Optimal. Leaf size=41 \[ \sqrt {x} \sqrt {2-b x}+\frac {2 \sin ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{\sqrt {b}} \]
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Rubi [A]
time = 0.01, antiderivative size = 41, 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*} \sqrt {x} \sqrt {2-b x}+\frac {2 \sin ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{\sqrt {b}} \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 56
Rule 222
Rubi steps
\begin {align*} \int \frac {\sqrt {2-b x}}{\sqrt {x}} \, dx &=\sqrt {x} \sqrt {2-b x}+\int \frac {1}{\sqrt {x} \sqrt {2-b x}} \, dx\\ &=\sqrt {x} \sqrt {2-b x}+2 \text {Subst}\left (\int \frac {1}{\sqrt {2-b x^2}} \, dx,x,\sqrt {x}\right )\\ &=\sqrt {x} \sqrt {2-b x}+\frac {2 \sin ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{\sqrt {b}}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 52, normalized size = 1.27 \begin {gather*} \sqrt {x} \sqrt {2-b x}-\frac {2 \log \left (-\sqrt {-b} \sqrt {x}+\sqrt {2-b x}\right )}{\sqrt {-b}} \end {gather*}
Antiderivative was successfully verified.
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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in
optimal.
time = 2.65, size = 100, normalized size = 2.44 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\frac {I \left (\sqrt {b} \sqrt {x} \left (-2+b x\right )-2 \text {ArcCosh}\left [\frac {\sqrt {2} \sqrt {b} \sqrt {x}}{2}\right ] \sqrt {-2+b x}\right )}{\sqrt {b} \sqrt {-2+b x}},\text {Abs}\left [b x\right ]>2\right \}\right \},\frac {2 \text {ArcSin}\left [\frac {\sqrt {2} \sqrt {b} \sqrt {x}}{2}\right ]}{\sqrt {b}}+\frac {2 \sqrt {x}}{\sqrt {2-b x}}-\frac {b x^{\frac {3}{2}}}{\sqrt {2-b x}}\right ] \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(62\) vs.
\(2(30)=60\).
time = 0.11, size = 63, normalized size = 1.54
method | result | size |
default | \(\sqrt {x}\, \sqrt {-b x +2}+\frac {\sqrt {\left (-b x +2\right ) x}\, \arctan \left (\frac {\sqrt {b}\, \left (x -\frac {1}{b}\right )}{\sqrt {-x^{2} b +2 x}}\right )}{\sqrt {-b x +2}\, \sqrt {x}\, \sqrt {b}}\) | \(63\) |
meijerg | \(\frac {\sqrt {-b}\, \left (-\sqrt {\pi }\, \sqrt {x}\, \sqrt {2}\, \sqrt {-b}\, \sqrt {-\frac {b x}{2}+1}-\frac {2 \sqrt {\pi }\, \sqrt {-b}\, \arcsin \left (\frac {\sqrt {b}\, \sqrt {x}\, \sqrt {2}}{2}\right )}{\sqrt {b}}\right )}{\sqrt {\pi }\, b}\) | \(63\) |
risch | \(-\frac {\sqrt {x}\, \left (b x -2\right ) \sqrt {\left (-b x +2\right ) x}}{\sqrt {-x \left (b x -2\right )}\, \sqrt {-b x +2}}+\frac {\sqrt {\left (-b x +2\right ) x}\, \arctan \left (\frac {\sqrt {b}\, \left (x -\frac {1}{b}\right )}{\sqrt {-x^{2} b +2 x}}\right )}{\sqrt {-b x +2}\, \sqrt {x}\, \sqrt {b}}\) | \(89\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.36, size = 49, normalized size = 1.20 \begin {gather*} -\frac {2 \, \arctan \left (\frac {\sqrt {-b x + 2}}{\sqrt {b} \sqrt {x}}\right )}{\sqrt {b}} + \frac {2 \, \sqrt {-b x + 2}}{{\left (b - \frac {b x - 2}{x}\right )} \sqrt {x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.31, size = 89, normalized size = 2.17 \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}, \frac {\sqrt {-b x + 2} b \sqrt {x} - 2 \, \sqrt {b} \arctan \left (\frac {\sqrt {-b x + 2}}{\sqrt {b} \sqrt {x}}\right )}{b}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.86, size = 119, normalized size = 2.90 \begin {gather*} \begin {cases} \frac {i b x^{\frac {3}{2}}}{\sqrt {b x - 2}} - \frac {2 i \sqrt {x}}{\sqrt {b x - 2}} - \frac {2 i \operatorname {acosh}{\left (\frac {\sqrt {2} \sqrt {b} \sqrt {x}}{2} \right )}}{\sqrt {b}} & \text {for}\: \left |{b x}\right | > 2 \\- \frac {b x^{\frac {3}{2}}}{\sqrt {- b x + 2}} + \frac {2 \sqrt {x}}{\sqrt {- b x + 2}} + \frac {2 \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {b} \sqrt {x}}{2} \right )}}{\sqrt {b}} & \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. 70 vs.
\(2 (30) = 60\).
time = 1.14, size = 91, normalized size = 2.22 \begin {gather*} \frac {b^{2} \left (\frac {\frac {1}{2}\cdot 2 \sqrt {-b x+2} \sqrt {-b \left (-b x+2\right )+2 b}}{b}+\frac {2 \ln \left |\sqrt {-b \left (-b x+2\right )+2 b}-\sqrt {-b} \sqrt {-b x+2}\right |}{\sqrt {-b}}\right )}{\left |b\right | b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.56, size = 42, normalized size = 1.02 \begin {gather*} \sqrt {x}\,\sqrt {2-b\,x}-\frac {4\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {x}}{\sqrt {2}-\sqrt {2-b\,x}}\right )}{\sqrt {b}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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