Optimal. Leaf size=42 \[ -\frac {2 \sqrt {2-b x}}{\sqrt {x}}-2 \sqrt {b} \sin ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 42, 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 = {49, 56, 222}
\begin {gather*} -\frac {2 \sqrt {2-b x}}{\sqrt {x}}-2 \sqrt {b} \sin ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 49
Rule 56
Rule 222
Rubi steps
\begin {align*} \int \frac {\sqrt {2-b x}}{x^{3/2}} \, dx &=-\frac {2 \sqrt {2-b x}}{\sqrt {x}}-b \int \frac {1}{\sqrt {x} \sqrt {2-b x}} \, dx\\ &=-\frac {2 \sqrt {2-b x}}{\sqrt {x}}-(2 b) \text {Subst}\left (\int \frac {1}{\sqrt {2-b x^2}} \, dx,x,\sqrt {x}\right )\\ &=-\frac {2 \sqrt {2-b x}}{\sqrt {x}}-2 \sqrt {b} \sin ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 53, normalized size = 1.26 \begin {gather*} -\frac {2 \sqrt {2-b x}}{\sqrt {x}}-2 \sqrt {-b} \log \left (-\sqrt {-b} \sqrt {x}+\sqrt {2-b x}\right ) \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 = 2.93, size = 111, normalized size = 2.64 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\frac {2 I \left (\sqrt {b} \sqrt {x} \text {ArcCosh}\left [\frac {\sqrt {2} \sqrt {b} \sqrt {x}}{2}\right ] \left (-2+b x\right )-b x \sqrt {-2+b x}+2 \sqrt {-2+b x}\right )}{\sqrt {x} \left (-2+b x\right )},\text {Abs}\left [b x\right ]>2\right \}\right \},-2 \sqrt {b} \text {ArcSin}\left [\frac {\sqrt {2} \sqrt {b} \sqrt {x}}{2}\right ]-\frac {4}{\sqrt {x} \sqrt {2-b x}}+\frac {2 b \sqrt {x}}{\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. \(63\) vs.
\(2(31)=62\).
time = 0.11, size = 64, normalized size = 1.52
method | result | size |
meijerg | \(\frac {\left (-b \right )^{\frac {3}{2}} \left (\frac {4 \sqrt {\pi }\, \sqrt {2}\, \sqrt {-\frac {b x}{2}+1}}{\sqrt {x}\, \sqrt {-b}}+\frac {4 \sqrt {\pi }\, \sqrt {b}\, \arcsin \left (\frac {\sqrt {b}\, \sqrt {x}\, \sqrt {2}}{2}\right )}{\sqrt {-b}}\right )}{2 \sqrt {\pi }\, b}\) | \(64\) |
risch | \(\frac {2 \left (b x -2\right ) \sqrt {\left (-b x +2\right ) x}}{\sqrt {-x \left (b x -2\right )}\, \sqrt {x}\, \sqrt {-b x +2}}-\frac {\sqrt {b}\, \arctan \left (\frac {\sqrt {b}\, \left (x -\frac {1}{b}\right )}{\sqrt {-x^{2} b +2 x}}\right ) \sqrt {\left (-b x +2\right ) x}}{\sqrt {x}\, \sqrt {-b x +2}}\) | \(90\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.34, size = 35, normalized size = 0.83 \begin {gather*} 2 \, \sqrt {b} \arctan \left (\frac {\sqrt {-b x + 2}}{\sqrt {b} \sqrt {x}}\right ) - \frac {2 \, \sqrt {-b x + 2}}{\sqrt {x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.31, size = 90, normalized size = 2.14 \begin {gather*} \left [\frac {\sqrt {-b} x \log \left (-b x + \sqrt {-b x + 2} \sqrt {-b} \sqrt {x} + 1\right ) - 2 \, \sqrt {-b x + 2} \sqrt {x}}{x}, \frac {2 \, {\left (\sqrt {b} x \arctan \left (\frac {\sqrt {-b x + 2}}{\sqrt {b} \sqrt {x}}\right ) - \sqrt {-b x + 2} \sqrt {x}\right )}}{x}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.78, size = 122, normalized size = 2.90 \begin {gather*} \begin {cases} 2 i \sqrt {b} \operatorname {acosh}{\left (\frac {\sqrt {2} \sqrt {b} \sqrt {x}}{2} \right )} - \frac {2 i b \sqrt {x}}{\sqrt {b x - 2}} + \frac {4 i}{\sqrt {x} \sqrt {b x - 2}} & \text {for}\: \left |{b x}\right | > 2 \\- 2 \sqrt {b} \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {b} \sqrt {x}}{2} \right )} + \frac {2 b \sqrt {x}}{\sqrt {- b x + 2}} - \frac {4}{\sqrt {x} \sqrt {- b x + 2}} & \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. 69 vs.
\(2 (31) = 62\).
time = 1.14, size = 102, normalized size = 2.43 \begin {gather*} -\frac {b b^{2} \left (\frac {2 \sqrt {-b x+2} \sqrt {-b \left (-b x+2\right )+2 b}}{-b \left (-b x+2\right )+2 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 [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {\sqrt {2-b\,x}}{x^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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