Optimal. Leaf size=40 \[ \frac {x^2 \sqrt {a+\frac {b}{x^4}}}{a^2}-\frac {x^2}{2 a \sqrt {a+\frac {b}{x^4}}} \]
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Rubi [A] time = 0.01, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {273, 264} \[ \frac {x^2 \sqrt {a+\frac {b}{x^4}}}{a^2}-\frac {x^2}{2 a \sqrt {a+\frac {b}{x^4}}} \]
Antiderivative was successfully verified.
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Rule 264
Rule 273
Rubi steps
\begin {align*} \int \frac {x}{\left (a+\frac {b}{x^4}\right )^{3/2}} \, dx &=-\frac {x^2}{2 a \sqrt {a+\frac {b}{x^4}}}+\frac {2 \int \frac {x}{\sqrt {a+\frac {b}{x^4}}} \, dx}{a}\\ &=-\frac {x^2}{2 a \sqrt {a+\frac {b}{x^4}}}+\frac {\sqrt {a+\frac {b}{x^4}} x^2}{a^2}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 30, normalized size = 0.75 \[ \frac {a x^4+2 b}{2 a^2 x^2 \sqrt {a+\frac {b}{x^4}}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.84, size = 42, normalized size = 1.05 \[ \frac {{\left (a x^{6} + 2 \, b x^{2}\right )} \sqrt {\frac {a x^{4} + b}{x^{4}}}}{2 \, {\left (a^{3} x^{4} + a^{2} b\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 42, normalized size = 1.05 \[ \frac {\frac {\sqrt {a x^{4} + b}}{a} + \frac {b}{\sqrt {a x^{4} + b} a}}{2 \, a} - \frac {\sqrt {b}}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 38, normalized size = 0.95 \[ \frac {\left (a \,x^{4}+b \right ) \left (a \,x^{4}+2 b \right )}{2 \left (\frac {a \,x^{4}+b}{x^{4}}\right )^{\frac {3}{2}} a^{2} x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.82, size = 36, normalized size = 0.90 \[ \frac {\sqrt {a + \frac {b}{x^{4}}} x^{2}}{2 \, a^{2}} + \frac {b}{2 \, \sqrt {a + \frac {b}{x^{4}}} a^{2} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.39, size = 24, normalized size = 0.60 \[ \frac {\frac {a\,x^4}{2}+b}{a^2\,x^2\,\sqrt {a+\frac {b}{x^4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.21, size = 42, normalized size = 1.05 \[ \frac {x^{4}}{2 a \sqrt {b} \sqrt {\frac {a x^{4}}{b} + 1}} + \frac {\sqrt {b}}{a^{2} \sqrt {\frac {a x^{4}}{b} + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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