Optimal. Leaf size=101 \[ \frac {2 p^2 x (f x)^{-1+n}}{n}-\frac {2 p x^{1-n} (f x)^{-1+n} \left (d+e x^n\right ) \log \left (c \left (d+e x^n\right )^p\right )}{e n}+\frac {x^{1-n} (f x)^{-1+n} \left (d+e x^n\right ) \log ^2\left (c \left (d+e x^n\right )^p\right )}{e n} \]
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Rubi [A]
time = 0.06, antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {2506, 2504,
2436, 2333, 2332} \begin {gather*} \frac {x^{1-n} (f x)^{n-1} \left (d+e x^n\right ) \log ^2\left (c \left (d+e x^n\right )^p\right )}{e n}-\frac {2 p x^{1-n} (f x)^{n-1} \left (d+e x^n\right ) \log \left (c \left (d+e x^n\right )^p\right )}{e n}+\frac {2 p^2 x (f x)^{n-1}}{n} \end {gather*}
Antiderivative was successfully verified.
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Rule 2332
Rule 2333
Rule 2436
Rule 2504
Rule 2506
Rubi steps
\begin {align*} \int (f x)^{-1+n} \log ^2\left (c \left (d+e x^n\right )^p\right ) \, dx &=\left (x^{1-n} (f x)^{-1+n}\right ) \int x^{-1+n} \log ^2\left (c \left (d+e x^n\right )^p\right ) \, dx\\ &=\frac {\left (x^{1-n} (f x)^{-1+n}\right ) \text {Subst}\left (\int \log ^2\left (c (d+e x)^p\right ) \, dx,x,x^n\right )}{n}\\ &=\frac {\left (x^{1-n} (f x)^{-1+n}\right ) \text {Subst}\left (\int \log ^2\left (c x^p\right ) \, dx,x,d+e x^n\right )}{e n}\\ &=\frac {x^{1-n} (f x)^{-1+n} \left (d+e x^n\right ) \log ^2\left (c \left (d+e x^n\right )^p\right )}{e n}-\frac {\left (2 p x^{1-n} (f x)^{-1+n}\right ) \text {Subst}\left (\int \log \left (c x^p\right ) \, dx,x,d+e x^n\right )}{e n}\\ &=\frac {2 p^2 x (f x)^{-1+n}}{n}-\frac {2 p x^{1-n} (f x)^{-1+n} \left (d+e x^n\right ) \log \left (c \left (d+e x^n\right )^p\right )}{e n}+\frac {x^{1-n} (f x)^{-1+n} \left (d+e x^n\right ) \log ^2\left (c \left (d+e x^n\right )^p\right )}{e n}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 74, normalized size = 0.73 \begin {gather*} \frac {x^{-n} (f x)^n \left (2 e p^2 x^n-2 p \left (d+e x^n\right ) \log \left (c \left (d+e x^n\right )^p\right )+\left (d+e x^n\right ) \log ^2\left (c \left (d+e x^n\right )^p\right )\right )}{e f n} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.08, size = 0, normalized size = 0.00 \[\int \left (f x \right )^{-1+n} \ln \left (c \left (d +e \,x^{n}\right )^{p}\right )^{2}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 146, normalized size = 1.45 \begin {gather*} -\frac {2 \, 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 )} \log \left ({\left (e x^{n} + d\right )}^{p} c\right )}{f} + \frac {\left (f x\right )^{n} \log \left ({\left (e x^{n} + d\right )}^{p} c\right )^{2}}{f n} - \frac {{\left (d f^{n} \log \left (e x^{n} + d\right )^{2} - 2 \, e f^{n} x^{n} - 2 \, {\left (f^{n} \log \left (e\right ) - f^{n}\right )} d \log \left (e x^{n} + d\right )\right )} p^{2}}{e f n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 128, normalized size = 1.27 \begin {gather*} \frac {{\left ({\left (2 \, p^{2} e - 2 \, p e \log \left (c\right ) + e \log \left (c\right )^{2}\right )} f^{n - 1} x^{n} + {\left (f^{n - 1} p^{2} x^{n} e + d f^{n - 1} p^{2}\right )} \log \left (x^{n} e + d\right )^{2} - 2 \, {\left ({\left (p^{2} e - p e \log \left (c\right )\right )} f^{n - 1} x^{n} + {\left (d p^{2} - d p \log \left (c\right )\right )} f^{n - 1}\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(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 )}^2\,{\left (f\,x\right )}^{n-1} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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