Optimal. Leaf size=120 \[ -\frac{(f x)^{-2 n} \log \left (c \left (d+e x^n\right )^p\right )}{2 f n}-\frac{e^2 p x^{2 n} \log (x) (f x)^{-2 n}}{2 d^2 f}+\frac{e^2 p x^{2 n} (f x)^{-2 n} \log \left (d+e x^n\right )}{2 d^2 f n}-\frac{e p x^n (f x)^{-2 n}}{2 d f n} \]
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Rubi [A] time = 0.0578244, antiderivative size = 120, normalized size of antiderivative = 1., 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 = {2455, 20, 266, 44} \[ -\frac{(f x)^{-2 n} \log \left (c \left (d+e x^n\right )^p\right )}{2 f n}-\frac{e^2 p x^{2 n} \log (x) (f x)^{-2 n}}{2 d^2 f}+\frac{e^2 p x^{2 n} (f x)^{-2 n} \log \left (d+e x^n\right )}{2 d^2 f n}-\frac{e p x^n (f x)^{-2 n}}{2 d f n} \]
Antiderivative was successfully verified.
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Rule 2455
Rule 20
Rule 266
Rule 44
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-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 ) \operatorname{Subst}\left (\int \frac{1}{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 ) \operatorname{Subst}\left (\int \left (\frac{1}{d x^2}-\frac{e}{d^2 x}+\frac{e^2}{d^2 (d+e x)}\right ) \, dx,x,x^n\right )}{2 f n}\\ &=-\frac{e p x^n (f x)^{-2 n}}{2 d f n}-\frac{e^2 p x^{2 n} (f x)^{-2 n} \log (x)}{2 d^2 f}+\frac{e^2 p x^{2 n} (f x)^{-2 n} \log \left (d+e x^n\right )}{2 d^2 f n}-\frac{(f x)^{-2 n} \log \left (c \left (d+e x^n\right )^p\right )}{2 f n}\\ \end{align*}
Mathematica [A] time = 0.0481357, size = 76, normalized size = 0.63 \[ -\frac{(f x)^{-2 n} \left (d \left (d \log \left (c \left (d+e x^n\right )^p\right )+e p x^n\right )-e^2 p x^{2 n} \log \left (d+e x^n\right )+e^2 n p x^{2 n} \log (x)\right )}{2 d^2 f n} \]
Antiderivative was successfully verified.
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Maple [F] time = 1.854, size = 0, normalized size = 0. \begin{align*} \int \left ( fx \right ) ^{-1-2\,n}\ln \left ( c \left ( d+e{x}^{n} \right ) ^{p} \right ) \, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.07245, size = 240, normalized size = 2. \begin{align*} -\frac{e^{2} f^{-2 \, n - 1} n p x^{2 \, n} \log \left (x\right ) + d e f^{-2 \, n - 1} p x^{n} + d^{2} f^{-2 \, n - 1} \log \left (c\right ) -{\left (e^{2} f^{-2 \, n - 1} p x^{2 \, n} - d^{2} f^{-2 \, n - 1} p\right )} \log \left (e x^{n} + d\right )}{2 \, d^{2} n x^{2 \, n}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (f x\right )^{-2 \, n - 1} \log \left ({\left (e x^{n} + d\right )}^{p} c\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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