Optimal. Leaf size=16 \[ -\frac {\sqrt {2+b x}}{\sqrt {x}} \]
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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 = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {37}
\begin {gather*} -\frac {\sqrt {b x+2}}{\sqrt {x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rubi steps
\begin {align*} \int \frac {1}{x^{3/2} \sqrt {2+b x}} \, dx &=-\frac {\sqrt {2+b x}}{\sqrt {x}}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 16, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {2+b x}}{\sqrt {x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 13, normalized size = 0.81
method | result | size |
gosper | \(-\frac {\sqrt {b x +2}}{\sqrt {x}}\) | \(13\) |
default | \(-\frac {\sqrt {b x +2}}{\sqrt {x}}\) | \(13\) |
risch | \(-\frac {\sqrt {b x +2}}{\sqrt {x}}\) | \(13\) |
meijerg | \(-\frac {\sqrt {2}\, \sqrt {\frac {b x}{2}+1}}{\sqrt {x}}\) | \(17\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 12, normalized size = 0.75 \begin {gather*} -\frac {\sqrt {b x + 2}}{\sqrt {x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.44, size = 12, normalized size = 0.75 \begin {gather*} -\frac {\sqrt {b x + 2}}{\sqrt {x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.45, size = 15, normalized size = 0.94 \begin {gather*} - \sqrt {b} \sqrt {1 + \frac {2}{b x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 29 vs.
\(2 (12) = 24\).
time = 0.97, size = 29, normalized size = 1.81 \begin {gather*} -\frac {\sqrt {b x + 2} b^{2}}{\sqrt {{\left (b x + 2\right )} b - 2 \, b} {\left | b \right |}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.33, size = 12, normalized size = 0.75 \begin {gather*} -\frac {\sqrt {b\,x+2}}{\sqrt {x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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