Optimal. Leaf size=45 \[ \frac {2 \sqrt {x}}{b \sqrt {2-b x}}-\frac {2 \sin ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{b^{3/2}} \]
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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 = {49, 56, 222}
\begin {gather*} \frac {2 \sqrt {x}}{b \sqrt {2-b x}}-\frac {2 \text {ArcSin}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{b^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 49
Rule 56
Rule 222
Rubi steps
\begin {align*} \int \frac {\sqrt {x}}{(2-b x)^{3/2}} \, dx &=\frac {2 \sqrt {x}}{b \sqrt {2-b x}}-\frac {\int \frac {1}{\sqrt {x} \sqrt {2-b x}} \, dx}{b}\\ &=\frac {2 \sqrt {x}}{b \sqrt {2-b x}}-\frac {2 \text {Subst}\left (\int \frac {1}{\sqrt {2-b x^2}} \, dx,x,\sqrt {x}\right )}{b}\\ &=\frac {2 \sqrt {x}}{b \sqrt {2-b x}}-\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 = 0.10, size = 56, normalized size = 1.24 \begin {gather*} \frac {2 \sqrt {x}}{b \sqrt {2-b x}}-\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 = 0.10, size = 67, normalized size = 1.49
method | result | size |
meijerg | \(-\frac {2 \left (\frac {\sqrt {\pi }\, \sqrt {x}\, \sqrt {2}\, \left (-b \right )^{\frac {3}{2}}}{2 b \sqrt {-\frac {b x}{2}+1}}-\frac {\sqrt {\pi }\, \left (-b \right )^{\frac {3}{2}} \arcsin \left (\frac {\sqrt {b}\, \sqrt {x}\, \sqrt {2}}{2}\right )}{b^{\frac {3}{2}}}\right )}{\sqrt {-b}\, \sqrt {\pi }\, b}\) | \(67\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 38, normalized size = 0.84 \begin {gather*} \frac {2 \, \arctan \left (\frac {\sqrt {-b x + 2}}{\sqrt {b} \sqrt {x}}\right )}{b^{\frac {3}{2}}} + \frac {2 \, \sqrt {x}}{\sqrt {-b x + 2} b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.09, size = 122, normalized size = 2.71 \begin {gather*} \left [-\frac {{\left (b x - 2\right )} \sqrt {-b} \log \left (-b x - \sqrt {-b x + 2} \sqrt {-b} \sqrt {x} + 1\right ) + 2 \, \sqrt {-b x + 2} b \sqrt {x}}{b^{3} x - 2 \, b^{2}}, \frac {2 \, {\left ({\left (b x - 2\right )} \sqrt {b} \arctan \left (\frac {\sqrt {-b x + 2}}{\sqrt {b} \sqrt {x}}\right ) - \sqrt {-b x + 2} b \sqrt {x}\right )}}{b^{3} x - 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 = 0.85, size = 90, normalized size = 2.00 \begin {gather*} \begin {cases} - \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 {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 [B] Leaf count of result is larger than twice the leaf count of optimal. 92 vs.
\(2 (34) = 68\).
time = 5.23, size = 92, normalized size = 2.04 \begin {gather*} -\frac {{\left (\frac {\log \left ({\left (\sqrt {-b x + 2} \sqrt {-b} - \sqrt {{\left (b x - 2\right )} b + 2 \, b}\right )}^{2}\right )}{\sqrt {-b}} + \frac {8 \, \sqrt {-b}}{{\left (\sqrt {-b x + 2} \sqrt {-b} - \sqrt {{\left (b x - 2\right )} b + 2 \, b}\right )}^{2} - 2 \, b}\right )} {\left | b \right |}}{b^{2}} \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 {x}}{{\left (2-b\,x\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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