Optimal. Leaf size=45 \[ -\frac {\sqrt {x} \sqrt {2-b x}}{b}+\frac {2 \sin ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{b^{3/2}} \]
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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 = {52, 56, 222}
\begin {gather*} \frac {2 \text {ArcSin}\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 52
Rule 56
Rule 222
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 \text {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 = 0.06, size = 56, normalized size = 1.24 \begin {gather*} -\frac {\sqrt {x} \sqrt {2-b x}}{b}+\frac {2 \log \left (-\sqrt {-b} \sqrt {x}+\sqrt {2-b x}\right )}{(-b)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.13, size = 67, normalized size = 1.49
method | result | size |
meijerg | \(-\frac {2 \left (-\frac {\sqrt {\pi }\, \sqrt {x}\, \sqrt {2}\, \left (-b \right )^{\frac {3}{2}} \sqrt {-\frac {b x}{2}+1}}{2 b}+\frac {\sqrt {\pi }\, \left (-b \right )^{\frac {3}{2}} \arcsin \left (\frac {\sqrt {b}\, \sqrt {x}\, \sqrt {2}}{2}\right )}{b^{\frac {3}{2}}}\right )}{\sqrt {-b}\, \sqrt {\pi }\, b}\) | \(66\) |
default | \(-\frac {\sqrt {x}\, \sqrt {-b x +2}}{b}+\frac {\sqrt {\left (-b x +2\right ) x}\, \arctan \left (\frac {\sqrt {b}\, \left (x -\frac {1}{b}\right )}{\sqrt {-x^{2} b +2 x}}\right )}{b^{\frac {3}{2}} \sqrt {-b x +2}\, \sqrt {x}}\) | \(67\) |
risch | \(\frac {\sqrt {x}\, \left (b x -2\right ) \sqrt {\left (-b x +2\right ) x}}{b \sqrt {-x \left (b x -2\right )}\, \sqrt {-b x +2}}+\frac {\sqrt {\left (-b x +2\right ) x}\, \arctan \left (\frac {\sqrt {b}\, \left (x -\frac {1}{b}\right )}{\sqrt {-x^{2} b +2 x}}\right )}{b^{\frac {3}{2}} \sqrt {-b x +2}\, \sqrt {x}}\) | \(91\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, 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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Fricas [A]
time = 1.54, 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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Sympy [C] Result contains complex when optimal does not.
time = 1.06, size = 119, normalized size = 2.64 \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}\: \left |{b x}\right | > 2 \\\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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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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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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