Optimal. Leaf size=10 \[ \frac {\sin ^{-1}(b x+1)}{b} \]
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Rubi [A] time = 0.01, antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {53, 619, 216} \begin {gather*} \frac {\sin ^{-1}(b x+1)}{b} \end {gather*}
Antiderivative was successfully verified.
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Rule 53
Rule 216
Rule 619
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {-b x} \sqrt {2+b x}} \, dx &=\int \frac {1}{\sqrt {-2 b x-b^2 x^2}} \, dx\\ &=-\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{4 b^2}}} \, dx,x,-2 b-2 b^2 x\right )}{2 b^2}\\ &=\frac {\sin ^{-1}(1+b x)}{b}\\ \end {align*}
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Mathematica [B] time = 0.01, size = 51, normalized size = 5.10 \begin {gather*} \frac {2 \sqrt {x} \sqrt {b x+2} \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{\sqrt {b} \sqrt {-b x (b x+2)}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [B] time = 0.04, size = 24, normalized size = 2.40 \begin {gather*} \frac {2 \tan ^{-1}\left (\frac {\sqrt {b x+2}}{\sqrt {-b x}}\right )}{b} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 1.12, size = 26, normalized size = 2.60 \begin {gather*} -\frac {2 \, \arctan \left (\frac {\sqrt {b x + 2} \sqrt {-b x}}{b x}\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.93, size = 18, normalized size = 1.80 \begin {gather*} \frac {2 \, \arcsin \left (\frac {1}{2} \, \sqrt {2} \sqrt {b x + 2}\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.00, size = 58, normalized size = 5.80 \begin {gather*} \frac {\sqrt {-\left (b x +2\right ) b x}\, \arctan \left (\frac {\sqrt {b^{2}}\, \left (x +\frac {1}{b}\right )}{\sqrt {-b^{2} x^{2}-2 b x}}\right )}{\sqrt {-b x}\, \sqrt {b x +2}\, \sqrt {b^{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.14, size = 18, normalized size = 1.80 \begin {gather*} -\frac {\arcsin \left (-\frac {b^{2} x + b}{b}\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.29, size = 34, normalized size = 3.40 \begin {gather*} -\frac {4\,\mathrm {atan}\left (\frac {b\,\left (\sqrt {2}-\sqrt {b\,x+2}\right )}{\sqrt {-b\,x}\,\sqrt {b^2}}\right )}{\sqrt {b^2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 1.28, size = 24, normalized size = 2.40 \begin {gather*} - \frac {2 i \operatorname {asinh}{\left (\frac {\sqrt {2} \sqrt {b} \sqrt {x}}{2} \right )}}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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