Optimal. Leaf size=32 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {\frac {a}{x^2}+b x^2}}\right )}{2 \sqrt {b}} \]
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Rubi [A] time = 0.02, antiderivative size = 32, 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 = {1979, 2008, 206} \begin {gather*} \frac {\tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {\frac {a}{x^2}+b x^2}}\right )}{2 \sqrt {b}} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 1979
Rule 2008
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {\frac {a+b x^4}{x^2}}} \, dx &=\int \frac {1}{\sqrt {\frac {a}{x^2}+b x^2}} \, dx\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x}{\sqrt {\frac {a}{x^2}+b x^2}}\right )\\ &=\frac {\tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {\frac {a}{x^2}+b x^2}}\right )}{2 \sqrt {b}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 59, normalized size = 1.84 \begin {gather*} \frac {\sqrt {a+b x^4} \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a+b x^4}}\right )}{2 \sqrt {b} x \sqrt {\frac {a+b x^4}{x^2}}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 3.54, size = 58, normalized size = 1.81 \begin {gather*} \frac {x \sqrt {\frac {a+b x^4}{x^2}} \log \left (\sqrt {a+b x^4}+\sqrt {b} x^2\right )}{2 \sqrt {b} \sqrt {a+b x^4}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 80, normalized size = 2.50 \begin {gather*} \left [\frac {\log \left (-2 \, b x^{4} - 2 \, \sqrt {b} x^{3} \sqrt {\frac {b x^{4} + a}{x^{2}}} - a\right )}{4 \, \sqrt {b}}, -\frac {\sqrt {-b} \arctan \left (\frac {\sqrt {-b} x^{3} \sqrt {\frac {b x^{4} + a}{x^{2}}}}{b x^{4} + a}\right )}{2 \, b}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 40, normalized size = 1.25 \begin {gather*} \frac {\log \left ({\left | a \right |}\right ) \mathrm {sgn}\relax (x)}{4 \, \sqrt {b}} - \frac {\log \left ({\left | -\sqrt {b} x^{2} + \sqrt {b x^{4} + a} \right |}\right )}{2 \, \sqrt {b} \mathrm {sgn}\relax (x)} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.19, size = 49, normalized size = 1.53 \begin {gather*} \frac {\sqrt {b \,x^{4}+a}\, \ln \left (\sqrt {b}\, x^{2}+\sqrt {b \,x^{4}+a}\right )}{2 \sqrt {\frac {b \,x^{4}+a}{x^{2}}}\, \sqrt {b}\, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} b \int \frac {x^{5}}{{\left (b x^{4} + a\right )}^{\frac {3}{2}}}\,{d x} + \frac {x^{2}}{2 \, \sqrt {b x^{4} + a}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {1}{\sqrt {\frac {b\,x^4+a}{x^2}}} \,d x \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 \frac {1}{\sqrt {\frac {a + b x^{4}}{x^{2}}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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