Optimal. Leaf size=11 \[ \frac{\sin ^{-1}(2 b x+3)}{b} \]
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Rubi [A] time = 0.0100564, antiderivative size = 11, 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} \[ \frac{\sin ^{-1}(2 b x+3)}{b} \]
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{-2-3 b x-b^2 x^2}} \, dx\\ &=-\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{x^2}{b^2}}} \, dx,x,-3 b-2 b^2 x\right )}{b^2}\\ &=\frac{\sin ^{-1}(3+2 b x)}{b}\\ \end{align*}
Mathematica [B] time = 0.0138349, size = 49, normalized size = 4.45 \[ \frac{2 \sqrt{b x+1} \sqrt{b x+2} \sinh ^{-1}\left (\sqrt{b x+1}\right )}{b \sqrt{-(b x+1) (b x+2)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.006, size = 66, normalized size = 6. \begin{align*}{\sqrt{ \left ( -bx-1 \right ) \left ( bx+2 \right ) }\arctan \left ({\sqrt{{b}^{2}} \left ( x+{\frac{3}{2\,b}} \right ){\frac{1}{\sqrt{-{b}^{2}{x}^{2}-3\,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 = 2.05474, size = 107, normalized size = 9.73 \begin{align*} -\frac{\arctan \left (\frac{{\left (2 \, b x + 3\right )} \sqrt{b x + 2} \sqrt{-b x - 1}}{2 \,{\left (b^{2} x^{2} + 3 \, 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 = 1.0648, size = 18, normalized size = 1.64 \begin{align*} \frac{2 \, \arcsin \left (\sqrt{b x + 2}\right )}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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