Optimal. Leaf size=21 \[ -\frac {2}{a \sqrt {x} \sqrt {a+\frac {b}{x}}} \]
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Rubi [A] time = 0.01, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {264} \[ -\frac {2}{a \sqrt {x} \sqrt {a+\frac {b}{x}}} \]
Antiderivative was successfully verified.
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Rule 264
Rubi steps
\begin {align*} \int \frac {1}{\left (a+\frac {b}{x}\right )^{3/2} x^{3/2}} \, dx &=-\frac {2}{a \sqrt {a+\frac {b}{x}} \sqrt {x}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 21, normalized size = 1.00 \[ -\frac {2}{a \sqrt {x} \sqrt {a+\frac {b}{x}}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.78, size = 27, normalized size = 1.29 \[ -\frac {2 \, \sqrt {x} \sqrt {\frac {a x + b}{x}}}{a^{2} x + a b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 21, normalized size = 1.00 \[ -\frac {2}{\sqrt {a x + b} a} + \frac {2}{a \sqrt {b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 25, normalized size = 1.19 \[ -\frac {2 \left (a x +b \right )}{\left (\frac {a x +b}{x}\right )^{\frac {3}{2}} a \,x^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.10, size = 17, normalized size = 0.81 \[ -\frac {2}{\sqrt {a + \frac {b}{x}} a \sqrt {x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.43, size = 25, normalized size = 1.19 \[ -\frac {2\,\sqrt {x}\,\sqrt {a+\frac {b}{x}}}{x\,a^2+b\,a} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 5.40, size = 19, normalized size = 0.90 \[ - \frac {2}{a \sqrt {b} \sqrt {\frac {a x}{b} + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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