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 = {45, 37} \[ \frac {1}{\sqrt {x} \sqrt {2-b x}}-\frac {\sqrt {2-b x}}{\sqrt {x}} \]
Antiderivative was successfully verified.
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Rule 37
Rule 45
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 = 0.01, size = 21, normalized size = 0.62 \[ \frac {b x-1}{\sqrt {x} \sqrt {2-b x}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 29, normalized size = 0.85 \[ -\frac {{\left (b x - 1\right )} \sqrt {-b x + 2} \sqrt {x}}{b x^{2} - 2 \, x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.09, size = 83, normalized size = 2.44 \[ -\frac {\sqrt {-b x + 2} b^{2}}{2 \, \sqrt {{\left (b x - 2\right )} b + 2 \, b} {\left | b \right |}} - \frac {2 \, \sqrt {-b} b^{2}}{{\left ({\left (\sqrt {-b x + 2} \sqrt {-b} - \sqrt {{\left (b x - 2\right )} b + 2 \, b}\right )}^{2} - 2 \, b\right )} {\left | b \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 18, normalized size = 0.53 \[ \frac {b x -1}{\sqrt {-b x +2}\, \sqrt {x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.26, size = 28, normalized size = 0.82 \[ \frac {b \sqrt {x}}{2 \, \sqrt {-b x + 2}} - \frac {\sqrt {-b x + 2}}{2 \, \sqrt {x}} \]
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 \[ \frac {b\,\sqrt {x}}{\sqrt {2-b\,x}}-\frac {1}{\sqrt {x}\,\sqrt {2-b\,x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.61, size = 90, normalized size = 2.65 \[ \begin {cases} \frac {\sqrt {b}}{\sqrt {-1 + \frac {2}{b x}}} - \frac {1}{\sqrt {b} x \sqrt {-1 + \frac {2}{b x}}} & \text {for}\: \frac {2}{\left |{b x}\right |} > 1 \\\frac {i b^{\frac {5}{2}} x \sqrt {1 - \frac {2}{b x}}}{- b^{2} x + 2 b} - \frac {i b^{\frac {3}{2}} \sqrt {1 - \frac {2}{b x}}}{- b^{2} x + 2 b} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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