Optimal. Leaf size=16 \[ \frac {\sqrt {x}}{\sqrt {2-b x}} \]
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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 = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {37}
\begin {gather*} \frac {\sqrt {x}}{\sqrt {2-b x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {x} (2-b x)^{3/2}} \, dx &=\frac {\sqrt {x}}{\sqrt {2-b x}}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 16, normalized size = 1.00 \begin {gather*} \frac {\sqrt {x}}{\sqrt {2-b x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.11, size = 13, normalized size = 0.81
method | result | size |
gosper | \(\frac {\sqrt {x}}{\sqrt {-b x +2}}\) | \(13\) |
default | \(\frac {\sqrt {x}}{\sqrt {-b x +2}}\) | \(13\) |
meijerg | \(\frac {\sqrt {x}\, \sqrt {2}}{2 \sqrt {-\frac {b x}{2}+1}}\) | \(17\) |
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 {x}}{\sqrt {-b x + 2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.07, size = 20, normalized size = 1.25 \begin {gather*} -\frac {\sqrt {-b x + 2} \sqrt {x}}{b x - 2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 0.45, size = 41, normalized size = 2.56 \begin {gather*} \begin {cases} \frac {1}{\sqrt {b} \sqrt {-1 + \frac {2}{b x}}} & \text {for}\: \frac {1}{\left |{b x}\right |} > \frac {1}{2} \\- \frac {i}{\sqrt {b} \sqrt {1 - \frac {2}{b x}}} & \text {otherwise} \end {cases} \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. 50 vs.
\(2 (12) = 24\).
time = 2.26, size = 50, normalized size = 3.12 \begin {gather*} -\frac {4 \, \sqrt {-b} b}{{\left ({\left (\sqrt {-b x + 2} \sqrt {-b} - \sqrt {{\left (b x - 2\right )} b + 2 \, b}\right )}^{2} - 2 \, b\right )} {\left | b \right |}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.30, size = 12, normalized size = 0.75 \begin {gather*} \frac {\sqrt {x}}{\sqrt {2-b\,x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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