Optimal. Leaf size=30 \[ \frac{x \left (c x^n\right )^{-1/n} \log \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )}{b} \]
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Rubi [A] time = 0.007532, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {254, 31} \[ \frac{x \left (c x^n\right )^{-1/n} \log \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )}{b} \]
Antiderivative was successfully verified.
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Rule 254
Rule 31
Rubi steps
\begin{align*} \int \frac{1}{a+b \left (c x^n\right )^{\frac{1}{n}}} \, dx &=\left (x \left (c x^n\right )^{-1/n}\right ) \operatorname{Subst}\left (\int \frac{1}{a+b x} \, dx,x,\left (c x^n\right )^{\frac{1}{n}}\right )\\ &=\frac{x \left (c x^n\right )^{-1/n} \log \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )}{b}\\ \end{align*}
Mathematica [A] time = 0.0034746, size = 30, normalized size = 1. \[ \frac{x \left (c x^n\right )^{-1/n} \log \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )}{b} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.093, size = 214, normalized size = 7.1 \begin{align*}{\frac{1}{b\sqrt [n]{c}}\ln \left ( b{{\rm e}^{-{\frac{i\pi \,{\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ){\it csgn} \left ( i{x}^{n} \right ) -i\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( i{x}^{n} \right ) -i\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) +i\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}+2\,n\ln \left ( x \right ) -2\,\ln \left ( c \right ) -2\,\ln \left ({x}^{n} \right ) }{2\,n}}}}x+a \right ){{\rm e}^{{\frac{i\pi \,{\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ){\it csgn} \left ( i{x}^{n} \right ) -i\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( i{x}^{n} \right ) -i\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) +i\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}+2\,n\ln \left ( x \right ) -2\,\ln \left ({x}^{n} \right ) }{2\,n}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (c x^{n}\right )^{\left (\frac{1}{n}\right )} b + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.49074, size = 46, normalized size = 1.53 \begin{align*} \frac{\log \left (b c^{\left (\frac{1}{n}\right )} x + a\right )}{b c^{\left (\frac{1}{n}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{a + b \left (c x^{n}\right )^{\frac{1}{n}}}\, dx \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{1}{\left (c x^{n}\right )^{\left (\frac{1}{n}\right )} b + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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