Optimal. Leaf size=42 \[ -\frac {\sqrt {a+b x^2}}{x}+\sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {283, 223, 212}
\begin {gather*} \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )-\frac {\sqrt {a+b x^2}}{x} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 223
Rule 283
Rubi steps
\begin {align*} \int \frac {\sqrt {a+b x^2}}{x^2} \, dx &=-\frac {\sqrt {a+b x^2}}{x}+b \int \frac {1}{\sqrt {a+b x^2}} \, dx\\ &=-\frac {\sqrt {a+b x^2}}{x}+b \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right )\\ &=-\frac {\sqrt {a+b x^2}}{x}+\sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 45, normalized size = 1.07 \begin {gather*} -\frac {\sqrt {a+b x^2}}{x}-\sqrt {b} \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.03, size = 60, normalized size = 1.43
method | result | size |
risch | \(-\frac {\sqrt {b \,x^{2}+a}}{x}+\sqrt {b}\, \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )\) | \(36\) |
default | \(-\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{a x}+\frac {2 b \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{a}\) | \(60\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 28, normalized size = 0.67 \begin {gather*} \sqrt {b} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right ) - \frac {\sqrt {b x^{2} + a}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.22, size = 88, normalized size = 2.10 \begin {gather*} \left [\frac {\sqrt {b} x \log \left (-2 \, b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) - 2 \, \sqrt {b x^{2} + a}}{2 \, x}, -\frac {\sqrt {-b} x \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) + \sqrt {b x^{2} + a}}{x}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.74, size = 56, normalized size = 1.33 \begin {gather*} - \frac {\sqrt {a}}{x \sqrt {1 + \frac {b x^{2}}{a}}} + \sqrt {b} \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )} - \frac {b x}{\sqrt {a} \sqrt {1 + \frac {b x^{2}}{a}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.53, size = 57, normalized size = 1.36 \begin {gather*} -\frac {1}{2} \, \sqrt {b} \log \left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2}\right ) + \frac {2 \, a \sqrt {b}}{{\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.82, size = 56, normalized size = 1.33 \begin {gather*} -\frac {\sqrt {b\,x^2+a}}{x}-\frac {\sqrt {b}\,\mathrm {asin}\left (\frac {\sqrt {b}\,x\,1{}\mathrm {i}}{\sqrt {a}}\right )\,\sqrt {b\,x^2+a}\,1{}\mathrm {i}}{\sqrt {a}\,\sqrt {\frac {b\,x^2}{a}+1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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