Optimal. Leaf size=42 \[ x \sqrt {a+\frac {b}{x^2}}-\sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b}}{x \sqrt {a+\frac {b}{x^2}}}\right ) \]
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Rubi [A] time = 0.02, antiderivative size = 42, 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, 277, 217, 206} \[ x \sqrt {a+\frac {b}{x^2}}-\sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b}}{x \sqrt {a+\frac {b}{x^2}}}\right ) \]
Antiderivative was successfully verified.
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Rule 206
Rule 217
Rule 242
Rule 277
Rubi steps
\begin {align*} \int \sqrt {a+\frac {b}{x^2}} \, dx &=-\operatorname {Subst}\left (\int \frac {\sqrt {a+b x^2}}{x^2} \, dx,x,\frac {1}{x}\right )\\ &=\sqrt {a+\frac {b}{x^2}} x-b \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\frac {1}{x}\right )\\ &=\sqrt {a+\frac {b}{x^2}} x-b \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {1}{\sqrt {a+\frac {b}{x^2}} x}\right )\\ &=\sqrt {a+\frac {b}{x^2}} x-\sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b}}{\sqrt {a+\frac {b}{x^2}} x}\right )\\ \end {align*}
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Mathematica [A] time = 0.03, size = 62, normalized size = 1.48 \[ x \sqrt {a+\frac {b}{x^2}}-\frac {\sqrt {b} x \sqrt {a+\frac {b}{x^2}} \tanh ^{-1}\left (\frac {\sqrt {a x^2+b}}{\sqrt {b}}\right )}{\sqrt {a x^2+b}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.92, size = 108, normalized size = 2.57 \[ \left [x \sqrt {\frac {a x^{2} + b}{x^{2}}} + \frac {1}{2} \, \sqrt {b} \log \left (-\frac {a x^{2} - 2 \, \sqrt {b} x \sqrt {\frac {a x^{2} + b}{x^{2}}} + 2 \, b}{x^{2}}\right ), x \sqrt {\frac {a x^{2} + b}{x^{2}}} + \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x \sqrt {\frac {a x^{2} + b}{x^{2}}}}{a x^{2} + b}\right )\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.16, size = 69, normalized size = 1.64 \[ \frac {b \arctan \left (\frac {\sqrt {a x^{2} + b}}{\sqrt {-b}}\right ) \mathrm {sgn}\relax (x)}{\sqrt {-b}} + \sqrt {a x^{2} + b} \mathrm {sgn}\relax (x) - \frac {{\left (b \arctan \left (\frac {\sqrt {b}}{\sqrt {-b}}\right ) + \sqrt {-b} \sqrt {b}\right )} \mathrm {sgn}\relax (x)}{\sqrt {-b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 63, normalized size = 1.50 \[ -\frac {\sqrt {\frac {a \,x^{2}+b}{x^{2}}}\, \left (\sqrt {b}\, \ln \left (\frac {2 b +2 \sqrt {a \,x^{2}+b}\, \sqrt {b}}{x}\right )-\sqrt {a \,x^{2}+b}\right ) x}{\sqrt {a \,x^{2}+b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.97, size = 53, normalized size = 1.26 \[ \sqrt {a + \frac {b}{x^{2}}} x + \frac {1}{2} \, \sqrt {b} \log \left (\frac {\sqrt {a + \frac {b}{x^{2}}} x - \sqrt {b}}{\sqrt {a + \frac {b}{x^{2}}} x + \sqrt {b}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.14, size = 55, normalized size = 1.31 \[ x\,\sqrt {a+\frac {b}{x^2}}+\frac {\sqrt {b}\,\mathrm {asin}\left (\frac {\sqrt {b}\,1{}\mathrm {i}}{\sqrt {a}\,x}\right )\,\sqrt {a+\frac {b}{x^2}}\,1{}\mathrm {i}}{\sqrt {a}\,\sqrt {\frac {b}{a\,x^2}+1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.58, size = 56, normalized size = 1.33 \[ \frac {\sqrt {a} x}{\sqrt {1 + \frac {b}{a x^{2}}}} - \sqrt {b} \operatorname {asinh}{\left (\frac {\sqrt {b}}{\sqrt {a} x} \right )} + \frac {b}{\sqrt {a} x \sqrt {1 + \frac {b}{a x^{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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