Optimal. Leaf size=112 \[ -\frac {p (f x)^{2 n}}{4 f n}+\frac {d p x^{-n} (f x)^{2 n}}{2 e f n}-\frac {d^2 p x^{-2 n} (f x)^{2 n} \log \left (d+e x^n\right )}{2 e^2 f n}+\frac {(f x)^{2 n} \log \left (c \left (d+e x^n\right )^p\right )}{2 f n} \]
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Rubi [A]
time = 0.04, antiderivative size = 112, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {2505, 20, 272,
45} \begin {gather*} \frac {(f x)^{2 n} \log \left (c \left (d+e x^n\right )^p\right )}{2 f n}-\frac {d^2 p x^{-2 n} (f x)^{2 n} \log \left (d+e x^n\right )}{2 e^2 f n}+\frac {d p x^{-n} (f x)^{2 n}}{2 e f n}-\frac {p (f x)^{2 n}}{4 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+2 n} \log \left (c \left (d+e x^n\right )^p\right ) \, dx &=\frac {(f x)^{2 n} \log \left (c \left (d+e x^n\right )^p\right )}{2 f n}-\frac {(e p) \int \frac {x^{-1+n} (f x)^{2 n}}{d+e x^n} \, dx}{2 f}\\ &=\frac {(f x)^{2 n} \log \left (c \left (d+e x^n\right )^p\right )}{2 f n}-\frac {\left (e p x^{-2 n} (f x)^{2 n}\right ) \int \frac {x^{-1+3 n}}{d+e x^n} \, dx}{2 f}\\ &=\frac {(f x)^{2 n} \log \left (c \left (d+e x^n\right )^p\right )}{2 f n}-\frac {\left (e p x^{-2 n} (f x)^{2 n}\right ) \text {Subst}\left (\int \frac {x^2}{d+e x} \, dx,x,x^n\right )}{2 f n}\\ &=\frac {(f x)^{2 n} \log \left (c \left (d+e x^n\right )^p\right )}{2 f n}-\frac {\left (e p x^{-2 n} (f x)^{2 n}\right ) \text {Subst}\left (\int \left (-\frac {d}{e^2}+\frac {x}{e}+\frac {d^2}{e^2 (d+e x)}\right ) \, dx,x,x^n\right )}{2 f n}\\ &=-\frac {p (f x)^{2 n}}{4 f n}+\frac {d p x^{-n} (f x)^{2 n}}{2 e f n}-\frac {d^2 p x^{-2 n} (f x)^{2 n} \log \left (d+e x^n\right )}{2 e^2 f n}+\frac {(f x)^{2 n} \log \left (c \left (d+e x^n\right )^p\right )}{2 f n}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 74, normalized size = 0.66 \begin {gather*} -\frac {x^{-2 n} (f x)^{2 n} \left (2 d^2 p \log \left (d+e x^n\right )+e x^n \left (-2 d p+e p x^n-2 e x^n \log \left (c \left (d+e x^n\right )^p\right )\right )\right )}{4 e^2 f n} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.05, size = 0, normalized size = 0.00 \[\int \left (f x \right )^{-1+2 n} \ln \left (c \left (d +e \,x^{n}\right )^{p}\right )\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 95, normalized size = 0.85 \begin {gather*} -\frac {e p {\left (\frac {2 \, d^{2} f^{2 \, n} \log \left (\frac {e x^{n} + d}{e}\right )}{e^{3} n} + \frac {e f^{2 \, n} x^{2 \, n} - 2 \, d f^{2 \, n} x^{n}}{e^{2} n}\right )}}{4 \, f} + \frac {\left (f x\right )^{2 \, n} \log \left ({\left (e x^{n} + d\right )}^{p} c\right )}{2 \, f n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 90, normalized size = 0.80 \begin {gather*} \frac {{\left (2 \, d f^{2 \, n - 1} p x^{n} e - {\left (p e^{2} - 2 \, e^{2} \log \left (c\right )\right )} f^{2 \, n - 1} x^{2 \, n} - 2 \, {\left (d^{2} f^{2 \, n - 1} p - f^{2 \, n - 1} p x^{2 \, n} e^{2}\right )} \log \left (x^{n} e + d\right )\right )} e^{\left (-2\right )}}{4 \, 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 )\,{\left (f\,x\right )}^{2\,n-1} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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