Optimal. Leaf size=65 \[ -\frac {\sqrt {x} \sqrt {2-b x}}{2 b}+\frac {1}{2} x^{3/2} \sqrt {2-b x}+\frac {\sin ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{b^{3/2}} \]
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Rubi [A]
time = 0.01, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps
used = 4, 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 {\text {ArcSin}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{b^{3/2}}+\frac {1}{2} x^{3/2} \sqrt {2-b x}-\frac {\sqrt {x} \sqrt {2-b x}}{2 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 56
Rule 222
Rubi steps
\begin {align*} \int \sqrt {x} \sqrt {2-b x} \, dx &=\frac {1}{2} x^{3/2} \sqrt {2-b x}+\frac {1}{2} \int \frac {\sqrt {x}}{\sqrt {2-b x}} \, dx\\ &=-\frac {\sqrt {x} \sqrt {2-b x}}{2 b}+\frac {1}{2} x^{3/2} \sqrt {2-b x}+\frac {\int \frac {1}{\sqrt {x} \sqrt {2-b x}} \, dx}{2 b}\\ &=-\frac {\sqrt {x} \sqrt {2-b x}}{2 b}+\frac {1}{2} x^{3/2} \sqrt {2-b x}+\frac {\text {Subst}\left (\int \frac {1}{\sqrt {2-b x^2}} \, dx,x,\sqrt {x}\right )}{b}\\ &=-\frac {\sqrt {x} \sqrt {2-b x}}{2 b}+\frac {1}{2} x^{3/2} \sqrt {2-b x}+\frac {\sin ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{b^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.08, size = 62, normalized size = 0.95 \begin {gather*} \frac {\sqrt {x} \sqrt {2-b x} (-1+b x)}{2 b}+\frac {\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 = 0.12, size = 85, normalized size = 1.31
method | result | size |
meijerg | \(\frac {\frac {\sqrt {\pi }\, \sqrt {x}\, \sqrt {2}\, \left (-b \right )^{\frac {3}{2}} \left (-3 b x +3\right ) \sqrt {-\frac {b x}{2}+1}}{6 b}-\frac {\sqrt {\pi }\, \left (-b \right )^{\frac {3}{2}} \arcsin \left (\frac {\sqrt {b}\, \sqrt {x}\, \sqrt {2}}{2}\right )}{b^{\frac {3}{2}}}}{\sqrt {-b}\, \sqrt {\pi }\, b}\) | \(73\) |
default | \(-\frac {\sqrt {x}\, \left (-b x +2\right )^{\frac {3}{2}}}{2 b}+\frac {\sqrt {x}\, \sqrt {-b x +2}+\frac {\sqrt {\left (-b x +2\right ) x}\, \arctan \left (\frac {\sqrt {b}\, \left (x -\frac {1}{b}\right )}{\sqrt {-x^{2} b +2 x}}\right )}{\sqrt {-b x +2}\, \sqrt {x}\, \sqrt {b}}}{2 b}\) | \(85\) |
risch | \(-\frac {\left (b x -1\right ) \sqrt {x}\, \left (b x -2\right ) \sqrt {\left (-b x +2\right ) x}}{2 b \sqrt {-x \left (b x -2\right )}\, \sqrt {-b x +2}}+\frac {\arctan \left (\frac {\sqrt {b}\, \left (x -\frac {1}{b}\right )}{\sqrt {-x^{2} b +2 x}}\right ) \sqrt {\left (-b x +2\right ) x}}{2 b^{\frac {3}{2}} \sqrt {x}\, \sqrt {-b x +2}}\) | \(98\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 81, normalized size = 1.25 \begin {gather*} \frac {\frac {\sqrt {-b x + 2} b}{\sqrt {x}} - \frac {{\left (-b x + 2\right )}^{\frac {3}{2}}}{x^{\frac {3}{2}}}}{b^{3} - \frac {2 \, {\left (b x - 2\right )} b^{2}}{x} + \frac {{\left (b x - 2\right )}^{2} b}{x^{2}}} - \frac {\arctan \left (\frac {\sqrt {-b x + 2}}{\sqrt {b} \sqrt {x}}\right )}{b^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.48, size = 107, normalized size = 1.65 \begin {gather*} \left [\frac {{\left (b^{2} x - b\right )} \sqrt {-b x + 2} \sqrt {x} - \sqrt {-b} \log \left (-b x + \sqrt {-b x + 2} \sqrt {-b} \sqrt {x} + 1\right )}{2 \, b^{2}}, \frac {{\left (b^{2} x - b\right )} \sqrt {-b x + 2} \sqrt {x} - 2 \, \sqrt {b} \arctan \left (\frac {\sqrt {-b x + 2}}{\sqrt {b} \sqrt {x}}\right )}{2 \, 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 = 1.82, size = 155, normalized size = 2.38 \begin {gather*} \begin {cases} \frac {i b x^{\frac {5}{2}}}{2 \sqrt {b x - 2}} - \frac {3 i x^{\frac {3}{2}}}{2 \sqrt {b x - 2}} + \frac {i \sqrt {x}}{b \sqrt {b x - 2}} - \frac {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 {b x^{\frac {5}{2}}}{2 \sqrt {- b x + 2}} + \frac {3 x^{\frac {3}{2}}}{2 \sqrt {- b x + 2}} - \frac {\sqrt {x}}{b \sqrt {- b x + 2}} + \frac {\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 = 0.10, size = 53, normalized size = 0.82 \begin {gather*} \sqrt {x}\,\left (\frac {x}{2}-\frac {1}{2\,b}\right )\,\sqrt {2-b\,x}-\frac {\ln \left (\sqrt {-b}\,\sqrt {x}\,\sqrt {2-b\,x}-b\,x+1\right )}{2\,{\left (-b\right )}^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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