Optimal. Leaf size=37 \[ \frac{2 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a x^n+b x^2}}\right )}{\sqrt{b} (2-n)} \]
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Rubi [A] time = 0.0193817, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {1979, 2008, 206} \[ \frac{2 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a x^n+b x^2}}\right )}{\sqrt{b} (2-n)} \]
Antiderivative was successfully verified.
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Rule 1979
Rule 2008
Rule 206
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{x^2 \left (b+a x^{-2+n}\right )}} \, dx &=\int \frac{1}{\sqrt{b x^2+a x^n}} \, dx\\ &=\frac{2 \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{x}{\sqrt{b x^2+a x^n}}\right )}{2-n}\\ &=\frac{2 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{b x^2+a x^n}}\right )}{\sqrt{b} (2-n)}\\ \end{align*}
Mathematica [B] time = 0.021973, size = 78, normalized size = 2.11 \[ -\frac{2 \sqrt{a} x^{n/2} \sqrt{\frac{b x^{2-n}}{a}+1} \sinh ^{-1}\left (\frac{\sqrt{b} x^{1-\frac{n}{2}}}{\sqrt{a}}\right )}{\sqrt{b} (n-2) \sqrt{a x^n+b x^2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 2.16, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{\sqrt{{x}^{2} \left ( b+a{x}^{-2+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 (a x^{n - 2} + b\right )} x^{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.02494, size = 252, normalized size = 6.81 \begin{align*} \left [\frac{\sqrt{b} \log \left (\frac{a x x^{n - 2} + 2 \, b x - 2 \, \sqrt{a x^{2} x^{n - 2} + b x^{2}} \sqrt{b}}{x x^{n - 2}}\right )}{b n - 2 \, b}, \frac{2 \, \sqrt{-b} \arctan \left (\frac{\sqrt{a x^{2} x^{n - 2} + b x^{2}} \sqrt{-b}}{b x}\right )}{b n - 2 \, b}\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 (a x^{n - 2} + b\right )} x^{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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