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 = 0.0332318, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ -\frac{2 \tan ^{-1}\left (\frac{\sqrt{-b x-3}}{\sqrt{b x+2}}\right )}{b} \]
Antiderivative was successfully verified.
[In] Int[1/(Sqrt[-3 - b*x]*Sqrt[2 + b*x]),x]
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Rubi in Sympy [A] time = 4.589, size = 22, normalized size = 0.85 \[ \frac{2 \operatorname{atan}{\left (\frac{\sqrt{b x + 2}}{\sqrt{- b x - 3}} \right )}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(-b*x-3)**(1/2)/(b*x+2)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0132892, size = 33, normalized size = 1.27 \[ -\frac{2 \tan ^{-1}\left (\frac{\sqrt{-b x-3} \sqrt{b x+2}}{b x+3}\right )}{b} \]
Antiderivative was successfully verified.
[In] Integrate[1/(Sqrt[-3 - b*x]*Sqrt[2 + b*x]),x]
[Out]
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Maple [B] time = 0.013, size = 66, normalized size = 2.5 \[{1\sqrt{ \left ( -bx-3 \right ) \left ( bx+2 \right ) }\arctan \left ({1\sqrt{{b}^{2}} \left ( x+{\frac{5}{2\,b}} \right ){\frac{1}{\sqrt{-{b}^{2}{x}^{2}-5\,bx-6}}}} \right ){\frac{1}{\sqrt{-bx-3}}}{\frac{1}{\sqrt{bx+2}}}{\frac{1}{\sqrt{{b}^{2}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(-b*x-3)^(1/2)/(b*x+2)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(b*x + 2)*sqrt(-b*x - 3)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.215201, size = 38, normalized size = 1.46 \[ \frac{\arctan \left (\frac{2 \, b x + 5}{2 \, \sqrt{b x + 2} \sqrt{-b x - 3}}\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(b*x + 2)*sqrt(-b*x - 3)),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{- b x - 3} \sqrt{b x + 2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(-b*x-3)**(1/2)/(b*x+2)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.216893, size = 20, normalized size = 0.77 \[ -\frac{2 i \, \arcsin \left (i \, \sqrt{b x + 2}\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(b*x + 2)*sqrt(-b*x - 3)),x, algorithm="giac")
[Out]