Optimal. Leaf size=16 \[ -\frac{\sin ^{-1}\left (\frac{1}{3} (-2 b x-1)\right )}{b} \]
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Rubi [A] time = 0.0133358, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {53, 619, 216} \[ -\frac{\sin ^{-1}\left (\frac{1}{3} (-2 b x-1)\right )}{b} \]
Antiderivative was successfully verified.
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Rule 53
Rule 619
Rule 216
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*}
Mathematica [A] time = 0.0124177, size = 22, normalized size = 1.38 \[ -\frac{2 \sin ^{-1}\left (\frac{\sqrt{1-b x}}{\sqrt{3}}\right )}{b} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.007, size = 66, normalized size = 4.1 \begin{align*}{\sqrt{ \left ( -bx+1 \right ) \left ( bx+2 \right ) }\arctan \left ({\sqrt{{b}^{2}} \left ( x+{\frac{1}{2\,b}} \right ){\frac{1}{\sqrt{-{b}^{2}{x}^{2}-bx+2}}}} \right ){\frac{1}{\sqrt{-bx+1}}}{\frac{1}{\sqrt{bx+2}}}{\frac{1}{\sqrt{{b}^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.01138, size = 104, normalized size = 6.5 \begin{align*} -\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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{- b x + 1} \sqrt{b x + 2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.07573, size = 24, normalized size = 1.5 \begin{align*} \frac{2 \, \arcsin \left (\frac{1}{3} \, \sqrt{3} \sqrt{b x + 2}\right )}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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