Optimal. Leaf size=35 \[ \sqrt{x+1} \sqrt{3 x+2}-\frac{\sinh ^{-1}\left (\sqrt{3 x+2}\right )}{\sqrt{3}} \]
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Rubi [A] time = 0.0294906, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ \sqrt{x+1} \sqrt{3 x+2}-\frac{\sinh ^{-1}\left (\sqrt{3 x+2}\right )}{\sqrt{3}} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[2 + 3*x]/Sqrt[1 + x],x]
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Rubi in Sympy [A] time = 2.045, size = 31, normalized size = 0.89 \[ \sqrt{x + 1} \sqrt{3 x + 2} - \frac{\sqrt{3} \operatorname{asinh}{\left (\sqrt{3 x + 2} \right )}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((2+3*x)**(1/2)/(1+x)**(1/2),x)
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Mathematica [A] time = 0.0253311, size = 45, normalized size = 1.29 \[ \sqrt{x+1} \sqrt{3 x+2}-\frac{\log \left (3 \sqrt{x+1}+\sqrt{9 x+6}\right )}{\sqrt{3}} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[2 + 3*x]/Sqrt[1 + x],x]
[Out]
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Maple [B] time = 0.008, size = 67, normalized size = 1.9 \[ \sqrt{1+x}\sqrt{2+3\,x}-{\frac{\sqrt{3}}{6}\sqrt{ \left ( 1+x \right ) \left ( 2+3\,x \right ) }\ln \left ({\frac{\sqrt{3}}{3} \left ({\frac{5}{2}}+3\,x \right ) }+\sqrt{3\,{x}^{2}+5\,x+2} \right ){\frac{1}{\sqrt{1+x}}}{\frac{1}{\sqrt{2+3\,x}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((2+3*x)^(1/2)/(1+x)^(1/2),x)
[Out]
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Maxima [A] time = 0.759216, size = 55, normalized size = 1.57 \[ -\frac{1}{6} \, \sqrt{3} \log \left (2 \, \sqrt{3} \sqrt{3 \, x^{2} + 5 \, x + 2} + 6 \, x + 5\right ) + \sqrt{3 \, x^{2} + 5 \, x + 2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(3*x + 2)/sqrt(x + 1),x, algorithm="maxima")
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Fricas [A] time = 0.27843, size = 78, normalized size = 2.23 \[ \frac{1}{12} \, \sqrt{3}{\left (4 \, \sqrt{3} \sqrt{3 \, x + 2} \sqrt{x + 1} + \log \left (-12 \,{\left (6 \, x + 5\right )} \sqrt{3 \, x + 2} \sqrt{x + 1} + \sqrt{3}{\left (72 \, x^{2} + 120 \, x + 49\right )}\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(3*x + 2)/sqrt(x + 1),x, algorithm="fricas")
[Out]
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Sympy [A] time = 5.777, size = 97, normalized size = 2.77 \[ \begin{cases} \frac{3 \left (x + 1\right )^{\frac{3}{2}}}{\sqrt{3 x + 2}} - \frac{\sqrt{x + 1}}{\sqrt{3 x + 2}} - \frac{\sqrt{3} \operatorname{acosh}{\left (\sqrt{3} \sqrt{x + 1} \right )}}{3} & \text{for}\: 3 \left |{x + 1}\right | > 1 \\i \sqrt{- 3 x - 2} \sqrt{x + 1} + \frac{\sqrt{3} i \operatorname{asin}{\left (\sqrt{3} \sqrt{x + 1} \right )}}{3} & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2+3*x)**(1/2)/(1+x)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.268893, size = 53, normalized size = 1.51 \[ \frac{1}{3} \, \sqrt{3}{\left (\sqrt{3 \, x + 3} \sqrt{3 \, x + 2} +{\rm ln}\left (\sqrt{3 \, x + 3} - \sqrt{3 \, x + 2}\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(3*x + 2)/sqrt(x + 1),x, algorithm="giac")
[Out]