Optimal. Leaf size=16 \[ -\frac {\sin ^{-1}\left (\frac {1}{3} (-2 b x-1)\right )}{b} \]
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Rubi [A] time = 0.01, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {53, 619, 216} \begin {gather*} -\frac {\sin ^{-1}\left (\frac {1}{3} (-2 b x-1)\right )}{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 {1-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}{9 b^2}}} \, dx,x,-b-2 b^2 x\right )}{3 b^2}\\ &=-\frac {\sin ^{-1}\left (\frac {1}{3} (-1-2 b x)\right )}{b}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 22, normalized size = 1.38 \begin {gather*} -\frac {2 \sin ^{-1}\left (\frac {\sqrt {1-b x}}{\sqrt {3}}\right )}{b} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.05, size = 26, normalized size = 1.62 \begin {gather*} -\frac {2 \tan ^{-1}\left (\frac {\sqrt {1-b x}}{\sqrt {b x+2}}\right )}{b} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 1.07, size = 43, normalized size = 2.69 \begin {gather*} -\frac {\arctan \left (\frac {{\left (2 \, b x + 1\right )} \sqrt {b x + 2} \sqrt {-b x + 1}}{2 \, {\left (b^{2} x^{2} + b x - 2\right )}}\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.05, size = 18, normalized size = 1.12 \begin {gather*} \frac {2 \, \arcsin \left (\frac {1}{3} \, \sqrt {3} \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 = 0.01, size = 66, normalized size = 4.12 \begin {gather*} \frac {\sqrt {\left (-b x +1\right ) \left (b x +2\right )}\, \arctan \left (\frac {\sqrt {b^{2}}\, \left (x +\frac {1}{2 b}\right )}{\sqrt {-b^{2} x^{2}-b x +2}}\right )}{\sqrt {-b x +1}\, \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 = 3.01, size = 19, normalized size = 1.19 \begin {gather*} -\frac {\arcsin \left (-\frac {2 \, b^{2} x + b}{3 \, b}\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.32, size = 40, normalized size = 2.50 \begin {gather*} -\frac {4\,\mathrm {atan}\left (\frac {b\,\left (\sqrt {2}-\sqrt {b\,x+2}\right )}{\left (\sqrt {1-b\,x}-1\right )\,\sqrt {b^2}}\right )}{\sqrt {b^2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {- b x + 1} \sqrt {b x + 2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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