Optimal. Leaf size=46 \[ \frac{1}{2} x \sqrt{a+b x^2}+\frac{a \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{2 \sqrt{b}} \]
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Rubi [A] time = 0.0103711, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {195, 217, 206} \[ \frac{1}{2} x \sqrt{a+b x^2}+\frac{a \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{2 \sqrt{b}} \]
Antiderivative was successfully verified.
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Rule 195
Rule 217
Rule 206
Rubi steps
\begin{align*} \int \sqrt{a+b x^2} \, dx &=\frac{1}{2} x \sqrt{a+b x^2}+\frac{1}{2} a \int \frac{1}{\sqrt{a+b x^2}} \, dx\\ &=\frac{1}{2} x \sqrt{a+b x^2}+\frac{1}{2} a \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{x}{\sqrt{a+b x^2}}\right )\\ &=\frac{1}{2} x \sqrt{a+b x^2}+\frac{a \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{2 \sqrt{b}}\\ \end{align*}
Mathematica [A] time = 0.0197803, size = 49, normalized size = 1.07 \[ \frac{1}{2} x \sqrt{a+b x^2}+\frac{a \log \left (\sqrt{b} \sqrt{a+b x^2}+b x\right )}{2 \sqrt{b}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0., size = 36, normalized size = 0.8 \begin{align*}{\frac{x}{2}\sqrt{b{x}^{2}+a}}+{\frac{a}{2}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){\frac{1}{\sqrt{b}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.57063, size = 232, normalized size = 5.04 \begin{align*} \left [\frac{2 \, \sqrt{b x^{2} + a} b x + a \sqrt{b} \log \left (-2 \, b x^{2} - 2 \, \sqrt{b x^{2} + a} \sqrt{b} x - a\right )}{4 \, b}, \frac{\sqrt{b x^{2} + a} b x - a \sqrt{-b} \arctan \left (\frac{\sqrt{-b} x}{\sqrt{b x^{2} + a}}\right )}{2 \, b}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.79875, size = 41, normalized size = 0.89 \begin{align*} \frac{\sqrt{a} x \sqrt{1 + \frac{b x^{2}}{a}}}{2} + \frac{a \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{2 \sqrt{b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10883, size = 50, normalized size = 1.09 \begin{align*} \frac{1}{2} \, \sqrt{b x^{2} + a} x - \frac{a \log \left ({\left | -\sqrt{b} x + \sqrt{b x^{2} + a} \right |}\right )}{2 \, \sqrt{b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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