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

Optimal. Leaf size=43 \[ \frac{1}{2} \sqrt{x+1} x^{3/2}+\frac{1}{4} \sqrt{x+1} \sqrt{x}-\frac{1}{4} \sinh ^{-1}\left (\sqrt{x}\right ) \]

[Out]

(Sqrt[x]*Sqrt[1 + x])/4 + (x^(3/2)*Sqrt[1 + x])/2 - ArcSinh[Sqrt[x]]/4

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Rubi [A]  time = 0.0203311, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{1}{2} \sqrt{x+1} x^{3/2}+\frac{1}{4} \sqrt{x+1} \sqrt{x}-\frac{1}{4} \sinh ^{-1}\left (\sqrt{x}\right ) \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[x]*Sqrt[1 + x],x]

[Out]

(Sqrt[x]*Sqrt[1 + x])/4 + (x^(3/2)*Sqrt[1 + x])/2 - ArcSinh[Sqrt[x]]/4

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Rubi in Sympy [A]  time = 1.73303, size = 34, normalized size = 0.79 \[ \frac{\sqrt{x} \left (x + 1\right )^{\frac{3}{2}}}{2} - \frac{\sqrt{x} \sqrt{x + 1}}{4} - \frac{\operatorname{asinh}{\left (\sqrt{x} \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(1/2)*(1+x)**(1/2),x)

[Out]

sqrt(x)*(x + 1)**(3/2)/2 - sqrt(x)*sqrt(x + 1)/4 - asinh(sqrt(x))/4

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Mathematica [A]  time = 0.0250975, size = 31, normalized size = 0.72 \[ \frac{1}{4} \left (\sqrt{x} \sqrt{x+1} (2 x+1)-\sinh ^{-1}\left (\sqrt{x}\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[x]*Sqrt[1 + x],x]

[Out]

(Sqrt[x]*Sqrt[1 + x]*(1 + 2*x) - ArcSinh[Sqrt[x]])/4

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Maple [A]  time = 0.005, size = 50, normalized size = 1.2 \[{\frac{1}{2}\sqrt{x} \left ( 1+x \right ) ^{{\frac{3}{2}}}}-{\frac{1}{4}\sqrt{x}\sqrt{1+x}}-{\frac{1}{8}\sqrt{x \left ( 1+x \right ) }\ln \left ({\frac{1}{2}}+x+\sqrt{{x}^{2}+x} \right ){\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{1+x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(1/2)*(1+x)^(1/2),x)

[Out]

1/2*x^(1/2)*(1+x)^(3/2)-1/4*x^(1/2)*(1+x)^(1/2)-1/8*(x*(1+x))^(1/2)/(1+x)^(1/2)/
x^(1/2)*ln(1/2+x+(x^2+x)^(1/2))

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Maxima [A]  time = 1.36398, size = 96, normalized size = 2.23 \[ \frac{\frac{{\left (x + 1\right )}^{\frac{3}{2}}}{x^{\frac{3}{2}}} + \frac{\sqrt{x + 1}}{\sqrt{x}}}{4 \,{\left (\frac{{\left (x + 1\right )}^{2}}{x^{2}} - \frac{2 \,{\left (x + 1\right )}}{x} + 1\right )}} - \frac{1}{8} \, \log \left (\frac{\sqrt{x + 1}}{\sqrt{x}} + 1\right ) + \frac{1}{8} \, \log \left (\frac{\sqrt{x + 1}}{\sqrt{x}} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x + 1)*sqrt(x),x, algorithm="maxima")

[Out]

1/4*((x + 1)^(3/2)/x^(3/2) + sqrt(x + 1)/sqrt(x))/((x + 1)^2/x^2 - 2*(x + 1)/x +
1) - 1/8*log(sqrt(x + 1)/sqrt(x) + 1) + 1/8*log(sqrt(x + 1)/sqrt(x) - 1)

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Fricas [A]  time = 0.211217, size = 158, normalized size = 3.67 \[ \frac{128 \, x^{4} + 256 \, x^{3} - 4 \,{\left (32 \, x^{3} + 48 \, x^{2} + 18 \, x + 1\right )} \sqrt{x + 1} \sqrt{x} + 152 \, x^{2} + 4 \,{\left (4 \,{\left (2 \, x + 1\right )} \sqrt{x + 1} \sqrt{x} - 8 \, x^{2} - 8 \, x - 1\right )} \log \left (2 \, \sqrt{x + 1} \sqrt{x} - 2 \, x - 1\right ) + 24 \, x - 1}{32 \,{\left (4 \,{\left (2 \, x + 1\right )} \sqrt{x + 1} \sqrt{x} - 8 \, x^{2} - 8 \, x - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x + 1)*sqrt(x),x, algorithm="fricas")

[Out]

1/32*(128*x^4 + 256*x^3 - 4*(32*x^3 + 48*x^2 + 18*x + 1)*sqrt(x + 1)*sqrt(x) + 1
52*x^2 + 4*(4*(2*x + 1)*sqrt(x + 1)*sqrt(x) - 8*x^2 - 8*x - 1)*log(2*sqrt(x + 1)
*sqrt(x) - 2*x - 1) + 24*x - 1)/(4*(2*x + 1)*sqrt(x + 1)*sqrt(x) - 8*x^2 - 8*x -
1)

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Sympy [A]  time = 3.96857, size = 119, normalized size = 2.77 \[ \begin{cases} - \frac{\operatorname{acosh}{\left (\sqrt{x + 1} \right )}}{4} + \frac{\left (x + 1\right )^{\frac{5}{2}}}{2 \sqrt{x}} - \frac{3 \left (x + 1\right )^{\frac{3}{2}}}{4 \sqrt{x}} + \frac{\sqrt{x + 1}}{4 \sqrt{x}} & \text{for}\: \left |{x + 1}\right | > 1 \\\frac{i \operatorname{asin}{\left (\sqrt{x + 1} \right )}}{4} - \frac{i \left (x + 1\right )^{\frac{5}{2}}}{2 \sqrt{- x}} + \frac{3 i \left (x + 1\right )^{\frac{3}{2}}}{4 \sqrt{- x}} - \frac{i \sqrt{x + 1}}{4 \sqrt{- x}} & \text{otherwise} \end{cases} \]

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))/4 + (x + 1)**(5/2)/(2*sqrt(x)) - 3*(x + 1)**(3/2)
/(4*sqrt(x)) + sqrt(x + 1)/(4*sqrt(x)), Abs(x + 1) > 1), (I*asin(sqrt(x + 1))/4
- I*(x + 1)**(5/2)/(2*sqrt(-x)) + 3*I*(x + 1)**(3/2)/(4*sqrt(-x)) - I*sqrt(x + 1
)/(4*sqrt(-x)), True))

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GIAC/XCAS [A]  time = 0.227168, size = 42, normalized size = 0.98 \[ \frac{1}{4} \,{\left (2 \, x + 1\right )} \sqrt{x + 1} \sqrt{x} + \frac{1}{4} \,{\rm ln}\left ({\left | -\sqrt{x + 1} + \sqrt{x} \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x + 1)*sqrt(x),x, algorithm="giac")

[Out]

1/4*(2*x + 1)*sqrt(x + 1)*sqrt(x) + 1/4*ln(abs(-sqrt(x + 1) + sqrt(x)))