Optimal. Leaf size=16 \[ \frac {x}{a \sqrt {a+b x^2}} \]
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Rubi [A] time = 0.00, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {191} \begin {gather*} \frac {x}{a \sqrt {a+b x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 191
Rubi steps
\begin {align*} \int \frac {1}{\left (a+b x^2\right )^{3/2}} \, dx &=\frac {x}{a \sqrt {a+b x^2}}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 16, normalized size = 1.00 \begin {gather*} \frac {x}{a \sqrt {a+b x^2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.00, size = 16, normalized size = 1.00 \begin {gather*} \frac {x}{a \sqrt {a+b x^2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.96, size = 23, normalized size = 1.44 \begin {gather*} \frac {\sqrt {b x^{2} + a} x}{a b x^{2} + a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.60, size = 14, normalized size = 0.88 \begin {gather*} \frac {x}{\sqrt {b x^{2} + a} a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 15, normalized size = 0.94 \begin {gather*} \frac {x}{\sqrt {b \,x^{2}+a}\, a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.32, size = 14, normalized size = 0.88 \begin {gather*} \frac {x}{\sqrt {b x^{2} + a} a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 14, normalized size = 0.88 \begin {gather*} \frac {x}{a\,\sqrt {b\,x^2+a}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.61, size = 17, normalized size = 1.06 \begin {gather*} \frac {x}{a^{\frac {3}{2}} \sqrt {1 + \frac {b x^{2}}{a}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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