Optimal. Leaf size=26 \[ \frac{2 \sqrt{\sqrt{a+b^2 x^2}+b x}}{b} \]
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Rubi [A] time = 0.0970069, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.057, Rules used = {2122, 30} \[ \frac{2 \sqrt{\sqrt{a+b^2 x^2}+b x}}{b} \]
Antiderivative was successfully verified.
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Rule 2122
Rule 30
Rubi steps
\begin{align*} \int \frac{\sqrt{b x+\sqrt{a+b^2 x^2}}}{\sqrt{a+b^2 x^2}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt{x}} \, dx,x,b x+\sqrt{a+b^2 x^2}\right )}{b}\\ &=\frac{2 \sqrt{b x+\sqrt{a+b^2 x^2}}}{b}\\ \end{align*}
Mathematica [A] time = 0.0170815, size = 26, normalized size = 1. \[ \frac{2 \sqrt{\sqrt{a+b^2 x^2}+b x}}{b} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.015, size = 0, normalized size = 0. \begin{align*} \int{\sqrt{bx+\sqrt{{b}^{2}{x}^{2}+a}}{\frac{1}{\sqrt{{b}^{2}{x}^{2}+a}}}}\, 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{\sqrt{b x + \sqrt{b^{2} x^{2} + a}}}{\sqrt{b^{2} x^{2} + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.97501, size = 47, normalized size = 1.81 \begin{align*} \frac{2 \, \sqrt{b x + \sqrt{b^{2} x^{2} + a}}}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.879158, size = 27, normalized size = 1.04 \begin{align*} \begin{cases} \frac{2 \sqrt{b x + \sqrt{a + b^{2} x^{2}}}}{b} & \text{for}\: b \neq 0 \\\frac{x}{\sqrt [4]{a}} & \text{otherwise} \end{cases} \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{\sqrt{b x + \sqrt{b^{2} x^{2} + a}}}{\sqrt{b^{2} x^{2} + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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