Optimal. Leaf size=49 \[ \frac {1}{2} x^2 \sqrt {a+\frac {b}{x^4}}-\frac {1}{2} \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b}}{x^2 \sqrt {a+\frac {b}{x^4}}}\right ) \]
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Rubi [A] time = 0.04, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {335, 275, 277, 217, 206} \[ \frac {1}{2} x^2 \sqrt {a+\frac {b}{x^4}}-\frac {1}{2} \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b}}{x^2 \sqrt {a+\frac {b}{x^4}}}\right ) \]
Antiderivative was successfully verified.
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Rule 206
Rule 217
Rule 275
Rule 277
Rule 335
Rubi steps
\begin {align*} \int \sqrt {a+\frac {b}{x^4}} x \, dx &=-\operatorname {Subst}\left (\int \frac {\sqrt {a+b x^4}}{x^3} \, dx,x,\frac {1}{x}\right )\\ &=-\left (\frac {1}{2} \operatorname {Subst}\left (\int \frac {\sqrt {a+b x^2}}{x^2} \, dx,x,\frac {1}{x^2}\right )\right )\\ &=\frac {1}{2} \sqrt {a+\frac {b}{x^4}} x^2-\frac {1}{2} b \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\frac {1}{x^2}\right )\\ &=\frac {1}{2} \sqrt {a+\frac {b}{x^4}} x^2-\frac {1}{2} b \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {1}{\sqrt {a+\frac {b}{x^4}} x^2}\right )\\ &=\frac {1}{2} \sqrt {a+\frac {b}{x^4}} x^2-\frac {1}{2} \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b}}{\sqrt {a+\frac {b}{x^4}} x^2}\right )\\ \end {align*}
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Mathematica [A] time = 0.06, size = 66, normalized size = 1.35 \[ \frac {x^2 \sqrt {a+\frac {b}{x^4}} \left (\sqrt {a x^4+b}-\sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {a x^4+b}}{\sqrt {b}}\right )\right )}{2 \sqrt {a x^4+b}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.19, size = 112, normalized size = 2.29 \[ \left [\frac {1}{2} \, x^{2} \sqrt {\frac {a x^{4} + b}{x^{4}}} + \frac {1}{4} \, \sqrt {b} \log \left (\frac {a x^{4} - 2 \, \sqrt {b} x^{2} \sqrt {\frac {a x^{4} + b}{x^{4}}} + 2 \, b}{x^{4}}\right ), \frac {1}{2} \, x^{2} \sqrt {\frac {a x^{4} + b}{x^{4}}} + \frac {1}{2} \, \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x^{2} \sqrt {\frac {a x^{4} + b}{x^{4}}}}{b}\right )\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 36, normalized size = 0.73 \[ \frac {b \arctan \left (\frac {\sqrt {a x^{4} + b}}{\sqrt {-b}}\right )}{2 \, \sqrt {-b}} + \frac {1}{2} \, \sqrt {a x^{4} + b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 64, normalized size = 1.31 \[ \frac {\sqrt {\frac {a \,x^{4}+b}{x^{4}}}\, \left (-\sqrt {b}\, \ln \left (\frac {2 b +2 \sqrt {a \,x^{4}+b}\, \sqrt {b}}{x^{2}}\right )+\sqrt {a \,x^{4}+b}\right ) x^{2}}{2 \sqrt {a \,x^{4}+b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.04, size = 60, normalized size = 1.22 \[ \frac {1}{2} \, \sqrt {a + \frac {b}{x^{4}}} x^{2} + \frac {1}{4} \, \sqrt {b} \log \left (\frac {\sqrt {a + \frac {b}{x^{4}}} x^{2} - \sqrt {b}}{\sqrt {a + \frac {b}{x^{4}}} x^{2} + \sqrt {b}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int x\,\sqrt {a+\frac {b}{x^4}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.79, size = 66, normalized size = 1.35 \[ \frac {\sqrt {a} x^{2}}{2 \sqrt {1 + \frac {b}{a x^{4}}}} - \frac {\sqrt {b} \operatorname {asinh}{\left (\frac {\sqrt {b}}{\sqrt {a} x^{2}} \right )}}{2} + \frac {b}{2 \sqrt {a} x^{2} \sqrt {1 + \frac {b}{a x^{4}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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