Optimal. Leaf size=45 \[ \frac{2 \sqrt{a+b x^n}}{n}-\frac{2 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+b x^n}}{\sqrt{a}}\right )}{n} \]
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Rubi [A] time = 0.0233814, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {266, 50, 63, 208} \[ \frac{2 \sqrt{a+b x^n}}{n}-\frac{2 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+b x^n}}{\sqrt{a}}\right )}{n} \]
Antiderivative was successfully verified.
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Rule 266
Rule 50
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{\sqrt{a+b x^n}}{x} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\sqrt{a+b x}}{x} \, dx,x,x^n\right )}{n}\\ &=\frac{2 \sqrt{a+b x^n}}{n}+\frac{a \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,x^n\right )}{n}\\ &=\frac{2 \sqrt{a+b x^n}}{n}+\frac{(2 a) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b x^n}\right )}{b n}\\ &=\frac{2 \sqrt{a+b x^n}}{n}-\frac{2 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+b x^n}}{\sqrt{a}}\right )}{n}\\ \end{align*}
Mathematica [A] time = 0.0116595, size = 43, normalized size = 0.96 \[ \frac{2 \sqrt{a+b x^n}-2 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+b x^n}}{\sqrt{a}}\right )}{n} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 36, normalized size = 0.8 \begin{align*}{\frac{1}{n} \left ( 2\,\sqrt{a+b{x}^{n}}-2\,\sqrt{a}{\it Artanh} \left ({\frac{\sqrt{a+b{x}^{n}}}{\sqrt{a}}} \right ) \right ) } \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 = 1.07658, size = 212, normalized size = 4.71 \begin{align*} \left [\frac{\sqrt{a} \log \left (\frac{b x^{n} - 2 \, \sqrt{b x^{n} + a} \sqrt{a} + 2 \, a}{x^{n}}\right ) + 2 \, \sqrt{b x^{n} + a}}{n}, \frac{2 \,{\left (\sqrt{-a} \arctan \left (\frac{\sqrt{b x^{n} + a} \sqrt{-a}}{a}\right ) + \sqrt{b x^{n} + a}\right )}}{n}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 1.54809, size = 76, normalized size = 1.69 \begin{align*} - \frac{2 \sqrt{a} \operatorname{asinh}{\left (\frac{\sqrt{a} x^{- \frac{n}{2}}}{\sqrt{b}} \right )}}{n} + \frac{2 a x^{- \frac{n}{2}}}{\sqrt{b} n \sqrt{\frac{a x^{- n}}{b} + 1}} + \frac{2 \sqrt{b} x^{\frac{n}{2}}}{n \sqrt{\frac{a x^{- n}}{b} + 1}} \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{\sqrt{b x^{n} + a}}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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