Optimal. Leaf size=43 \[ 4 \sqrt{a+b \sqrt{x}}-4 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{x}}}{\sqrt{a}}\right ) \]
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Rubi [A] time = 0.0217346, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235, Rules used = {266, 50, 63, 208} \[ 4 \sqrt{a+b \sqrt{x}}-4 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{x}}}{\sqrt{a}}\right ) \]
Antiderivative was successfully verified.
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Rule 266
Rule 50
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{\sqrt{a+b \sqrt{x}}}{x} \, dx &=2 \operatorname{Subst}\left (\int \frac{\sqrt{a+b x}}{x} \, dx,x,\sqrt{x}\right )\\ &=4 \sqrt{a+b \sqrt{x}}+(2 a) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,\sqrt{x}\right )\\ &=4 \sqrt{a+b \sqrt{x}}+\frac{(4 a) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b \sqrt{x}}\right )}{b}\\ &=4 \sqrt{a+b \sqrt{x}}-4 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{x}}}{\sqrt{a}}\right )\\ \end{align*}
Mathematica [A] time = 0.0106248, size = 43, normalized size = 1. \[ 4 \sqrt{a+b \sqrt{x}}-4 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{x}}}{\sqrt{a}}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 32, normalized size = 0.7 \begin{align*} -4\,{\it Artanh} \left ({\frac{\sqrt{a+b\sqrt{x}}}{\sqrt{a}}} \right ) \sqrt{a}+4\,\sqrt{a+b\sqrt{x}} \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 = 1.3125, size = 244, normalized size = 5.67 \begin{align*} \left [2 \, \sqrt{a} \log \left (\frac{b x - 2 \, \sqrt{b \sqrt{x} + a} \sqrt{a} \sqrt{x} + 2 \, a \sqrt{x}}{x}\right ) + 4 \, \sqrt{b \sqrt{x} + a}, 4 \, \sqrt{-a} \arctan \left (\frac{\sqrt{b \sqrt{x} + a} \sqrt{-a}}{a}\right ) + 4 \, \sqrt{b \sqrt{x} + a}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 1.94152, size = 75, normalized size = 1.74 \begin{align*} - 4 \sqrt{a} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} \sqrt [4]{x}} \right )} + \frac{4 a}{\sqrt{b} \sqrt [4]{x} \sqrt{\frac{a}{b \sqrt{x}} + 1}} + \frac{4 \sqrt{b} \sqrt [4]{x}}{\sqrt{\frac{a}{b \sqrt{x}} + 1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11227, size = 49, normalized size = 1.14 \begin{align*} \frac{4 \, a \arctan \left (\frac{\sqrt{b \sqrt{x} + a}}{\sqrt{-a}}\right )}{\sqrt{-a}} + 4 \, \sqrt{b \sqrt{x} + a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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