Optimal. Leaf size=26 \[ -\frac{2 \tan ^{-1}\left (\frac{\sqrt{-b x-3}}{\sqrt{b x+2}}\right )}{b} \]
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Rubi [A] time = 0.0137541, antiderivative size = 26, 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 = {63, 217, 203} \[ -\frac{2 \tan ^{-1}\left (\frac{\sqrt{-b x-3}}{\sqrt{b x+2}}\right )}{b} \]
Antiderivative was successfully verified.
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Rule 63
Rule 217
Rule 203
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{-3-b x} \sqrt{2+b x}} \, dx &=-\frac{2 \operatorname{Subst}\left (\int \frac{1}{\sqrt{-1-x^2}} \, dx,x,\sqrt{-3-b x}\right )}{b}\\ &=-\frac{2 \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\frac{\sqrt{-3-b x}}{\sqrt{2+b x}}\right )}{b}\\ &=-\frac{2 \tan ^{-1}\left (\frac{\sqrt{-3-b x}}{\sqrt{2+b x}}\right )}{b}\\ \end{align*}
Mathematica [B] time = 0.0153942, size = 53, normalized size = 2.04 \[ -\frac{2 \sqrt{-b x-3} \sqrt{-b x-2} \sin ^{-1}\left (\sqrt{b x+3}\right )}{b \sqrt{b x+2} \sqrt{b x+3}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.006, size = 66, normalized size = 2.5 \begin{align*}{\sqrt{ \left ( -bx-3 \right ) \left ( bx+2 \right ) }\arctan \left ({\sqrt{{b}^{2}} \left ( x+{\frac{5}{2\,b}} \right ){\frac{1}{\sqrt{-{b}^{2}{x}^{2}-5\,bx-6}}}} \right ){\frac{1}{\sqrt{-bx-3}}}{\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 [A] time = 2.04065, size = 107, normalized size = 4.12 \begin{align*} -\frac{\arctan \left (\frac{{\left (2 \, b x + 5\right )} \sqrt{b x + 2} \sqrt{-b x - 3}}{2 \,{\left (b^{2} x^{2} + 5 \, b x + 6\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 - 3} \sqrt{b x + 2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] time = 1.06312, size = 20, normalized size = 0.77 \begin{align*} -\frac{2 i \, \arcsin \left (i \, \sqrt{b x + 2}\right )}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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