3.575 \(\int \frac{\sqrt{x}}{\sqrt{1+x}} \, dx\)

Optimal. Leaf size=22 \[ \sqrt{x} \sqrt{x+1}-\sinh ^{-1}\left (\sqrt{x}\right ) \]

[Out]

Sqrt[x]*Sqrt[1 + x] - ArcSinh[Sqrt[x]]

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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]

[Out]

Sqrt[x]*Sqrt[1 + x] - ArcSinh[Sqrt[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]

sqrt(x)*sqrt(x + 1) - asinh(sqrt(x))

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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]

[Out]

(Sqrt[x/(1 + x)]*(Sqrt[x]*(1 + x) - Sqrt[1 + x]*ArcSinh[Sqrt[x]]))/Sqrt[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]

x^(1/2)*(1+x)^(1/2)-1/2*(x*(1+x))^(1/2)/x^(1/2)/(1+x)^(1/2)*ln(1/2+x+(x^2+x)^(1/
2))

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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]

sqrt(x + 1)/(sqrt(x)*((x + 1)/x - 1)) - 1/2*log(sqrt(x + 1)/sqrt(x) + 1) + 1/2*l
og(sqrt(x + 1)/sqrt(x) - 1)

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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")

[Out]

-1/4*(2*(4*x + 1)*sqrt(x + 1)*sqrt(x) - 8*x^2 - 2*(2*sqrt(x + 1)*sqrt(x) - 2*x -
1)*log(2*sqrt(x + 1)*sqrt(x) - 2*x - 1) - 6*x + 1)/(2*sqrt(x + 1)*sqrt(x) - 2*x
- 1)

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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)

[Out]

Piecewise((-acosh(sqrt(x + 1)) + (x + 1)**(3/2)/sqrt(x) - sqrt(x + 1)/sqrt(x), A
bs(x + 1) > 1), (I*sqrt(-x)*sqrt(x + 1) + I*asin(sqrt(x + 1)), True))

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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]

sqrt(x + 1)*sqrt(x) + ln(abs(-sqrt(x + 1) + sqrt(x)))