Optimal. Leaf size=69 \[ -\frac{b x^n \log (x) (c x)^{-n}}{a^2 c}+\frac{b x^n (c x)^{-n} \log \left (a+b x^n\right )}{a^2 c n}-\frac{(c x)^{-n}}{a c n} \]
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Rubi [A] time = 0.037959, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {268, 266, 44} \[ -\frac{b x^n \log (x) (c x)^{-n}}{a^2 c}+\frac{b x^n (c x)^{-n} \log \left (a+b x^n\right )}{a^2 c n}-\frac{(c x)^{-n}}{a c n} \]
Antiderivative was successfully verified.
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Rule 268
Rule 266
Rule 44
Rubi steps
\begin{align*} \int \frac{(c x)^{-1-n}}{a+b x^n} \, dx &=\frac{\left (x^n (c x)^{-n}\right ) \int \frac{x^{-1-n}}{a+b x^n} \, dx}{c}\\ &=\frac{\left (x^n (c x)^{-n}\right ) \operatorname{Subst}\left (\int \frac{1}{x^2 (a+b x)} \, dx,x,x^n\right )}{c n}\\ &=\frac{\left (x^n (c x)^{-n}\right ) \operatorname{Subst}\left (\int \left (\frac{1}{a x^2}-\frac{b}{a^2 x}+\frac{b^2}{a^2 (a+b x)}\right ) \, dx,x,x^n\right )}{c n}\\ &=-\frac{(c x)^{-n}}{a c n}-\frac{b x^n (c x)^{-n} \log (x)}{a^2 c}+\frac{b x^n (c x)^{-n} \log \left (a+b x^n\right )}{a^2 c n}\\ \end{align*}
Mathematica [A] time = 0.0266034, size = 42, normalized size = 0.61 \[ -\frac{(c x)^{-n} \left (-b x^n \log \left (a+b x^n\right )+a+b n x^n \log (x)\right )}{a^2 c n} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.062, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( cx \right ) ^{-1-n}}{a+b{x}^{n}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.96795, size = 85, normalized size = 1.23 \begin{align*} -\frac{b c^{-n - 1} \log \left (x\right )}{a^{2}} + \frac{b c^{-n - 1} \log \left (\frac{b x^{n} + a}{b}\right )}{a^{2} n} - \frac{c^{-n - 1}}{a n x^{n}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.4321, size = 123, normalized size = 1.78 \begin{align*} -\frac{b c^{-n - 1} n x^{n} \log \left (x\right ) - b c^{-n - 1} x^{n} \log \left (b x^{n} + a\right ) + a c^{-n - 1}}{a^{2} n x^{n}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 5.06794, size = 34, normalized size = 0.49 \begin{align*} - \frac{c^{- n} x^{- n}}{a c n} + \frac{b c^{- n} \log{\left (\frac{a x^{- n}}{b} + 1 \right )}}{a^{2} c n} \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 \frac{\left (c x\right )^{-n - 1}}{b x^{n} + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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