Optimal. Leaf size=34 \[ \frac {1}{\sqrt {x} \sqrt {2-b x}}-\frac {\sqrt {2-b x}}{\sqrt {x}} \]
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Rubi [A]
time = 0.00, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {47, 37}
\begin {gather*} \frac {1}{\sqrt {x} \sqrt {2-b x}}-\frac {\sqrt {2-b x}}{\sqrt {x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rule 47
Rubi steps
\begin {align*} \int \frac {1}{x^{3/2} (2-b x)^{3/2}} \, dx &=\frac {1}{\sqrt {x} \sqrt {2-b x}}+\int \frac {1}{x^{3/2} \sqrt {2-b x}} \, dx\\ &=\frac {1}{\sqrt {x} \sqrt {2-b x}}-\frac {\sqrt {2-b x}}{\sqrt {x}}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 21, normalized size = 0.62 \begin {gather*} \frac {-1+b x}{\sqrt {x} \sqrt {2-b x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.14, size = 28, normalized size = 0.82
method | result | size |
gosper | \(\frac {b x -1}{\sqrt {x}\, \sqrt {-b x +2}}\) | \(18\) |
meijerg | \(-\frac {\sqrt {2}\, \left (-b x +1\right )}{2 \sqrt {x}\, \sqrt {-\frac {b x}{2}+1}}\) | \(23\) |
default | \(-\frac {1}{\sqrt {x}\, \sqrt {-b x +2}}+\frac {b \sqrt {x}}{\sqrt {-b x +2}}\) | \(28\) |
risch | \(\frac {\left (b x -2\right ) \sqrt {\left (-b x +2\right ) x}}{2 \sqrt {-x \left (b x -2\right )}\, \sqrt {x}\, \sqrt {-b x +2}}+\frac {b \sqrt {x}\, \sqrt {\left (-b x +2\right ) x}}{2 \sqrt {-x \left (b x -2\right )}\, \sqrt {-b x +2}}\) | \(74\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 28, normalized size = 0.82 \begin {gather*} \frac {b \sqrt {x}}{2 \, \sqrt {-b x + 2}} - \frac {\sqrt {-b x + 2}}{2 \, \sqrt {x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.17, size = 29, normalized size = 0.85 \begin {gather*} -\frac {{\left (b x - 1\right )} \sqrt {-b x + 2} \sqrt {x}}{b x^{2} - 2 \, x} \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 = 0.82, size = 90, normalized size = 2.65 \begin {gather*} \begin {cases} - \frac {b^{\frac {5}{2}} x \sqrt {-1 + \frac {2}{b x}}}{b^{2} x - 2 b} + \frac {b^{\frac {3}{2}} \sqrt {-1 + \frac {2}{b x}}}{b^{2} x - 2 b} & \text {for}\: \frac {1}{\left |{b x}\right |} > \frac {1}{2} \\- \frac {i \sqrt {b}}{\sqrt {1 - \frac {2}{b x}}} + \frac {i}{\sqrt {b} x \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. 83 vs.
\(2 (26) = 52\).
time = 2.20, size = 83, normalized size = 2.44 \begin {gather*} -\frac {\sqrt {-b x + 2} b^{2}}{2 \, \sqrt {{\left (b x - 2\right )} b + 2 \, b} {\left | b \right |}} - \frac {2 \, \sqrt {-b} b^{2}}{{\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 = 0.32, size = 27, normalized size = 0.79 \begin {gather*} \frac {b\,\sqrt {x}}{\sqrt {2-b\,x}}-\frac {1}{\sqrt {x}\,\sqrt {2-b\,x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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