Optimal. Leaf size=15 \[ \frac {1}{b \sqrt {a+\frac {b}{x^2}}} \]
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Rubi [A] time = 0.01, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {261} \[ \frac {1}{b \sqrt {a+\frac {b}{x^2}}} \]
Antiderivative was successfully verified.
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Rule 261
Rubi steps
\begin {align*} \int \frac {1}{\left (a+\frac {b}{x^2}\right )^{3/2} x^3} \, dx &=\frac {1}{b \sqrt {a+\frac {b}{x^2}}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 15, normalized size = 1.00 \[ \frac {1}{b \sqrt {a+\frac {b}{x^2}}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.99, size = 29, normalized size = 1.93 \[ \frac {x^{2} \sqrt {\frac {a x^{2} + b}{x^{2}}}}{a b x^{2} + b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.19, size = 44, normalized size = 2.93 \[ \frac {1}{{\left ({\left (\sqrt {a} x^{2} - \sqrt {a x^{4} + b x^{2}}\right )} \sqrt {a} + b\right )} \sqrt {a}} - \frac {1}{\sqrt {a} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.00, size = 28, normalized size = 1.87 \[ \frac {a \,x^{2}+b}{\left (\frac {a \,x^{2}+b}{x^{2}}\right )^{\frac {3}{2}} b \,x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.88, size = 13, normalized size = 0.87 \[ \frac {1}{\sqrt {a + \frac {b}{x^{2}}} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.25, size = 18, normalized size = 1.20 \[ \frac {\sqrt {x^2}}{b\,\sqrt {a\,x^2+b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.37, size = 26, normalized size = 1.73 \[ \begin {cases} \frac {1}{b \sqrt {a + \frac {b}{x^{2}}}} & \text {for}\: b \neq 0 \\- \frac {1}{2 a^{\frac {3}{2}} x^{2}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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