Optimal. Leaf size=51 \[ -\frac{x^{-n} \sqrt{a+b x^n}}{n}-\frac{b \tanh ^{-1}\left (\frac{\sqrt{a+b x^n}}{\sqrt{a}}\right )}{\sqrt{a} n} \]
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Rubi [A] time = 0.0244006, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {266, 47, 63, 208} \[ -\frac{x^{-n} \sqrt{a+b x^n}}{n}-\frac{b \tanh ^{-1}\left (\frac{\sqrt{a+b x^n}}{\sqrt{a}}\right )}{\sqrt{a} n} \]
Antiderivative was successfully verified.
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Rule 266
Rule 47
Rule 63
Rule 208
Rubi steps
\begin{align*} \int x^{-1-n} \sqrt{a+b x^n} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\sqrt{a+b x}}{x^2} \, dx,x,x^n\right )}{n}\\ &=-\frac{x^{-n} \sqrt{a+b x^n}}{n}+\frac{b \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,x^n\right )}{2 n}\\ &=-\frac{x^{-n} \sqrt{a+b x^n}}{n}+\frac{\operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b x^n}\right )}{n}\\ &=-\frac{x^{-n} \sqrt{a+b x^n}}{n}-\frac{b \tanh ^{-1}\left (\frac{\sqrt{a+b x^n}}{\sqrt{a}}\right )}{\sqrt{a} n}\\ \end{align*}
Mathematica [A] time = 0.0394481, size = 62, normalized size = 1.22 \[ -\frac{x^{-n} \left (b x^n \sqrt{\frac{b x^n}{a}+1} \tanh ^{-1}\left (\sqrt{\frac{b x^n}{a}+1}\right )+a+b x^n\right )}{n \sqrt{a+b x^n}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.191, size = 0, normalized size = 0. \begin{align*} \int{x}^{-1-n}\sqrt{a+b{x}^{n}}\, dx \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 \sqrt{b x^{n} + a} x^{-n - 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.15808, size = 258, normalized size = 5.06 \begin{align*} \left [\frac{\sqrt{a} b x^{n} \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} a}{2 \, a n x^{n}}, \frac{\sqrt{-a} b x^{n} \arctan \left (\frac{\sqrt{b x^{n} + a} \sqrt{-a}}{a}\right ) - \sqrt{b x^{n} + a} a}{a n x^{n}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 131.743, size = 49, normalized size = 0.96 \begin{align*} - \frac{\sqrt{b} x^{- \frac{n}{2}} \sqrt{\frac{a x^{- n}}{b} + 1}}{n} - \frac{b \operatorname{asinh}{\left (\frac{\sqrt{a} x^{- \frac{n}{2}}}{\sqrt{b}} \right )}}{\sqrt{a} 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 \sqrt{b x^{n} + a} x^{-n - 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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