Optimal. Leaf size=45 \[ \frac {2 \sin ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{b^{3/2}}-\frac {\sqrt {x} \sqrt {2-b x}}{b} \]
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Rubi [A] time = 0.01, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {50, 54, 216} \begin {gather*} \frac {2 \sin ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{b^{3/2}}-\frac {\sqrt {x} \sqrt {2-b x}}{b} \end {gather*}
Antiderivative was successfully verified.
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Rule 50
Rule 54
Rule 216
Rubi steps
\begin {align*} \int \frac {\sqrt {x}}{\sqrt {2-b x}} \, dx &=-\frac {\sqrt {x} \sqrt {2-b x}}{b}+\frac {\int \frac {1}{\sqrt {x} \sqrt {2-b x}} \, dx}{b}\\ &=-\frac {\sqrt {x} \sqrt {2-b x}}{b}+\frac {2 \operatorname {Subst}\left (\int \frac {1}{\sqrt {2-b x^2}} \, dx,x,\sqrt {x}\right )}{b}\\ &=-\frac {\sqrt {x} \sqrt {2-b x}}{b}+\frac {2 \sin ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{b^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 45, normalized size = 1.00 \begin {gather*} \frac {2 \sin ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{b^{3/2}}-\frac {\sqrt {x} \sqrt {2-b x}}{b} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.07, size = 59, normalized size = 1.31 \begin {gather*} \frac {2 \sqrt {-b} \log \left (\sqrt {2-b x}-\sqrt {-b} \sqrt {x}\right )}{b^2}-\frac {\sqrt {x} \sqrt {2-b x}}{b} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.28, size = 90, normalized size = 2.00 \begin {gather*} \left [-\frac {\sqrt {-b x + 2} b \sqrt {x} + \sqrt {-b} \log \left (-b x + \sqrt {-b x + 2} \sqrt {-b} \sqrt {x} + 1\right )}{b^{2}}, -\frac {\sqrt {-b x + 2} b \sqrt {x} + 2 \, \sqrt {b} \arctan \left (\frac {\sqrt {-b x + 2}}{\sqrt {b} \sqrt {x}}\right )}{b^{2}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: NotImplementedError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 67, normalized size = 1.49 \begin {gather*} -\frac {\sqrt {-b x +2}\, \sqrt {x}}{b}+\frac {\sqrt {\left (-b x +2\right ) x}\, \arctan \left (\frac {\left (x -\frac {1}{b}\right ) \sqrt {b}}{\sqrt {-b \,x^{2}+2 x}}\right )}{\sqrt {-b x +2}\, b^{\frac {3}{2}} \sqrt {x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.93, size = 52, normalized size = 1.16 \begin {gather*} -\frac {2 \, \arctan \left (\frac {\sqrt {-b x + 2}}{\sqrt {b} \sqrt {x}}\right )}{b^{\frac {3}{2}}} - \frac {2 \, \sqrt {-b x + 2}}{{\left (b^{2} - \frac {{\left (b x - 2\right )} b}{x}\right )} \sqrt {x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.52, size = 46, normalized size = 1.02 \begin {gather*} -\frac {4\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {x}}{\sqrt {2}-\sqrt {2-b\,x}}\right )}{b^{3/2}}-\frac {\sqrt {x}\,\sqrt {2-b\,x}}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.97, size = 121, normalized size = 2.69 \begin {gather*} \begin {cases} - \frac {i x^{\frac {3}{2}}}{\sqrt {b x - 2}} + \frac {2 i \sqrt {x}}{b \sqrt {b x - 2}} - \frac {2 i \operatorname {acosh}{\left (\frac {\sqrt {2} \sqrt {b} \sqrt {x}}{2} \right )}}{b^{\frac {3}{2}}} & \text {for}\: \frac {\left |{b x}\right |}{2} > 1 \\\frac {x^{\frac {3}{2}}}{\sqrt {- b x + 2}} - \frac {2 \sqrt {x}}{b \sqrt {- b x + 2}} + \frac {2 \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {b} \sqrt {x}}{2} \right )}}{b^{\frac {3}{2}}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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