Optimal. Leaf size=37 \[ \frac{2 \tanh ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{a x^2+b x^n}}\right )}{\sqrt{a} (2-n)} \]
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Rubi [A] time = 0.015101, antiderivative size = 37, 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 = {2008, 206} \[ \frac{2 \tanh ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{a x^2+b x^n}}\right )}{\sqrt{a} (2-n)} \]
Antiderivative was successfully verified.
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Rule 2008
Rule 206
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{a x^2+b x^n}} \, dx &=\frac{2 \operatorname{Subst}\left (\int \frac{1}{1-a x^2} \, dx,x,\frac{x}{\sqrt{a x^2+b x^n}}\right )}{2-n}\\ &=\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{a x^2+b x^n}}\right )}{\sqrt{a} (2-n)}\\ \end{align*}
Mathematica [B] time = 0.0681361, size = 78, normalized size = 2.11 \[ -\frac{2 \sqrt{b} x^{n/2} \sqrt{\frac{a x^{2-n}}{b}+1} \sinh ^{-1}\left (\frac{\sqrt{a} x^{1-\frac{n}{2}}}{\sqrt{b}}\right )}{\sqrt{a} (n-2) \sqrt{a x^2+b x^n}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.326, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{\sqrt{a{x}^{2}+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 \frac{1}{\sqrt{a x^{2} + b x^{n}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \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}{\sqrt{a x^{2} + b x^{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}{\sqrt{a x^{2} + b x^{n}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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