Optimal. Leaf size=22 \[ \sqrt{x} \sqrt{x+1}-\sinh ^{-1}\left (\sqrt{x}\right ) \]
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Rubi [A] time = 0.0143864, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \sqrt{x} \sqrt{x+1}-\sinh ^{-1}\left (\sqrt{x}\right ) \]
Antiderivative was successfully verified.
[In] Int[Sqrt[x]/Sqrt[1 + x],x]
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Rubi in Sympy [A] time = 1.45867, size = 17, normalized size = 0.77 \[ \sqrt{x} \sqrt{x + 1} - \operatorname{asinh}{\left (\sqrt{x} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**(1/2)/(1+x)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0275221, size = 42, normalized size = 1.91 \[ \frac{\sqrt{\frac{x}{x+1}} \left (\sqrt{x} (x+1)-\sqrt{x+1} \sinh ^{-1}\left (\sqrt{x}\right )\right )}{\sqrt{x}} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[x]/Sqrt[1 + x],x]
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Maple [B] time = 0.005, size = 39, normalized size = 1.8 \[ \sqrt{x}\sqrt{1+x}-{\frac{1}{2}\sqrt{x \left ( 1+x \right ) }\ln \left ({\frac{1}{2}}+x+\sqrt{{x}^{2}+x} \right ){\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{1+x}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^(1/2)/(1+x)^(1/2),x)
[Out]
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Maxima [A] time = 0.720052, size = 66, normalized size = 3. \[ \frac{\sqrt{x + 1}}{\sqrt{x}{\left (\frac{x + 1}{x} - 1\right )}} - \frac{1}{2} \, \log \left (\frac{\sqrt{x + 1}}{\sqrt{x}} + 1\right ) + \frac{1}{2} \, \log \left (\frac{\sqrt{x + 1}}{\sqrt{x}} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(x)/sqrt(x + 1),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.273559, size = 104, normalized size = 4.73 \[ -\frac{2 \,{\left (4 \, x + 1\right )} \sqrt{x + 1} \sqrt{x} - 8 \, x^{2} - 2 \,{\left (2 \, \sqrt{x + 1} \sqrt{x} - 2 \, x - 1\right )} \log \left (2 \, \sqrt{x + 1} \sqrt{x} - 2 \, x - 1\right ) - 6 \, x + 1}{4 \,{\left (2 \, \sqrt{x + 1} \sqrt{x} - 2 \, x - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(x)/sqrt(x + 1),x, algorithm="fricas")
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Sympy [A] time = 5.70949, size = 60, normalized size = 2.73 \[ \begin{cases} - \operatorname{acosh}{\left (\sqrt{x + 1} \right )} + \frac{\left (x + 1\right )^{\frac{3}{2}}}{\sqrt{x}} - \frac{\sqrt{x + 1}}{\sqrt{x}} & \text{for}\: \left |{x + 1}\right | > 1 \\i \sqrt{- x} \sqrt{x + 1} + i \operatorname{asin}{\left (\sqrt{x + 1} \right )} & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**(1/2)/(1+x)**(1/2),x)
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GIAC/XCAS [A] time = 0.293066, size = 31, normalized size = 1.41 \[ \sqrt{x + 1} \sqrt{x} +{\rm ln}\left ({\left | -\sqrt{x + 1} + \sqrt{x} \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(x)/sqrt(x + 1),x, algorithm="giac")
[Out]