Optimal. Leaf size=39 \[ x \sqrt {a+\frac {b}{x}}+\frac {b \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{\sqrt {a}} \]
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Rubi [A] time = 0.02, antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {242, 47, 63, 208} \[ x \sqrt {a+\frac {b}{x}}+\frac {b \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{\sqrt {a}} \]
Antiderivative was successfully verified.
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Rule 47
Rule 63
Rule 208
Rule 242
Rubi steps
\begin {align*} \int \sqrt {a+\frac {b}{x}} \, dx &=-\operatorname {Subst}\left (\int \frac {\sqrt {a+b x}}{x^2} \, dx,x,\frac {1}{x}\right )\\ &=\sqrt {a+\frac {b}{x}} x-\frac {1}{2} b \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\frac {1}{x}\right )\\ &=\sqrt {a+\frac {b}{x}} x-\operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+\frac {b}{x}}\right )\\ &=\sqrt {a+\frac {b}{x}} x+\frac {b \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{\sqrt {a}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 39, normalized size = 1.00 \[ x \sqrt {a+\frac {b}{x}}+\frac {b \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{\sqrt {a}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.05, size = 99, normalized size = 2.54 \[ \left [\frac {2 \, a x \sqrt {\frac {a x + b}{x}} + \sqrt {a} b \log \left (2 \, a x + 2 \, \sqrt {a} x \sqrt {\frac {a x + b}{x}} + b\right )}{2 \, a}, \frac {a x \sqrt {\frac {a x + b}{x}} - \sqrt {-a} b \arctan \left (\frac {\sqrt {-a} \sqrt {\frac {a x + b}{x}}}{a}\right )}{a}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.20, size = 64, normalized size = 1.64 \[ -\frac {b \log \left ({\left | -2 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )} \sqrt {a} - b \right |}\right ) \mathrm {sgn}\relax (x)}{2 \, \sqrt {a}} + \frac {b \log \left ({\left | b \right |}\right ) \mathrm {sgn}\relax (x)}{2 \, \sqrt {a}} + \sqrt {a x^{2} + b x} \mathrm {sgn}\relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 74, normalized size = 1.90 \[ \frac {\sqrt {\frac {a x +b}{x}}\, \left (b \ln \left (\frac {2 a x +b +2 \sqrt {a \,x^{2}+b x}\, \sqrt {a}}{2 \sqrt {a}}\right )+2 \sqrt {a \,x^{2}+b x}\, \sqrt {a}\right ) x}{2 \sqrt {\left (a x +b \right ) x}\, \sqrt {a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.38, size = 50, normalized size = 1.28 \[ \sqrt {a + \frac {b}{x}} x - \frac {b \log \left (\frac {\sqrt {a + \frac {b}{x}} - \sqrt {a}}{\sqrt {a + \frac {b}{x}} + \sqrt {a}}\right )}{2 \, \sqrt {a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.09, size = 58, normalized size = 1.49 \[ x\,\sqrt {a\,x^2+b\,x}\,\sqrt {\frac {1}{x^2}}+\frac {b\,x\,\ln \left (\frac {\frac {b}{2}+a\,x+\sqrt {a}\,\sqrt {a\,x^2+b\,x}}{\sqrt {a}}\right )\,\sqrt {\frac {1}{x^2}}}{2\,\sqrt {a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.01, size = 42, normalized size = 1.08 \[ \sqrt {b} \sqrt {x} \sqrt {\frac {a x}{b} + 1} + \frac {b \operatorname {asinh}{\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}} \right )}}{\sqrt {a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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