)^n \log \left (c \left (d+e x^n\right )^p\right )}{f n}+\frac {d p x^{-n} (f x)^n \log \left (d+e x^n\right )}{e f n}-\frac {p (f x)^n}{f n} \end {gather*}
Antiderivative was successfully verified.
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Rule 20
Rule 45
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 {x^{-1+2 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 ) \text {Subst}\left (\int \frac {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 \left (\frac {1}{e}-\frac {d}{e (d+e x)}\right ) \, dx,x,x^n\right )}{f n}\\ &=-\frac {p (f x)^n}{f n}+\frac {d p x^{-n} (f x)^n \log \left (d+e x^n\right )}{e 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 = 0.03, size = 48, normalized size = 0.70 \begin {gather*} \frac {x^{1-n} (f x)^{-1+n} \left (-p x^n+\frac {\left (d+e x^n\right ) \log \left (c \left (d+e x^n\right )^p\right )}{e}\right )}{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 = 0.29, size = 70, normalized size = 1.01 \begin {gather*} -\frac {e p {\left (\frac {f^{n} x^{n}}{e n} - \frac {d f^{n} \log \left (\frac {e x^{n} + d}{e}\right )}{e^{2} n}\right )}}{f} + \frac {\left (f x\right )^{n} \log \left ({\left (e x^{n} + d\right )}^{p} c\right )}{f n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.48, size = 60, normalized size = 0.87 \begin {gather*} -\frac {{\left ({\left (p e - e \log \left (c\right )\right )} f^{n - 1} x^{n} - {\left (f^{n - 1} p x^{n} e + d f^{n - 1} p\right )} \log \left (x^{n} e + d\right )\right )} e^{\left (-1\right )}}{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 \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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