Optimal. Leaf size=47 \[ \frac {1}{2} \sqrt {a+\frac {b}{x^2}} x^2+\frac {b \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x^2}}}{\sqrt {a}}\right )}{2 \sqrt {a}} \]
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Rubi [A]
time = 0.02, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {272, 43, 65,
214} \begin {gather*} \frac {1}{2} x^2 \sqrt {a+\frac {b}{x^2}}+\frac {b \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x^2}}}{\sqrt {a}}\right )}{2 \sqrt {a}} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 65
Rule 214
Rule 272
Rubi steps
\begin {align*} \int \sqrt {a+\frac {b}{x^2}} x \, dx &=-\left (\frac {1}{2} \text {Subst}\left (\int \frac {\sqrt {a+b x}}{x^2} \, dx,x,\frac {1}{x^2}\right )\right )\\ &=\frac {1}{2} \sqrt {a+\frac {b}{x^2}} x^2-\frac {1}{4} b \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\frac {1}{x^2}\right )\\ &=\frac {1}{2} \sqrt {a+\frac {b}{x^2}} x^2-\frac {1}{2} \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+\frac {b}{x^2}}\right )\\ &=\frac {1}{2} \sqrt {a+\frac {b}{x^2}} x^2+\frac {b \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x^2}}}{\sqrt {a}}\right )}{2 \sqrt {a}}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 58, normalized size = 1.23 \begin {gather*} \frac {1}{2} \sqrt {a+\frac {b}{x^2}} x \left (x-\frac {b \log \left (-\sqrt {a} x+\sqrt {b+a x^2}\right )}{\sqrt {a} \sqrt {b+a x^2}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 62, normalized size = 1.32
method | result | size |
default | \(\frac {\sqrt {\frac {a \,x^{2}+b}{x^{2}}}\, x \left (x \sqrt {a \,x^{2}+b}\, \sqrt {a}+b \ln \left (x \sqrt {a}+\sqrt {a \,x^{2}+b}\right )\right )}{2 \sqrt {a \,x^{2}+b}\, \sqrt {a}}\) | \(62\) |
risch | \(\frac {\sqrt {\frac {a \,x^{2}+b}{x^{2}}}\, x^{2}}{2}+\frac {b \ln \left (x \sqrt {a}+\sqrt {a \,x^{2}+b}\right ) \sqrt {\frac {a \,x^{2}+b}{x^{2}}}\, x}{2 \sqrt {a}\, \sqrt {a \,x^{2}+b}}\) | \(65\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 53, normalized size = 1.13 \begin {gather*} \frac {1}{2} \, \sqrt {a + \frac {b}{x^{2}}} x^{2} - \frac {b \log \left (\frac {\sqrt {a + \frac {b}{x^{2}}} - \sqrt {a}}{\sqrt {a + \frac {b}{x^{2}}} + \sqrt {a}}\right )}{4 \, \sqrt {a}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 127, normalized size = 2.70 \begin {gather*} \left [\frac {2 \, a x^{2} \sqrt {\frac {a x^{2} + b}{x^{2}}} + \sqrt {a} b \log \left (-2 \, a x^{2} - 2 \, \sqrt {a} x^{2} \sqrt {\frac {a x^{2} + b}{x^{2}}} - b\right )}{4 \, a}, \frac {a x^{2} \sqrt {\frac {a x^{2} + b}{x^{2}}} - \sqrt {-a} b \arctan \left (\frac {\sqrt {-a} x^{2} \sqrt {\frac {a x^{2} + b}{x^{2}}}}{a x^{2} + b}\right )}{2 \, a}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.94, size = 41, normalized size = 0.87 \begin {gather*} \frac {\sqrt {b} x \sqrt {\frac {a x^{2}}{b} + 1}}{2} + \frac {b \operatorname {asinh}{\left (\frac {\sqrt {a} x}{\sqrt {b}} \right )}}{2 \sqrt {a}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.83, size = 52, normalized size = 1.11 \begin {gather*} \frac {b \log \left ({\left | b \right |}\right ) \mathrm {sgn}\left (x\right )}{4 \, \sqrt {a}} + \frac {1}{2} \, {\left (\sqrt {a x^{2} + b} x - \frac {b \log \left ({\left | -\sqrt {a} x + \sqrt {a x^{2} + b} \right |}\right )}{\sqrt {a}}\right )} \mathrm {sgn}\left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.38, size = 35, normalized size = 0.74 \begin {gather*} \frac {x^2\,\sqrt {a+\frac {b}{x^2}}}{2}+\frac {b\,\mathrm {atanh}\left (\frac {\sqrt {a+\frac {b}{x^2}}}{\sqrt {a}}\right )}{2\,\sqrt {a}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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