Optimal. Leaf size=61 \[ \frac{2 x \sqrt{\frac{a}{x^2}+b x^n}}{n+2}-\frac{2 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a}}{x \sqrt{\frac{a}{x^2}+b x^n}}\right )}{n+2} \]
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Rubi [A] time = 0.0834905, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {2007, 2029, 206} \[ \frac{2 x \sqrt{\frac{a}{x^2}+b x^n}}{n+2}-\frac{2 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a}}{x \sqrt{\frac{a}{x^2}+b x^n}}\right )}{n+2} \]
Antiderivative was successfully verified.
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Rule 2007
Rule 2029
Rule 206
Rubi steps
\begin{align*} \int \sqrt{\frac{a}{x^2}+b x^n} \, dx &=\frac{2 x \sqrt{\frac{a}{x^2}+b x^n}}{2+n}+a \int \frac{1}{x^2 \sqrt{\frac{a}{x^2}+b x^n}} \, dx\\ &=\frac{2 x \sqrt{\frac{a}{x^2}+b x^n}}{2+n}-\frac{(2 a) \operatorname{Subst}\left (\int \frac{1}{1-a x^2} \, dx,x,\frac{1}{x \sqrt{\frac{a}{x^2}+b x^n}}\right )}{2+n}\\ &=\frac{2 x \sqrt{\frac{a}{x^2}+b x^n}}{2+n}-\frac{2 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a}}{x \sqrt{\frac{a}{x^2}+b x^n}}\right )}{2+n}\\ \end{align*}
Mathematica [A] time = 0.0532376, size = 78, normalized size = 1.28 \[ \frac{x \sqrt{\frac{a}{x^2}+b x^n} \left (2 \sqrt{a+b x^{n+2}}-2 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+b x^{n+2}}}{\sqrt{a}}\right )\right )}{(n+2) \sqrt{a+b x^{n+2}}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.34, size = 0, normalized size = 0. \begin{align*} \int \sqrt{{\frac{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 \sqrt{b x^{n} + \frac{a}{x^{2}}}\,{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 \sqrt{\frac{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 \sqrt{b x^{n} + \frac{a}{x^{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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