Optimal. Leaf size=42 \[ x \sqrt {\frac {a}{x^2}+b}-\sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a}}{x \sqrt {\frac {a}{x^2}+b}}\right ) \]
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Rubi [A] time = 0.02, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1972, 242, 277, 217, 206} \begin {gather*} x \sqrt {\frac {a}{x^2}+b}-\sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a}}{x \sqrt {\frac {a}{x^2}+b}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 217
Rule 242
Rule 277
Rule 1972
Rubi steps
\begin {align*} \int \sqrt {\frac {a+b x^2}{x^2}} \, dx &=\int \sqrt {b+\frac {a}{x^2}} \, dx\\ &=-\operatorname {Subst}\left (\int \frac {\sqrt {b+a x^2}}{x^2} \, dx,x,\frac {1}{x}\right )\\ &=\sqrt {b+\frac {a}{x^2}} x-a \operatorname {Subst}\left (\int \frac {1}{\sqrt {b+a x^2}} \, dx,x,\frac {1}{x}\right )\\ &=\sqrt {b+\frac {a}{x^2}} x-a \operatorname {Subst}\left (\int \frac {1}{1-a x^2} \, dx,x,\frac {1}{\sqrt {b+\frac {a}{x^2}} x}\right )\\ &=\sqrt {b+\frac {a}{x^2}} x-\sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a}}{\sqrt {b+\frac {a}{x^2}} x}\right )\\ \end {align*}
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Mathematica [A] time = 0.03, size = 62, normalized size = 1.48 \begin {gather*} x \sqrt {\frac {a}{x^2}+b}-\frac {\sqrt {a} x \sqrt {\frac {a}{x^2}+b} \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{\sqrt {a+b x^2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 3.84, size = 61, normalized size = 1.45 \begin {gather*} \frac {x \sqrt {\frac {a}{x^2}+b} \left (\sqrt {a+b x^2}-\sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )\right )}{\sqrt {a+b x^2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 108, normalized size = 2.57 \begin {gather*} \left [x \sqrt {\frac {b x^{2} + a}{x^{2}}} + \frac {1}{2} \, \sqrt {a} \log \left (-\frac {b x^{2} - 2 \, \sqrt {a} x \sqrt {\frac {b x^{2} + a}{x^{2}}} + 2 \, a}{x^{2}}\right ), x \sqrt {\frac {b x^{2} + a}{x^{2}}} + \sqrt {-a} \arctan \left (\frac {\sqrt {-a} x \sqrt {\frac {b x^{2} + a}{x^{2}}}}{b x^{2} + a}\right )\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.17, size = 69, normalized size = 1.64 \begin {gather*} \frac {a \arctan \left (\frac {\sqrt {b x^{2} + a}}{\sqrt {-a}}\right ) \mathrm {sgn}\relax (x)}{\sqrt {-a}} + \sqrt {b x^{2} + a} \mathrm {sgn}\relax (x) - \frac {{\left (a \arctan \left (\frac {\sqrt {a}}{\sqrt {-a}}\right ) + \sqrt {-a} \sqrt {a}\right )} \mathrm {sgn}\relax (x)}{\sqrt {-a}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 61, normalized size = 1.45 \begin {gather*} \frac {\sqrt {\frac {b \,x^{2}+a}{x^{2}}}\, \left (-\sqrt {a}\, \ln \left (\frac {2 a +2 \sqrt {b \,x^{2}+a}\, \sqrt {a}}{x}\right )+\sqrt {b \,x^{2}+a}\right ) x}{\sqrt {b \,x^{2}+a}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.94, size = 53, normalized size = 1.26 \begin {gather*} \sqrt {b + \frac {a}{x^{2}}} x + \frac {1}{2} \, \sqrt {a} \log \left (\frac {\sqrt {b + \frac {a}{x^{2}}} x - \sqrt {a}}{\sqrt {b + \frac {a}{x^{2}}} x + \sqrt {a}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.57, size = 55, normalized size = 1.31 \begin {gather*} x\,\sqrt {b+\frac {a}{x^2}}+\frac {\sqrt {a}\,\mathrm {asin}\left (\frac {\sqrt {a}\,1{}\mathrm {i}}{\sqrt {b}\,x}\right )\,\sqrt {b+\frac {a}{x^2}}\,1{}\mathrm {i}}{\sqrt {b}\,\sqrt {\frac {a}{b\,x^2}+1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {\frac {a + b x^{2}}{x^{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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