Optimal. Leaf size=37 \[ \sqrt {a+b x^2}-\sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right ) \]
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Rubi [A] time = 0.02, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {266, 50, 63, 208} \[ \sqrt {a+b x^2}-\sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right ) \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 208
Rule 266
Rubi steps
\begin {align*} \int \frac {\sqrt {a+b x^2}}{x} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {\sqrt {a+b x}}{x} \, dx,x,x^2\right )\\ &=\sqrt {a+b x^2}+\frac {1}{2} a \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,x^2\right )\\ &=\sqrt {a+b x^2}+\frac {a \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x^2}\right )}{b}\\ &=\sqrt {a+b x^2}-\sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 37, normalized size = 1.00 \[ \sqrt {a+b x^2}-\sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.99, size = 77, normalized size = 2.08 \[ \left [\frac {1}{2} \, \sqrt {a} \log \left (-\frac {b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) + \sqrt {b x^{2} + a}, \sqrt {-a} \arctan \left (\frac {\sqrt {-a}}{\sqrt {b x^{2} + a}}\right ) + \sqrt {b x^{2} + a}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.59, size = 33, normalized size = 0.89 \[ \frac {a \arctan \left (\frac {\sqrt {b x^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a}} + \sqrt {b x^{2} + a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 39, normalized size = 1.05 \[ -\sqrt {a}\, \ln \left (\frac {2 a +2 \sqrt {b \,x^{2}+a}\, \sqrt {a}}{x}\right )+\sqrt {b \,x^{2}+a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.36, size = 27, normalized size = 0.73 \[ -\sqrt {a} \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right ) + \sqrt {b x^{2} + a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.66, size = 29, normalized size = 0.78 \[ \sqrt {b\,x^2+a}-\sqrt {a}\,\mathrm {atanh}\left (\frac {\sqrt {b\,x^2+a}}{\sqrt {a}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.41, size = 56, normalized size = 1.51 \[ - \sqrt {a} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x} \right )} + \frac {a}{\sqrt {b} x \sqrt {\frac {a}{b x^{2}} + 1}} + \frac {\sqrt {b} x}{\sqrt {\frac {a}{b x^{2}} + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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