Optimal. Leaf size=37 \[ \frac{2 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a x^{2-n}+b x^2}}\right )}{\sqrt{b} n} \]
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Rubi [A] time = 0.0239837, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {1979, 2008, 206} \[ \frac{2 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a x^{2-n}+b x^2}}\right )}{\sqrt{b} n} \]
Antiderivative was successfully verified.
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Rule 1979
Rule 2008
Rule 206
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{x^{2-n} \left (a+b x^n\right )}} \, dx &=\int \frac{1}{\sqrt{b x^2+a x^{2-n}}} \, dx\\ &=\frac{2 \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{x}{\sqrt{b x^2+a x^{2-n}}}\right )}{n}\\ &=\frac{2 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{b x^2+a x^{2-n}}}\right )}{\sqrt{b} n}\\ \end{align*}
Mathematica [B] time = 0.0387811, size = 76, normalized size = 2.05 \[ \frac{2 \sqrt{a} x^{1-\frac{n}{2}} \sqrt{\frac{b x^n}{a}+1} \sinh ^{-1}\left (\frac{\sqrt{b} x^{n/2}}{\sqrt{a}}\right )}{\sqrt{b} n \sqrt{x^{2-n} \left (a+b x^n\right )}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.358, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{\sqrt{{x}^{2-n} \left ( a+b{x}^{n} \right ) }}}\, 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 \frac{1}{\sqrt{{\left (b x^{n} + a\right )} x^{-n + 2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.911825, size = 216, normalized size = 5.84 \begin{align*} \left [\frac{\log \left (\frac{2 \, b x x^{n} + a x + 2 \, \sqrt{b} x^{n} \sqrt{\frac{b x^{2} x^{n} + a x^{2}}{x^{n}}}}{x}\right )}{\sqrt{b} n}, -\frac{2 \, \sqrt{-b} \arctan \left (\frac{\sqrt{-b} \sqrt{\frac{b x^{2} x^{n} + a x^{2}}{x^{n}}}}{b x}\right )}{b n}\right ] \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 \frac{1}{\sqrt{{\left (b x^{n} + a\right )} x^{-n + 2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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